My name is Toby Kavanaugh and I'm the Director of Radiant School of Ministry here, but my secret identity is for the last 15 years, we've led a ministry in China called Campus Target. So when I was 23 years old, I was finishing up grad school and I was in apartment with my best friend Matt, and one day as we were just deciding what we're going to do next in our life, we were talking in our living room about life and jobs we're going to get, and girls we're going to marry, and all this kind of stuff. As we're talking about that, I looked over at Matt and I said, Matt, before we get married, before we have kids, before we start our careers, what if we just gave a year in missions? Do you think God could use us? Matt was sitting next to me on the other couch and he goes, okay, let's do it. Where are we going? I said, I hadn't even thought that far ahead yet. He said, how about China? I said, okay. That was the beginning of our adventure. It was not like this since I was three years old.  It was not that kind of experience. It was an experience of us volunteering and offering ourselves up to the mission that he had given us. And so about six months later, we are on an airplane on our way into China and we have no idea what we're doing. We don't know Chinese. I don't know anything about Chinese culture. I just learned the name of the city that I was going to. I wish I could communicate just how ignorant we actually were. And so we land in China and there's a missionary that's hosting us and so he is trying to set us up in retrospect. Now I look back on it and I realize, I don't think he really knew what to do with us because we had no idea what we were doing.  Why don't you go see if you can meet someone there who speaks English? Okay, so first Friday we're in China. We walked through the south gate of this university and there's a huge sports building kind of looks like a beetle over here this huge kind of quad open square area and in this quad area, there's just hundreds and hundreds of students milling about in some kind of activity. And so I don't know if anybody can speak English in China. And so I walk over to a group and I tap a guy kind of on the edge of what's happened on the shoulder and I'm just like, what's going on here? Because I had not yet learned that very important missions principle that if they don't speak your language, it doesn't matter how slow or loud you say it. But the guy looks at me and he goes, oh, this is an English corner. This is where we get together to practice our English. So I was like, sweet.  And so I got a group of like 15 or 20 people around me peppering me, asking me questions about my life and what it was like. I remember, do you have any siblings because they were all from one child policy? The guy asked, is your life like the TV show Friends? Not exactly, you know, Ross like a little bit, but not really, right? And they just asked me all these kinds of questions and in the midst of it, one of the guys looks at me and he goes, hey, are you a Christian? Do you believe in Jesus? And I kind of freeze for a second because I don't know what I'm doing. I know I'm in a communist country. I know you gotta be careful, but this guy just asked me this question. And so I just start to share my testimony just about how Jesus had saved me and how he changed me and transformed my life. And as I'm sharing this story, I literally will never forget it to the day I die. Around this half circle of 15 or 20 students, it was like you could literally see spiritual hunger lighting up in their eyes. It was this lean forward and this look that was like, I've never heard what you're talking about and I really need it in my life. So that moment kind of passed. I didn't really share the gospel in that moment or anything like that, when that time was over, I kind of just started writing down people's names and phone numbers in a book ahead with me. And I just put like a little check mark next to their name if you kind of saw that look in their eye a little bit. And over those next few weeks, Matt and I just started calling up those friends and saying, hey, Gary, you wanna play basketball? Hey, Ren, you wanna grab lunch? We began to see person after person after person give their lives to Christ. It was absolutely unbelievable. I'd never seen any kind of openness like it. And God marked me and erected me during that season and what I was like, hey, I'm just gonna go for a year and do this thing. It was like, what else can I spend my life doing? These people are so hungry here on the universities of China, the leaders of the nation that's going to be holding the steering wheel for the world for the next 50 years. And it just erected me. And so missions is a huge part of my DNA. Most of my adult life, I've actually lived in China, bring another young adults over there as we plant churches on the college campuses. And you are just meeting me today, but many of you are familiar with Radiant being here. And you might not know that missions is a huge part of Radiant's DNA too. You see, when Pastor Lee and Jane were first married before Radiant was ever started and Jane was pregnant with their first baby, they were on their way to the Philippines. They were gonna be missionaries to the Philippines. Their church had already decided to send them out. They'd already said goodbye to their families. They were making plans for Jane to have the baby over in the Philippines. They'd raised all their support and they were going. And it was one of the last things they were doing before they left. They were going to Jane's parents. They had a cottage that was on a lake. And they were there just kind of spending some family time before they left. And Pastor Lee was sitting out by the lake on a lawn chair. Jane was sitting next to him on another chair. And he was reading a book by the persecuted Chinese theologian, Watchmen Ni, called Concerning Our Admissions. And when you hear Pastor Lee tell it, he remembers the details to this day. He says it like this. It was on page 47, left side of the book, bottom paragraph. It's etched in my memory forever. And he read these words. No amount of ambition, no affirmation from leaders and mensual attentions can substitute for the call of God. And all of a sudden Jesus said to him, I'm so glad that you're willing to go. But if you will plant a mission-sending church for me, you will impact this world even more than if you went yourself. And it was literally in that moment that the dream for Radiant was born. And so from the very beginning of this church and this movement being planted, it was on this idea that this would be a mission-sending place. And the very first check that Radiant Church wrote, check 1001, before any salaries, before rent for a building that they would use, was checked that was sent out to give to a missionary who was serving overseas. For many years, the sanctuary right over there, the sanctuary where it used to be, the whole back of the room was filled with flags, 65 flags of different nations of the world that were places where Radiant was sowing its money that the gospel would go forward there. Every week for decades in this church, they would pray and they would pray for a people group or they would pray for a nation that the cause of Christ would go forward among them on the earth. And that is a huge part, even till today, still on Sunday mornings. One of the few regular things that happens in this church in addition to worshiping, preaching and talking about the offering, is they pray over a people group that God would bring the gospel to the ends of the earth. And from the very beginning and the very deepest roots of our movement, missions is central to the DNA at Radiant. And my invitation for those of you here that work in different ministry roles or churches or even in your own life, is to take missions and put it at the very center of your DNA too. Oh, why? Why do we do this? The thing that Jesus wants more than anything else is worshiping sons and daughters from every people group he created. And if you will take time to read the scriptures through that lens, you will realize from the very beginning, the first page in Genesis, all the way to Revelation, this thread goes through all scripture that the thing that Jesus wants more than anything else is worshiping sons and daughters from every people group he created. Since I've gotten married, I've learned a few things over the years, trying to pick up on what are the kind of, how do you know what someone wants? What's the kind of gift they want? You can buy your wife a gift or you can buy your kid a gift. And here's some of the things I've seen. You look at what do they dream about, right? What does that person want for their future? That's a great gift you could buy for them. Well, here's what Jesus says as he describes his dream in Revelation chapter seven. After this I looked and there before me was a great multitude that no one could count. From every nation, tribe, people and language standing before the throne and before the lamb. They were wearing white robes and were holding palm branches in their hands and they cried out in a loud voice, salvation belongs to our God who sits on the throne and unto the lamb. And when Jesus looks at the future, the end of all things, what he's hoping for in the days to come is that there would be a crowd of his lost sons and daughters who've been saved that have all gathered together and they're worshiping him. And in that group there's those from every tribe, every tongue, every people and every nation. And that word nation is translated in English and the Greek the word is ethnos. And it's not what we think of today when we think of a nation like America or Mexico or China that has some kind of national boundaries. It's what we get our word ethnic group from. And the idea is a unique people group that has their own unique culture and their own unique language and a unique group of people. So for example, in China, as we've reached college students there over the years, one of the things we've done is we've launched them out on missions trips. And so one of the things they'll do is within their own country, one time we just totally based it on Luke 10. So they had no money with them and these college students have traveled up into the mountains of China together. And as they do, they come across these other people groups like the Tibetan people that you may have heard of. You know, one of the people groups they come across in these trips was this very, very small people group, only a couple thousand people called the Malimasa. And this group through all of history, the gospel had never come to them. And when our young people went up among that group and they were traveling up in there, they found and discovered that one of the people from this people group had intermarried with another tribe. They had some Christians in it. And this person had married a Christian. And so that family had converted to Christ. And so they came into this tribe and they found the first Christian in all of history among the Malimasa people. And so when the Bible uses this word nation, it's not talking about China or America, it's talking about the Han Chinese people and the Tibetans and even that little Malimasa people group. That's what Jesus is talking about. And that's what he says is his dream in the future that there would be rescued people from every one of those. And when you look even a little deeper at what is Jesus planning for? He says this in Matthew chapter 24, 14. This gospel of the kingdom will be preached in the whole world as a testimony to all nations. There's an ethnos word again. And then the end will come. Jesus is literally planning all of history around this issue. He says the gospel of the kingdom will first be preached to every one of these people groups. And then he will initiate and things. And he is in heaven waiting right now that the good news would go to every one of these groups that people will be rescued from every one of them that he could initiate and things and come back to the earth. He's planning all of history around it. This matters to him. It is the deepest part of his heart. It's the thing he longs for more than anything else. And man, you know, when you hear right now even as we're encountering this cry for revival and the service that God would move. One of the things when you look at all of history what happens when there is a move of God, his spirit is poured out is that mission bursts forth in a new way. Why? Because we are getting in touch fresh again with the heart of Jesus. Because it's the thing he wants more than anything else. You know, some of you may be like me and you're not really good at looking for clues or what kind of gifts somebody wants. You need someone to kind of tell you directly. So Jesus helps us too. He says this, Matthew 28. All authority in heaven and on earth has been given to me. Therefore go and make disciples of all nations. It's that same thing again. What he commands us with his mouth, what he dreams about, what he's planning history around and what he sprinkled in clues all throughout the Bible is this deep, deep desire of his heart that people would be rescued from all nations. And you know, as a pastor or a leader something can swell in size. It says, that's amazing. Yes, in our church we are doing the great commission. We're seeing people come to Christ in our community. It's amazing what's happening now. You should have seen the baptisms after Easter and all these things. And the Lord delights in that and he's so excited. But I wanna say something. Yes, Jesus wants his lost sons and daughters to be rescued. Yes, he wants to see them worshiping him and baptized and come home. But he specifically says throughout the pages of scripture he wants that happening in all people groups. And it's very important to understand that because it's a little counterintuitive actually. But let's think about it like this. Let's say us in here, we are the staff at a Coast Guard station, okay? And so we're on the East Coast of America. We got two giant boats that can go out to rescue people. We run the thing. And so we're just kind of on break while we're waiting for needs to happen. We're playing cards, we're eating lunch and we do all this kind of stuff. And then all of a sudden in the midst of it, doot, doot, doot, the alarm goes off. We all jump to attention. We run and grab our stuff and get our equipment ready. And we start running to the two boats loading everything on. And the noise comes over the loudspeaker. It says there's been a terrorist attack. Two miles offshore, a cruise ship has been struck by a torpedo. It is going down over the last hour. You're the only ones close enough to help. Besides who you rescue, all souls will be lost. So we are passionately running, hurrying, getting everything ready that we can rescue people from the ship. And as we're loading on the ship and just getting ready to take off, the alarm goes off again, doot, doot, doot.  It's going down unless you can help people, all souls will be lost. In that moment of our station with our two little boats, our leaders have a horrible decision to make, don't they? And the most efficient way to think about it for those leaders are, okay, we got one that's only two miles away and one that's 20 miles away. If we focus our resources on the one that's two miles away, we will be able to rescue more people. It's a horrible decision, but this is the right thing in the moment. We have to do that. We're gonna focus on the one ship. And so that is what is most intuitive, but it is not what Jesus asked for. Jesus asks that we would rescue people from both ships. You see, all around this world, there's thousands and thousands of people groups, each one of them a sinking ship on its way to a Christless oblivion. And as it heads down that direction, what Jesus is saying is, I want people rescued from every one of these ships. And the initial response is, but it's more difficult to get there. We'd have to learn a different language or we'd have to travel in all kinds of ways. We have so many people right here that need you. Shouldn't we just efficiently focus our resources right here? And Jesus says, no, that's not what I'm asking you. I want people rescued from every people group, from every ship. And when we begin to understand that, it changes the way we think about life and it changes the way we think about ministry. Because all of a sudden, I cannot just purely think through the lens of what is most efficient. I must think through the lens of what does Jesus's heart want? You know, I say it with shame a little bit, but in the first few years of my marriage, I knew what my wife would want for gifts for Christmas and birthday, and I didn't get it to her. Because I knew the kind of things she'd want were things that were like beautiful, like clothes and jewelry and stuff like that. And I just did not come from a family where we bought that kind of stuff for each other. We buy each other like gadgets and things that were useful and stuff like that. And so when I knew that was what it was in her heart and what she wanted, I was psyched out by it and I was intimidated by it. Because buying clothes for girls, that's like the worst thing ever.  It can go wrong in so many ways, okay? And so I knew in her heart that was what she wanted. But I was not willing to actually do it. I wanted to give her what I wanted to give instead. And for some of us in the church, even some of us in ministry, we actually find ourselves in that same position where we've oriented our lives around giving Jesus what we want to give instead of what his heart longs for. And the thing that Jesus desires, the thing his heart longs for is people from every people group rescued and worshiping before his throne. And so as that starts to wash over us, when we start to say, okay, if that is what is in Jesus's heart, I want that in the DNA of my church.   Okay, I understand. How do I begin to put that into the DNA of our church? That's what I want to talk about a little bit today, just kind of four practices that can help in the process of actually working that into your church. Because for some of you as leaders, you have it inside you, but you know it's not really built into your church. Here's how we do it. Number one, put missions into the rhythm of your church. Put missions into the rhythm of your church. What do I mean? Each of our church has rhythms, things we do every week, things we do every month, things we do every year. A rise shine is a part of this church's rhythm here. It's something that's built in, it happens every year. True North is another thing that's an annual thing. They have certain things they do in the service every week. These are the things that are part of the rhythm of our church. But let me say it like this. Statistics would say in your church, over the last month, probably 70% of the people that have attended have given money of some kind. They also say that in your church, 10% of the people that have attended have shared the gospel with somebody else. Sure, 70% of the people that have attended your church in the last month have given financially in some way. 10% of the people that have attended your church have actually shared the gospel with someone else. What's the massive difference in those two numbers about? Any thoughts? Fear, comfortability? What else? Yeah, you talk about money every single week. When was the last time you talked about sharing the gospel in your church service? Okay? Money is built into the rhythm of your church, sharing the gospel is not probably. It doesn't have to be that way though. Like when I think about our time in China, you know, I'm trying to see church playing movements that happen among college students in China. That's what we do. And so as I'm contending for that, I have to wrestle with a really big reality. There's 25 million college students in China. There's more college students in China than there are people in the whole continent of Australia.  So when you look at China's campuses, imagine Australia sinking into the sea. That is what's happening spiritually on China's campuses right now. And so as we engage with this reality in our team of 60 people that we're contending to see this stuff happen, we have to wrestle with something that's way bigger than us. And no matter how boldly we share the gospel, that's not enough. We have to mobilize the believers that are gonna reach to share the gospel. So they reach their friends and the gospel goes to their friends and the gospel goes to their family and friends and it spreads like that. But we had a problem early on in starting churches and that was the new believers were just depending on us to share the gospel and they weren't actually sharing the gospel with their friends and family. And so as we sought the Lord, I want to do about this, we realized we needed to somehow build it into the rhythm of the church. And so in that unique kind of house church setting of China, we made a plan that basically every single service when we gather together, someone stands up and they talk for just two or three minutes passionately and challenging people, encouraging people to share the gospel with their friends and family. Guess what happened? People started sharing the gospel with their friends and family. Why? Because we put it into the rhythm of the church. And if you want missions to be a part of the DNA of the church, you have to find a way to somehow put it into the rhythm of your church. You know, most people on our churches are not engaged with the Great Commission because they literally don't know what it is. Okay, got your statistic. Among millennials right now, only 10% know what the Great Commission is of those who go to church. 50% say they've never even heard of the term. Another 40% have heard the term and they don't know what it is. We're just not talking about it in church. They're not getting exposed to it. They're not getting open to it. And so there's no way for that DNA to get imparted in their heart that's just not getting talked about. And so how do we do that? There's different ways. It could be a weekly thing. You know, one of the things actually, Pastor Lee just started doing recently again after doing it for decades and that practice having stopped was he started having us on Sunday mornings actually pray over a nation or people group in the world. And so we do that. And so again, in addition to preaching, worship and offering, that's like the only thing that's happening on a Sunday morning. What is he saying? This matters. And so I'm putting it into the rhythm of the church. If this is the deepest desire of Jesus' heart, how can I leave it out of what we do together? And so he built it in that way. That's not the only way to do that. You could have monthly someone in your Sunday morning service get up and share a report from a missionary of something powerful God's doing around the earth from where you guys are investing your treasures and your finances and what's happening. That's amazing. You can have annual stuff. The movement of churches I grew up in, they'd have like a missions conference every October and there'd be like a missions emphasis for the month of October and everyone would talk about and people would commit to give for the year and they'd have a giant gathering around it. There's tons of ways to do it, whether it's annual, whether it's monthly, whether it's weekly, but you have to deal with the question, am I actually putting this into the rhythms of the church? And if I'm not, am I saying this is not important to me? Because what is important to me, I put into the rhythms of my life. If you care about exercise, you put it into the rhythm of your life. If you don't care about exercise, you look like me. And so if you care about this is something that you do. If you care about time with your family, you probably have some kind of family dinner during the week where you actually talk and share. We put into the rhythms of our life what matters to us. And here's my challenge for us today. We have to make what is most important to Jesus' heart, most important to our heart too. Okay, number two, put your money where your mouth is.  The second way to build missions to know our church's DNA is to invest our finances there. Jesus says this, where your treasure is, there your heart will be also. How we spend our money tells us where our priorities are. This is true in our personal life and it's true for us as churches. And if you survey across evangelical churches in America, less than 1% of the money that comes into a church's budget goes out to do anything reaching unreached people groups at all. And so that's us saying out of all that's been released to us that we're stewarding as leaders, Jesus to the thing that is the deepest desire of your heart, here's 1%. And that's the reality of where it is right now. I'm not saying your church is specifically there, but we have to invest, engage with the fact, where am I investing my finances? Commit yourself, that's my challenge. Commit yourself to give radically into missions. Think about it, if this is the deepest desire of Jesus' heart, what is the appropriate amount for us to spend on it together? Consider at least investing 10% of your church's budget, a tie that goes into missions outside of your church that's bringing the gospel to unreached people groups. Churches that we recognize as heroes, I think at John Piper's church, they give 30% of their budget into missions every year that's going out to reach unreached people groups. We have to be radical in our giving in it. And as we commit ourselves to give financially as churches, it will make a difference in the Great Commission being completed. I think of myself, when I was first called into missions, I go on to China for that first year and I was able to fund it out of money that I had when we went and then I got married to my wife and we felt the Lord calling us to go back to China. And so now we're going long term, like we have to have some kind of team that's financially supporting us to make this happen. And I don't know if any of you would feel this way, but I feel very, I felt very psyched out about that idea. That's like, I have to ask people to invest this, am I worthy of their investment, is this a good thing? And I just was in my own head wrestling over all that kind of stuff. And I was sitting at my desk at work one day and I got a phone call. I picked it up and it was a pastor named Bob Walls. And I'd met him for like 20 minutes, three or four months earlier, told him we were feeling called to China. I'd never been to his church. I'd never had a conversation longer than this 20 minutes.  I'm like, hey, he's like, hey Toby, how you doing? Shoot the breeze for a second. He said, hey, I was talking to the elders of our church and we know you and your wife are getting ready to go to China. And so we talked about it. And so we're going to begin supporting you for $425 a month. My jaw literally hit the desk in front of me. I started crying actually on the phone while I was talking to him. I said, thank you so much. So we're so excited. This has been so much to me.  And I walked away from that phone call as I went back to my apartment. And it said something to me that Jesus saw me and that he was sending me out. And that phone call from that pastor literally is what launched me into missions. I think I may have frozen up and never even taken that step, but it was the encouragement that came from that moment, from them investing in me that actually launched me. And now I've literally launched out hundreds of people into long term missions, at least a year. And God's used them in all kinds of incredible ways. And that may have gotten aborted at the very beginning before it ever started. But somebody was willing to give. I later went and visited that church for the first time since I'd never even been there before when they made that decision. I found this church that was an inner city that was great in Pennsylvania. Maybe 40 or 50 people in the room. 10 or 15 of them looked like homeless people that the church was reaching as they shared the gospel. The pastor himself was bivocational, was not even paid full time staff. And they made the decision to invest $425 a month into me. Not me, into missions, into Jesus' heart. When we make decisions to give radically, it furthers the cause of Christ and the world in ways that you cannot even imagine. The ultimate impact of some of those decisions to give are things you will not even be able to comprehend until you get to heaven. But we put our money where Jesus' heart is. Number three I wanna talk about here is to create a going culture. It can be very easy as we're trying to faithfully put missions into our church's DNA to work to build a sending culture because that's what most of our people are going to be doing. But what you actually want to do is build a going culture. A sending culture is good in many ways. We believe in missions, we give financially, we help those who go. But what I've found is when that's what's trying to be established, it's always missing the heartbeat of what Jesus wants.  And when that spirit gets released in the congregation, everything gets released to another level. And so one of the biggest ways this plays out is in giving short term missions trip opportunities to the people of your church. You know, there's been a lot of commentary over the years, is short term missions trips even good? Is it fruitful to do this? Is this efficient use of our money? And I honestly believe that entire conversation misses the point. The point is this, when someone goes on a short term missions trip, it changes them as they serve others. And their changed life advances the mission. What do I mean? Well, think about it like this. Let's say you got a young dad in your church and he spends $2,500 to go on a trip you guys take to Mexico together, and Jesus touches his life in the midst of it. You can pick at, oh, maybe that $2,500 could have been spent a little more efficiently if we gave it to a national missionary, this kind of thing. But what studies show is that people who go on these kind of trips actually give more radically to missions afterwards. And so this dad now gives $2,500 a year for the rest of his life into missions. Was that a good investment? Absolutely. Studies show those who go on short term missions trips pray for missions at a different level. For being in a movement here who is all in on the fact that prayer changes things, is that a worthy investment? I think absolutely. Those who go on short term mission trips are more likely to become long term missionaries in the future, which is an actual gain changer for the Great Commission. And so we want to engage people in this idea of short term missions trips because it changes things from the inside out in their heart and that changes the culture of the church. Short term missions trip change lives. When they survey people who have gone on short term trips, 75% point it out as a life changing experience that they've had. 75%. How many things can you say about life for three out of every four people? It will be life changing in their own words. Okay? When people experience that kind of thing, it changes them from the inside out. The problem though is only about 10% of church goers actually go on mission trips. And so we're 90% of the people in our churches, we're leaving on the table a 75% chance of a life changing experience for Christ. And so we want to encourage that and stir that up. Don't be satisfied with just a sending culture. Create a going culture. This is where the magic really happens. It's when people's hearts get activated and radical sacrifice starts happening. Just as kind of contrast, a sending culture gives out of its excess. A going culture gives sacrificially. A sending culture prays when you tell them to in a service. A going culture prays in the secret place from their own burden. A sending culture says, here's 10% of my stimulus. A going culture says, here I am, send me. And when you allow that spirit to get fostering your people, it does not just change what's happening overseas. It radically changes what flows in their life right here in the States too. And so this is the key for us as pastors and leaders in the church. Well, obviously, we want to allow our hearts to be connected to Jesus' heart and we want our church to be ordered and prioritized according to what's most important to him. But even beyond that, when I allow a mission's culture to come into my church, there is a release of blessing and God's presence that is unique. And we see it right in the Great Commission, right? Go and make disciples of all nations, baptizing them in the name of the Father, the Son, and the Holy Spirit and teaching them to obey everything that I have commanded you. Then what does it say next? And lo, I am with you always to the very end of the age. That promise of his presence is not in a random place. It's literally connected to the mission. And when we step into his mission with him, there is a release of his blessing and his provision and his presence to our churches that we cannot access any other way. And so we want to allow our churches to get this kind of spirit. And as we wrap up here, the last one and probably the most important one, but I think also the most difficult one is this. To put mission's DNA into our church, we actually have to put our children on the altar. The single most important thing you can do to put mission's DNA into your church is to actually communicate about it and launch your kids out into it, which becomes profoundly uncomfortable. Like my kids grew up in China, so like we're like on the one pretty far edge kind of how we engage with this kind of stuff. Well, not too long ago, I was sitting in my house with my family and my oldest daughter, Maya, who was nine at the time. She comes up to my wife and I and she goes, I guess you don't, guess where I want to be a missionary to? And that was like kind of a cool thing. Like, oh cool, our daughter's embraced us too, we're excited. And I'm thinking she's going to say India because we just watched a video about Amy Carmichael, who was a missionary in India, and I'm like, that's cool, she got this. And we go India, she goes, no, North Korea. I go, okay. I was pretty far in, I'm not sure if I was that far in, right? Okay. And I had to actually go back to my heart as a missionary and say, Lord, are my kids actually offered up for your purposes? Because I was in as far as China, I'm not sure if I was in for North Korea. Spurgeon, famous pastor in England, a mom came up to her son, was getting ready to actually go to China as a missionary. And she said to him, she said, would you talk to my son, he's gone nuts, he wants to go be a missionary to China.  And that perspective that there is no more noble calling than our children literally offering up their hearts and their lives, the thing that is most important to Jesus. And that can be very tough to swallow when your daughter tells you she wants to go to North Korea. But it is actually in that act of intentionally laying our kids on the altar before the Lord that our church becomes more powerful than you could possibly imagine. I think of a little church in Cortland, New York that I remember when it got planted maybe 15 years ago. It was reaching college students a little bit, but it never totally took off. They maybe only had 40 or 50 people that were in the church at a time and Hartford to grow, but it wasn't really happening. And during that maybe five years that I was interacting with that church, I saw them send 15 people out in missions in China. That's like a third of the size of the church. And so this church that some might look at on the outside and say, oh, it never quite reached the impact that the pastor wanted, never quite stepped into it. It literally is going to have an inheritance at the throne before Jesus of hundreds and hundreds and hundreds of Chinese college students that came to Christ. Why? Because they offered their children up on the altar. There was their young people that went. How do we do this practically? With our kids, I'm thinking like now, like grade school and younger, the key is this for them. Give them missionary heroes. There's curriculums that do this. There's a torchlighters video series that's really excellent on this. If any of you got exposed to that, you can prepare your own stuff on this, but don't let their heroes be people they see on TV. Don't let their heroes be the musicians they're listening to. Don't let their heroes be people that are influencers on social media. Give them heroes who have given their all for Christ.  Something begins growing in the seed of a child even as young as five or 60 years old that they begin saying this is something I aspire to. I wanna give my life unto Christ. I mean, that's where that stuff happened with my nine-year-old where she's making this crazy North Korean statement is that we have actually sowed those seeds into the soil of her heart over the years that we've parented her. This is cool and this isn't even us. My daughter, now she's 10, she's involved in the junior high group here and I was putting her to bed the other night. She'd been going to prayer meetings with the junior high group. I was putting her to bed the other night and I'm praying with her and I'm putting her to bed and all of a sudden she interrupts and goes, oh, wait, wait, wait. And she stands up and she climbs out of her bed and she walks into her closet. I'm like, what are you doing? Like we were in the middle of praying right there. And I see her get down on her knees and I didn't know it, but she has this little calendar in her closet that's for unreached people groups in the world. My little 10-year-old just starts praying for the people whose back is standing and she's crying out to God that he would move there and he would say, that wasn't even for me. I didn't buy her that calendar that was no influence. I don't think that I did. That was the people she's with in the church are fostering that kind of spirit in her that she would contend for that and that's what a church can do in the life of a child if they sow into this area. Second one are teens. For your students, do all that same stuff you do with your kids, give them heroes, that kind of stuff, but this is one of the most important decisions you can make in this area. Send your teenagers out on a short-term missions trip. Let it be absolutely annual, built into your rhythm unless COVID comes again, that's happening no matter what, we're doing this every single year. This is like the thing we're doing because when young people get out and they actually experience another culture, they experience being used in ministry, it changes them forever. All that other stuff I said about short-term trips in general for the church is like doubly true for teenagers. Catch this statistic. This was a study done between 2010 and 2016. 84% of new long-term missionaries indicated they had previously served on a one or two week missions trip prior to signing up for long-term service. 84% their first experience was on one of these short-term trips. I'll catch this one. In addition, 50% of all missionaries overseas serving long-term took a short-term trip to that country before they went there. And so when your kids go on a trip to Mexico, 50% of the future Mexican missionaries are there on a trip like that right now. It absolutely is game-changing for the Great Commission going forward, sending our teenagers out and this kind of stuff. My own testimony, I'm not gonna take time to share it right now.  It was because I experienced a short-term trip. Okay, our last point here we'll make is this. Talk about our kids, we talked about our teens, I wanna talk about our young adults for a second. Here is, I feel like, where a lot of churches do so much of the groundwork, but they actually fumble their inheritance at the last moment. And that is, they do talk to their kids about stuff. They do have a short-term trip for their things, but they don't offer their young adults a chance to take the next step. You see, we've awakened now this burden in the hearts of our young people as we're talking about this. And like that church I talked about in Cortland, if you actually will give your young adults to go and a challenge for them, take a gap year, give six months, give a year in missions, let God use you in some kind of way, that is the thing that absolutely changes the game. Catch the statistic. For us to finish the Great Commission in this generation, all we need is one out of every 200 American evangelicals aged 18 to 30. If one out of every 200 American evangelicals, not even talking about the missions forced through the rest of the world, not even talking about other wings of the church, 18 to 30 year olds, if one out of 200 would offer themselves up into missions, we would have every person we need to finish the Great Commission in this generation right now. It just takes us mobilizing our young adults to go. And so this is the challenge I give to you guys as church leaders, as believers, take the young adults of your church, the most precious treasure you have, your best and your brightest, and invest them into actually going overseas. Some of them will go over for a year and they'll end up staying overseas. Some of them will go over for a year, their life will be totally changed from the inside out and they'll come back and they'll sow into the church in ways that you never could have hoped for before. I think of people that have come with me to China over the years. I think of Alex Seidler, the punk Bible school kid who came to us in China and is now leading the missions expression of a whole denomination of churches. I think of Allison Chamberlain, the academic all star with three majors who gave a year to China and ended up spending 10 there and now helped us start Radiant School of Ministry this year. I think of Amanda Randolph was here right now who gave a year into China and God's used her as an evangelist over there and now she's gonna be training the whole Radiant Church and evangelism stuff sometime in the next month. These are people that took that little step in the beginning and God takes it and he does incredible things through that offering. And so this is even my challenge as we wrap up our time here today. If you feel like I don't know how to do that with young adults, this literally is what I've given my life for. And if we can help you guys in your church in any kind of way, help you mobilize your kids, your teens.


everybody to the conference entirely devoted to the concept of topos here at Yahshes which is the place where topos were originally introduced by Grotendik in his seminar the the geometry algebraic du buamari. So the aim of this conference is to illustrate the wide-ranging impact of the notion of topos across different mathematical areas and encourage further developments in this direction. Grotendik in fact was himself fully aware of the importance and the unifying nature of the concept of topos. In his text Rekolte Semai he compares the notion of topos to a double bed realizing an harmonious marriage between the continuous and the discrete or also to a large and peaceful very deep river in which all the kings horses could drink together. This explains our apparently eccentric poster. So this is what Grotendik thought about the toposis. He looked at the toposis as a unifying space and the more precise here is what he wrote. For those of you who are not familiar with French here is a translation. It is the theme of toposis which is this bed or this deep river in which come to be married geometry and algebra topology and arithmetic mathematical logic and category theory the word of the continuous and that of discontinues or discrete structures. It is what I have conceived of most broad to perceive with finance by the same language rich of geometric resonances in essence which is common to situations most distant from each other. So we hope that this conference will stimulate more people across the different areas to study and apply toposis in their work as Grotendik instilled and deserved and hoped for. So I would like to thank all of you for coming here so numerous and in particular the speakers for accepting to give talks that promise to be very interesting. And of course I would also like to thank the other organizers for joining me in this adventure and of course the IHS for hosting us and also for the financial support which was also given by the Foundation L'Oréal d'Unesco pour les familles la science and this is basically the funding that made it possible to organize this meeting. So thanks again everybody for being here we hope that the conference is going to be enjoyable and fruitful. So now I guess that it's time for me to leave the world to the first speaker Pierre Cartier who will give an historic introduction to the concept of topos. Thank you again. So I will give a historical account of the theory of sheaths. Sheaths came before Topoi and Topoi is just a synthesis of all different kinds of sheaths. So I will start I mean here are two Bibles the book by Godement Théoride Pesou and the Carton Seminar two Bibles and I will refer to them later on. So the first part will be prehistory of Riemann versus Weierstrass. So the problem which was tackled by both Riemann and Weierstrass in many in very different ways is a problem of multi-valued holomorphic functions. First of all, holomorphic functions can be defined as follows which is Cauchy, Riemann and I would like to add Diderot no no no equations. What we call the Riemann Cauchy equation which can be written as df over dz bar equals zero. We are written explicitly by Cauchy and then by Riemann but about 50 years before by D'Alembert in his study of the wind motion wind motion and so the Cauchy Riemann equation of an interpretation in full dynamics and D'Alembert was studying the motion of wind at a global scale of the earth and discovered that he could use complex variables. He could use complex variables. Of course the theory of complex variables were just in the preliminary form but he could use and so he could use for example sign of x plus or y y etc as other facts. So but of course it has been more or less forgotten for a long time and it's only recently that good connection with the score. So Cauchy Riemann equation so it's a kind, I will not write it more explicitly, it's a kind of differential equation first order system of first order differential equation if you replace f by e3 a part and imaginary part and the idea is that in slightly modern form you consider an open set u in the complex field and you look for a function which is defined on u f from u to c and which satisfies which is a solution of this differential equation and as in modern slang I mean the collection of all these for all open set forms a sheaf in a modern form and more generally if you can replace this by an arbitrary linear differential equation or even non-linear differential equation you can have the same situation but the point of view of course the challenge is to study the multi-valued function. The most simple example of multi-valued function is log z of course what is known is that by definition d log z is d z over z which means that if you consider z as a real positive variable and you make an initial condition log of 1 is 0 then you get a unique solution over the real axis but soon it became important to study log for complex number the first and it was done by Euler and Euler has a question if people repeatedly what is the value of log minus 1 of course if minus 1 to the square is plus 1 and if you use a functional equation log of ad is log of b you conclude that log of minus 1 is 0 but if you want to do to be brave enough to try to consider log of z as a complex or z as a complex variable you soon realize that and Euler knew it perfectly that you when you approach so if you remove if you remove the negative line minus infinity to zero the rest is simply connected so c minus which is to make a cut this is simply connected and the differential from the general principle of differential equation this differential equation d log z is d z over z as a unique solution provided the normalization log 1 is 0 which is defined in z but when you approach the two lips of this cut you'd get different different values why if you approach minus 1 from the from the from above you get pi i from below minus pi and there is a difference so that was a challenge and what is the true value of log z if you don't make a cut if you don't make a similar problems occur when you want to solve not differential equations very simple primitive but primitive but let's say if you want to calculate what to find what is the solution of y square is x 1 minus x of course if you if you consider the segment from 0 to 1 real sigma to 1 you have two solutions we differ by your side choose one let's say choose that y is square root of x 1 minus x for x between 0 and 1 okay now the problem the problem the problem is try to extend this for a complex number how can you calculate the square root of a complex number from a modern point of view of course if again if you make two cuts if you remove now the real line from minus infinity to 0 and 1 to plus infinity the rest is simply connected and it's a general principle that if you have a continuous function on a simply connected space you can take the you can take the square root provide but there are two branches plus or minus so but if you want to have agreement over the real over the real part 0 to 1 you have only one solution so you have a solution you have again a complex complex solution of this equation so when x is real and complex provided it not on the cut y is well defined but you have two branches plus or minus one but then again when you have the same problem if you approach if you approach from from the from the bottom from the top or from the bottom the cuts then you get different regions if you start with one determination let's say and you approach and you approach it from the top then you meet the other real is the other branch minus one which comes from the from below so that was a that was a challenge and and similar similar things occur for differential equation the most conspicuous example is about the uh about hyper geometric function f2 2f alpha beta gamma z i will not repeat it's given by your certain power series in z which converges with ranges of convergence is is one so you have a function which is well defined on the which is well defined on the unit disk but then you want to extend it and you have a certain differential equation no need to write it in detail but now as oiler discovered and kumar exploited this equates this function as an integral representation and the integral representation is for the type 0 to 1 t alpha 1 alpha prime 1 minus t beta prime 1 minus tz gamma prime dt don't have well there are there are some relations between alpha beta gamma and alpha but it's not important but the point is that if you consider this if you consider this integral and you consider that you went uh yes then you can you can you can see it's well defined because it's a real integral integral over z1 but of course the difficulty is about this t alpha t alpha prime okay and then if you choose z which is not on the cut from 1 to infinity then you can choose a proper branch of 1 minus t z power gamma prime and integrate and again you have you have a solution which is defined outside of this but it's a second order differential equation and then you have two solutions to independently independent solution the other solution will not give the precise formula the other solution appears to be uh holomorphic except on the cut minus infinity to zero so in order to come to have all the solution you make these two cuts again you have a if you make the two cuts you have a simply connected domain and according to general result a second order linear differential equation has two linearly independent solution and so but the same problem which was solved by kumar if you if you take one branch that means one solution one solution and if you approach a cut from below or from above because the differential equation remains valid one solution will approach a linear combination of two one solution from the top two certain linear combination from the bottom which means that when if you have a if you have a vector space v of s of solution which is two-dimensional then when you approach from the top of the top that means if you make a full turn around zero you have a linear transformation which is exactly what we call monodromes but since there are two singular points for this differential equation there are two monodromes one around one one around zero and they generate a certain group of linear transformation which was defined given in explicit form by kumar so that's that's and now what do we do what do we do Riemann motivated by this example introduced the idea of putting putting a sheet and gluing them together for example for the square root you have two sheets on one sheet which is a copy which is a copy of this simply connected domain you have the you take the the branch of the function which is positive on this interval 0 to 1 and then the other and then you put minus one on the other sheet so you have two sheets and in order to to to describe the the connection the connection rules i mean the boundary conduction which brings one solution to the other one when you approach the cut then you make a you make a geometrical picture by taking two sheets of paper cutting and pasting of course this is highly intuitive and which was advocated by by Riemann but the criticism came from Bayerstrass and it's also a political a political fight the two major centers of mathematics in germany at the time were berlin and heidelberg and guttingen and there was a bit of fight between the people in and in the guttingen mostly around around felix ryan and heidelberg and in berlin mostly around bayerstrass there was recently in berlin a celebration of the 200th birthday of an anniversary birthday of bayerstrass and many good explanations were given at that occasion and i was there so now this is riemann which is quite intuitive and it's not by it's not by chance that one of the books by heidelberg is called unshowlish geometry which is translated that geometry in the imagination is not a good translation it's more geometry as you see it unshowlish geometry unshowlish in philosophy has a very strict sense which is not the imagination okay so then but on the other end bayerstrass came with a different solution his solution was the following if you consider if you consider a function holomorphic function what did you say a solution of the coche riemann solution equation are the given point z zero then you know that you can expand it as a power series sigma a and z minus z zero power n for n zero are positive suppose that you have two points z prime one z prime zero and z double prime zero and similar expenses are put in and then coefficient in n a double prime n okay you have two you have two solution two two power series expansion for instance they represent solution of algebraic or differential equation how can you say that it is the same solution how can do which is a problem of analytic continuation of course analytic continuation as a sound solution in the complex domain but people now try to do holomorphic function in a period is much more complicated and it would require you had three hours to explain how do you do it in the period case okay so now you have this solution and then the proposal of the proposal of of various tasks and that you trace a path from z prime zero to z double prime zero and you follow by continuity this power this power series this power series well more precisely inside a circle of convergence if you have another point then you can transform this this power series expansion into another one which is not centered at zero but another point and then you follow along the path and you get so you get a transformation which is analytic continuation if you give me a path and you start from you start with the power series mentioned around the origin of the path can you go until well of course it may be obstaculate you can assume that there are no obstaculate and if you have a differential equation or algebraic equation only the singular point can be can be can prevent you from following so now the proposal in slightly modern form slightly modernized form is the following by various tasks you consider the complex plane and your running point in is called z and then you consider you consider the following you consider and you consider some power series some a and t and and positive we call it phi phi of t phi of t so a series correspond to a collection of coefficient with some aboundness condition with some gross condition which which will give you that radius of convergence is not zero so now what you have if you call if you call s the set of all this series you are really dealing with c times s c times s you give a point and a series now analytic continuation can be as follows if i give you a point z zero and a phi zero and a phi zero and that's what i explained you consider the radius of convergence of this series and if you try and if you have a power series which is centered at z zero by transformation which are made completely explicit by habel you can generate a family of of power series so for this and so it means that given a point in this space you can have a neighborhood and you take this you take this as a neighbor so in the product c as the ordinary topology and s has no topology at the moment and then you can you can have this you can lift if you can if you start from this you lift this a small disk to this and of course it says that in the collection c times s you have a you have a collection of neighborhood and it generates a topology it generates a topology which is the one showcase use that in this you often in this lecture on analysis this presentation which is just a small variation but then if you now you have a topology on c times s and now if you take this this this topology is disconnected there are many components and the Riemann surface a Riemann surface is just a connected component of this so it's a connected Riemann surface called it r is a connected component but it has obviously a projection over c because it's part of this product space and it has also yes c which is z but also you can have you can have another projection you have another projection well not a projection this is projection which i call it z and then you have f of z which is the function which is a function will give corresponding to the Riemann surface so a function or homomorphic function in this setup is not a function over an open set in the complex domain you are first to consider a covering which is of course et al covering as we saw we need we know now and then the function is not defined on c but it's defined on this that's the idea of Riemann I put a Riemann surface okay so now it can be well it can be it can make the connection between the two approaches and in modern terms it corresponds to consider a shift by its section over open set or by its et al space corresponding et al space so but this is the pre-story and already all the ideas are there and they were quite explicit by Hermann Weid in 1911 in his book the idea of Riemann surface the idea of the Riemann shunt flesher that is more more philosophical okay so that's the pre-history of shapes now what come next second second is Leuré Leuré from 1941 to basically 1950 well as it is well known Leuré was he's made a work prisoner and he spent five years in a camp which was called an off-large officer lager which is camp for officer in German and which was rather my rather my conditions rather my condition and there was a kind of home rule in such camps that means the French the French officer elected their leaders elected their leaders and their leaders had to negotiate with the German leader with the German German commander of the camp a little kind of home rule and Leuré was able to organize to organize a so-called in the in university within the camp and which was a way to well officer were not forced to work as ordinary soldier were sometimes forced to work in feats or in factories but they had to they had to find something to do okay Leuré organized that I suppose to have been the French leader in such a situation would have been sometimes difficult situation I can imagine it is known I know from family witnesses that the Jews were not usually chased in such camps except if some Frenchman pretended that they were Jews and they don't send them to the open okay that's another story so Leuré who was an expert in fluid dynamics was approached by especially Hasse to work on different projects for the German for instance to be editor of central blood or maybe to work in the field in fluid dynamics but then in order not to be able he stayed which was very brave he stayed in his camp he stayed in his camp and developed topology it's not my chance because there was already the fixed point of theorem of Schauder and Leuré which was already topological in spirit but which was used to find have to find prove uniqueness or existence in uniqueness solution of non-linear differential equation especially in fluid dynamics so what Leuré invented so Leuré invented these sheets and his main motivation was the following suppose that you have two two topological spaces and a map from the space x to the space y okay now you want to calculate the homology of x and at that time homology means check homology the homology of x in terms of the homology of y and of the mapping so basically the idea of Leuré was the following if I take a point y in this small y then I consider the fiber the pre-image fiber f minus one y I consider the the the homology group of this fiber and I study how they change when y moves okay in under reasonable condition the reasonable condition for instance if it's a locally trivial bundle then of course it's locally constant because in a small neighborhood around why you can identify the fiber with it's also and the common with it's also but what's happened when you move out of course the idea of a local system was already known by I think it's with Whitney will introduce this I still know the Whitney I'm not sure but at least it was known to both so the idea of local system and all the topologies that you have a system of coefficients which vary from one point to another one okay now you want to reconstruct the new data of that you have for each fiber it's homology group and you want to reconstruct the homology of the total space out of this information and that's where Leuré invented the so-called spectral sequence which sometimes is called the Leuré-Cosul spectral sequence for the reason that Leuré invented it but in a specific situation and Cosul used it in another situation he used it for the homology and common of Li algebras but then if Cosul fed necessary to give a general formulation of what is a spectral sequence and then there are sequence and he gave the solution that it's a collection of it's a collection of complexes each one being the common of the previous one which is a very very good very good and sometimes misleading very good but sometimes so this and then this spectral sequence is something which enabled you to reconcile the common of X from the common of Y and the common of the fiber and Leuré and those many other people I mean develop this idea and especially Borel, use it to consider the common g of groups because if you have a group and a subgroup if your d group h and g then you have the projection of g over gh homogenous space and it connects the common g of g with the common g of the subgroup and the common g of homogenous space and Borel used this with great anesthesia with great benefit okay so that's that was already known and but Leuré did not well because of course of the conditions he published his final account only only in 1950 so it will not enter into the I think he was motivated by his study in functional okay so that's what we had then the final account was published into only 1950 and then in between so the Armand Borel used immediately Armand Borel who was at this time in Zurich and the Ecole Polytechnique ETH in Armand Borel used that for common g of groups and that's the main one is of his best contributions okay then in between then other people joined and Carton Henri Carton studied that already starting from 46 47 and Carton was invited in the Harvard in the in the beginning of 47 48 so beginning of 48 and there is a story that his plane was delayed and he could not miss a very well okay you can write a novel with that okay so and the the daughter of Rodrigo has written a novel about okay so Henri Carton and then Henri Carton had another approach another approach at this occasion he developed the idea of a greeting or which is in French carapace and I will come back to this idea later gating so it was another approach but then Carton and you have also to mention André Vey who entered into the game and entered into the game and André Vey was obsessed by the theorem of of Durham, Durham theorems and he repeatedly said that the core of geometry the Durham theorem is a core of geometry so and André Vey published a version of a proof of the of the theorem of Durham which is by far the best possible the best the best and which influence which influence Carton of course André Vey would pretend I don't know what is a sheath it's just a stenography would say what that's what's part of his way of of thing but is when you look at his proof it's clear that he perfectly understood what is a sheath okay now that was so there was a period of hesitation a period of hesitation and Carton when coming back from from Harvard in 47 48 and 40 not 48 49 in his first seminar in his official seminar which is available in this collection of books collection of books this was his collection then Carton explained both in his lectures in the in the Harvard and his seminar a way to define sheath grating etc but he was not happy with that he was not happy with this and this part of his seminar has been dropped from the published version of the seminar and it started again in 1950 1951 in a long seminar which was a collaboration with Samuel Ailenberg and it's my the first time I attended a mathematics seminar I was a first-year student that he called normal at the time and then it started again it started again to develop this in the last part of his one this recording in this volume then he started with a completely new point of view completely new point of view and then he gave a general definition of sheaths he gave a general definition of sheaths but mostly under the influence of Michel Nazard who is credited for that he developed this idea starting not from the sheaths collection of section of open set but the corresponding et al space and it's it's it's clear that it is it's dominant it's dominant it is exposition and so it comes at the variance from the modern exposition well of course I will come back to that later to the to the various views about sheath but then he started again as that's Henri Carton and but he was influenced by Samuel Ailenberg who gave in the first part of the seminar an account of the homology and homology of group in the axiomatic spirit of course Samuel Ailenberg and McLean invented the homogyre and other people invented the homogyre groups of also but for the first time Samuel Ailenberg presented it into an axiomatic way and he gave a series of axiom for the homogyre group which could be more more less formally well more or less immediately translated as axiom for the homogyre of spaces but the point of view is changing the point of view is changing because so far when you where people develop the homogyre of spaces where there was Czech Spaniard there was Durham they choose coefficients a space and coefficients they kept the coefficient and let the space vary for it so you take the homogyre with real coefficient and associated to the various spaces the point of view of space is a dramatic change of point of view now you have one space but you take the collection you take the coefficient as variables that means you consider one space you consider the category of old sheaths or coefficient whatever whatever over that space and the homogyre is defined as depending on the coefficient the space is filled by the coefficient right which is a totally new point of view of course and then we have also to remember that about the same time to pose a historical purpose to make that Ailenberg and Stienrod published in 1951 foundations of algebraic topology where they gave an axiomatic description of the homology group of a space with with a coefficient which are fixed not to call not to shift and and then they prove that for reasonable category of space comprising all the standard spaces the homogy is uniquely defined by this axiom but one crucial axiom is that the homogy of a port is trivial h0 exists it's a coefficient of the homology h1 h2 it's r0 and this small but later on later on it was this condition was removed and the so-called extraordinary commodity were invented were invented mostly by Krillin and other people can't Krillin and so on to get account of this so but it's not it's not the point so the change of emphasis is the following the space is there but it's just the place to put all the shifts okay now about the development so Henri Carton in his seminar and I spent yesterday evening I'm reading again this seminar and so give an axiomatic description prove a uniqueness existence and uniqueness theorem but with some restriction on the space is used well Henri Carton was obsessed by the idea of treating commodity with compact support and commodity with ordinary support at the same time so he introduced a certain collection he has a space x but he introduced a suitable collection of close subsets close subset of the space and he considers this the commodity with support in five five maybe the set of compact subset of a localized compact space or all the open or closed subset of a compact space compact spaces are a slight extension of locally spaces which was invented by Geudonné in 42 or 43 for the purpose of decomposing of a continuous function in two variables as a sum of product of one variable so that makes the exposition slightly slightly disturbing but the point is that there is a restriction this restriction on five forces you if you want to have general commodity with a close support that the space be by compact what by compact as I say so slight extension of locally compact but of important reason is that any metering metric space is by compact okay and that was enough for applications for various applications but there was a restriction there was a restriction and soon after that sir came as a dog in the garden twice first of all in his thesis which was defended in 1951 he applied this idea to to to homotopy problem so and the main object was the loops the loop space or the loop surpass space of if you have a space x then you consider the space of all paths originating in a given point zero and you consider this with the proper topology and you study the this is certainly not a locally compact space and you study the commodity of that in order to get information on the homotopy of the given space that is thesis that is thesis and with great success because you had a great success with a homotopy group of spheres with the hope that we could calculate all the homotopy group of sphere but we are still far from that unfortunately okay so sir came and he had to but he had to so he needed both commodity for these new spaces which were not locally compact not by a compact and also he needed yes and he needed these spaces and he needed the spectral sequence because the specter you can consider the state of pass is the vibration over x over space x what is called now a cell vibration which is characterized by the lifting possibility of lifting of polyhedral then he had but in order to calculate p the point is that p is a contractible space also commodity of p is nothing except for one is the same commodity as one point but you want to have information about the homotopy and homotopy then you can consider that the loop space is a fiber of this of this projection so the strategy is to combine the commodity of a space and the commodity of the fiber to get the commodity of the space that's what it is doing for instance in the in the in Borat but here the commodity of this p is no one it's a contract but so you get an interconnection between x and omega and the point which gives you information about the homotopy group but for unfortunately both the both the few of shifts and the commodity of shifts and the notion of fiber did not work so he had to invent he had to invent and he explained to me recently that he listed this he made he made a comment i have to reprove when he was cautious about not yet not not disturbing loray and i had to reprove some statement or make a new connection etc and he explained to me that it was quite delicate not to offend loray who was not a music character okay so loray did that and obviously one needed and then he invented ad hoc method ad hoc method he in order to fit with the vibration he introduced not a singular homology but not with more than which are simplices but tubes now we have much better much better term not better tools in the in the in simple it's implicit homology and we can deal but we're using by simplisher island there's it burns so on we can do this directly but he was forced to use chemical homology but he had to prove that chemical homology was the same as simplisher homology so a lot of discussion of course great great great progress but at the cost of costing the boundaries now sir did a second time did cost a boundary a second time where he used well first of all in between there was the work of carton and sir on complex variable many complex variable and then they use civc homology with a great success to deal with problem of many complex what holomorphic function of many complex variable where the topology is quite delicate a remand surface on one variable is rather easy to discard a remand surface in many complex variable is an awful object an awful object which is really not completely understood okay so but then carton and sir used they use civc homology they did not create a new homology they refer on the carton homology which was defined as I mentioned first in 46 47 and in new form in 50 51 okay but when after that they obtained the so-called theorem a and b which are famous theorems in the chronology of coherent shift of holomorphic functions but then carton sir developed application to projective spaces and using a certain trick which was given to him by someone in robot someone in the group and then then he used civc homology but he used for projective space common geofrojective space and then he realized that his method were basically algebraic basically algebraic but what was needed what was needed is the risk topology by the way the risk topology as we know it today is not what inventor was I just invented invented the variant of that topology on the net of all local of all valuation rings of a given fun or given feet which is not exactly the same but who was used sometime by other people's ice kid topology but the size kid topology the rumor is that this was invented by chauvalet in a seminar given by the risky of no real proof but it's it's quite full quite you it seems quite reasonable to assume that but the series kid topology was first used by on the way on the way in 1948 gave a course in gave a course in in Chicago which has not yet been published unfortunately I tried repeatedly to have it published but I did not succeed he gave a course about what are fiber spaces in algebraic geometry of course and then if you have a fiber space the main property is that it should be locally trivial but locally the human that you have a new need of topology and from various conditions it was clear that the good topology was the risky topology so it's really on the way who invented what were used for the first time seriously and also he was connected with so-called cousin problem which is to reconstruct a meromorphic function in many complex variables out of the set of the poles and divisor of the function and zeros and poles okay so Andre Bay was the first to insist on the risky topology as a tool to imitate topology in algebraic geometry and then sir knew that perfectly sir wrote a report for the Boba key seminar about this work of of Andre Bay and then and he immediately tried to imitate what he had done with the help of carton for a complex complex projective space viewed with common of shifts over projective complex positive space and he as he explained to me many times I mean it was so easy now it was so easy now so easy now compared to what he had done with carton in complex variables so it's sir who invented and his famous paper which was finally published I don't remember it well it was it was written 53 but published in 59 55 that's there and and then I remember my surprise when I presented to me part of his manuscript an algebraic variety is a topological space in algebra it was a surprise to many people okay then but he had to invent he had to to use shift homology but Zeissky topology is a terrible object compared to ordinary topology it's it's completely exotic compared to ordinary topology and there was no shift homology available for that and then he had to reinvent it to reinvent it in a slightly different way so sir twice cause the border cause the border showing the necessity of going beyond ordinary spaces and all the chiefs over ordinary spaces but this time you had to the challenge was to invent a homology of shift without any restriction on the space uh at the time got and it was back to Paris and there were many exchanges at the time with Grotonnik and it's Grotonnik who provides the first solution he provided in his typical way first well of course the main challenge was the following from homological algebra non-metal of homological algebra as the book of carton and island man was just printed at the time he appeared just at the time in order to define homology you need resolution well if you have if you have a module for instance that the modules is a quotient of a projective module this is easily a quotient of a free module so you can define a projective resolution quite easily but duly injective resolution injective resolution at first injective objects were invented just by duality because island man was obsessed by the duality and obsessed if you take a diagram and reverse all the arrow what happens if you were defined by diagrams of property of being a projective reversal the arrow you have the definition of injective but it was a quite exotic object an injective module and it's not easy to prove not easy to prove that the modules can be embedded in an injective module buyer buyer gave a proof which is quite quite sophisticated then okay so it was clear that for shift you could you don't have projective shifts you don't have free shift projective keys whatever you don't have that so the need is you have to go to the other side and take the injective shifts but to to to to construct an injective shift were not obvious and no one knew how to make it okay then got and came from the sky I mean he said what in order to prove that any object can be embedded into an injective object I use a buyer construction the buyer construction and I look at what are the formal properties using this proof and the formal properties are basically or the basic the crucial idea is that if you have a module M and a connection of some module M alpha and another module it's a distributivity law which is intersection let's see intersection of M alpha plus M is something intersection of M alpha M alpha plus M something like that something like that I'm sorry ah okay okay okay okay you are right you are right well one basic ingredient in the proof was this so Golden Dick invented the idea come from the sky I will consider an abstract category and at this point invented the notion of of of a billion category which was about the same time invented by your books bomb under the name of exact category so that was more or less natural natural idea the new idea was to introduce this as an axiom somewhat corresponding abstract form of this action of this action and then repeating the buyer construction you can prove that any object can be embedded into an injective object in a given category so now the next step is to prove that for shifts it works well but it's a rather easy exercise it's rather easy exercise so in this way Golden Dick could prove absolute from abstract principle that a shift can be embedded in an injective shift and then you have injective resolution resolution of shifts okay but almost immediately Godamon came with a more explicit solution Godamon came with almost almost immediately with an explicit solution which happened which is the following take a shift f over space x now I consider the various the various fiber fx for all small small x into it and take the product and take the product of the fx which amounts to the same you have x space and you have x delta which is the same space with a discrete topology at the time you have a you have a natural map of course natural map we discontinue the right direction call it pi and then you play with the factor pi star and pi lower star of direct image and inverse image but for this map from x delta to x and then it's easy to show it's easy to when you have a discrete space what is a shift over discrete space which is injective each fiber is injective so it was easy so that's the that's the solution of that was provided by Godamon and the other variants of various form of these conditions so here we are here we are gotten dick invented well by his new category category called theory of a billion category is able to prove that you have injective can apply it to the shift you have injective see if you can repeat exactly word for word what was given in the carton rebe for modules and get in one in one or two pages all the necessary properties okay and then as I said Godamon in his textbook repeated that with a slight variable so now as we are well checked yesterday that Godamon book was published in 1958 now the famous paper of golden dick which is called talk because it was published in the talk mathematical journal was written in 55 and the copies were circulated and discussed among burbaki at the time but it was published only in 58 so the book of Godamon and the and and and and the good and the papers appear at the same time and they refer what Godamon refers to to Godamon okay so that's now what we have that's now what we have are at the moment you have a now a completely general theory of shifts and shift homology over an arbitrary space and you can cover both the point of view of sir i mean the sir vibration is and the path space is covered by this new new theory and also the application to a gemite to a gemite geometry so that's what it was around 58 now right no no no no no no no no no no no no no no no no no no no no no carton carton carton gives us in for certain spaces the common g is a check homology and already already learned at few rams comparing his chronology with check homology spanner singular and uh what was the last one spanner singular and uh doram so we are already many many many in with wait wait wait but already well loray already compared to take well loray loray come defined check homology for shifts and compare it with his common and understandable condition is the thing okay pardon see see see see loray invented the chief commodity no problem and then carton gave another version but it was well another presentation i would say but no no no no he doesn't define it as check homology no he defined it in general and then he proved that by by resolution by resolution i will come to that at that at that moment okay so that's exactly the idea of grading so okay so at that time since that what we had of course are what came what became very important and what precedes topos topos is that first of all there is a comparison between uh well the comparison between the two point of view it's either a trial space or it's a collection of section over the open in in carton seminar it takes a definition for a trial space gives one in a few sentences he said what we could also use it in this way we use it in this way but i think carton was uh dealing with uh complex variable and he was in full by the point of view of viostras okay so uh yes but uh the notion of pre-chief for instance which is so familiar today doesn't appear explicitly in both in eyes of case so of course there are some remarks if you start with what we call now a pre-chief you can generate etc but usually what was what was done is that you start when you had a pre-chief you associated to it an et al space and et al space was considered a shift but the problem of shiftification starting from a pre-chief to get a shift was not really really i mean the present in this day it's really good and we introduce it in this sg for this point of for more general situation so uh another another problem is that about another problem is about a a continuous map so if you have if you have a if you have a shift as here you can define an inverse image x star g which is x star g which is here and you can define the direct image x star f there and then there is these two factors are adjoint to each other in a proper way and by the way when the the adjointness is not as mentioned in the both both carton and the one is not the right one they take a right adjoint instead of a left adjoint etc so but then it's uh this was not realized as such import such import but now of course the point is that if you want to define f the direct image is very easy using the definition of the cheapest collection of section of open set because a section of this u is simply f of f1 v but the the the inverse image is better using the entire space okay so it shows that you have to to do to do with both now the so there was this problem there was this problem and it was important that when you have a continuous map from x into y you have two factors for shift one which is the inverse image and other one directly made and they are adjoint to each other in the proper way and and the existence of this adjointness is considered as a crucial property by Rotary the the new idea of Rotary is to replace the space x by its collection of sheets by this category of sheets and now instead of having a continuous map from x to y you have you replace fx by its topos which is this category of sheets y by its topos so called x the topos these are now categories and then instead of having f you have to find the f lower star and f upper star which are adjoint to each other with suitable property so that was now that gave the definition of maps between what we call geometric factor between both topos then there was another important fact which is when you have when you have a pre-shift a pre-shift and as I said it a pre-shift is what it's just a factor already in modern terms you have a space x you have the collection you have the category let's call it's top tx topology of x which is a collection of open sets so that's the object of this category and the map are just simply the inclusion if you within v you have the injection of inclusion in u of v and that's only maps which exist so this call and now a factor called trivial factor from tx tool it says category of set category of things category of whatever category of cows if you want to have chief of cows and then then it's a it's a covariant so it's a factor tx op tool let's say sets if you want to all those things okay then it how can you generate how can you generate a shift it the idea is the following in godemort carton etc the idea is that first of first research a factor you associate a space e which is ital over x and then you take the section of this space to generate a shift but the point the point the disadvantage is that you have to use the point of your spaces and it's contrary it's opposite to the to the spirit of categories the spirit of category you should never mention rarely mention the set the point of a set or space but only what maps etc so this was not very satisfying and as a result it's not very mentioned it's good and it who really understood the importance of it and the point of the now the what is important is the shiftification which is so called pre-shift pre-shift pre-shift of pre-shift are just the I mean this is a collection of this this co-travagant factor with no with no relation and then you have shifts which are shifts shifts which are of course a special class of pre-shift now you have the inclusion map a pre-shift is a pre-shift and upon the shiftification the shiftification factor is what here you have the inject injection of this category into this category and there is an adjoint there is an adjoint which is the one called called hs and adjoint with suitable properties the point is that an adjoint of a factor satisfies automatically certain conditions with respect to limits and you put another limit so core limit general core limit sorry preserved by s but you want you have to assume a little more so now the point what what was really important for Grotendi is realizing the analogy between these two situations first of all you have if you replace the space by the top of all this shift then a map is correspond to a collection of two to a data of two adjoint factors and again here but it's completely different you have a you have a shift you have a category of pre-shift and you have a sub-category and again two adjoint factors and this is really really the crucial discovery of Grotendi in this case and then from starting from these two examples you can formulate the general definition of a of a shift which is the static sorry a topos well I elementary topos if you want but these are special these are special case Grotendi Grotendi topos which are a particular case of the of the of the of the elementary topos okay so I just finish that and that's a short historical account the last part should be much more developed because but then we are very close to Grotendi or you already got in Grotendi so I wanted to describe the pre well the pre-history the pre-history the pre-history and then well the history begins there and related it's a different story I would just wanted to show you that the notion of shift was natural natural after Riemann and Weierstrass the two point of view then it was developed in the 14 and 15 by especially Loré, Carton, Borel and Seyre that the people who should mention and then after that well after this big synthesis both by Carton and by by Godemard then what started but of course shifts are many many other applications which I did not mention for instance my colorful theory of shifts perverse shifts so I don't have time to to to develop all these the development of shifts I just wanted to tell the creation of shifts thank you very much


I'm Bjorn Grenade, you're watching Bjorn Grenade today and today is part two of our Nashville Yee Adventure Diaries. We have it in part one, go ahead and click right here so you can watch that first. But the adventure continues, let's do this. Good morning! More Yee Adventures! We are about to go get some boots, some cowboy boots. I've never had cowboy boots before. It's my very first pair, so they're super cute. I want them to have a lot of bling, I might be asking too much, but we'll see. I think I have to get a cowboy hat as well, right? I don't know if you want to cage those curls. I know, my curls might not fit a true cowboy hat. Do they have satin lines, cowboy hats? Probably. They're pretty dope. So cute. True joy. I'm blushing, soft. Just a little bit. Wow. The possibilities are endless. This is how my closet needs to look, just like the whole downstairs. Go. Yeah, baby, today we are saying yes to shooting some guns. The neck. Just trying to bring that down there. Bianca's doing the big boy. One down, baby. Now you're all warmed up and everything. Can I try this one? I haven't tried that one. We're about to eat after the shooting range, Pinewood Social. Awesome. We're with what, Lauren? We're going on Mint Shulip tours to go see the Jack Daniels Distillery. Wow, man. You're gonna love it. Driver, can you give us a quick review?  Oh, about four or five times. It's time to... Yeah, couple friends that would love this. There's gonna be two chairs by his grave. Friends could come and mourn his death after the fact. Sit on the grave, you know. Pay the respects. Why didn't he get six chairs then? You know, I mean, we can't ask him that. Gentlemen's Jack, I don't know about that one. Sounds like a side-chick fight waiting to happen. Yeah, his grave side-chicks are really going hard. It's not what you want to be. That's about the worst thing you could do with a grave side-chick. So you can hold your drink and have fun with it. Make it your own, so have a good time. So if you mess up the moves, just boogie and then hop back in. That's what I do. And that concludes part two of our Nashville Yee Adventure Diary. Believe it or not, I've got one more day to show you guys, which is my meet and greet at Jewel Salon. So make sure you come back next week to see part three. I'll see you guys next time. Thanks for watching Bianca Renee today.