Hello everybody and welcome back to the average, today I have got an exciting package and it is some alcohol markers from Artex that they kindly sent me to review for free which is very nice, thank you Artex. That is really nice, look at this box that it comes in, let's have a look at it properly. So it comes with this little carry case which is quite cool, so it has a proper class pier and it looks like this on the front, this comes off so we can reveal what's on the inside, unfortunately it is a little bit dented here but you know I'm not too bothered about that. I never really get annoyed about stuff that happens in shipping because I live quite far away from probably where these are posted so it's kind of understandable. Let's open them up, whoa, these are awesome. So it comes with a thank you, a nice little postcard which is quite pretty actually I like that and ooh is there an artist, no it just says Artex so yeah and it comes with its own little sheet for what's the word, why can't I think of the word, wow I am the worst artist in the history, I cannot even, scamping, sketching, swatches, they come with a little paper for swatching the colours and it tells you each one, let's just look at them, oh my god, they're beautiful, they are beautiful, they feel pretty solid plastic which is quite nice, they come with the lids with the colours on and the number and then it has this little like, I don't know if you could see that little cloud detail around the lid which is really cute and original so we'll give them points for that, that's very nice and it says the number on both ends so it doesn't matter which way you put it back in and I think it does sort of just rest like this if you want it to which is pretty neat because then you can just take the colours like this and I think that is amazing so I don't even know, I don't think I've ever owned this many alcohol markers maybe I do in Copic form but never as like a package like this so I think if you bought these in Copics they would be lots of money like I know that a Copic set of like this many are roughly that many is probably about 100 quid so you're getting for 40 pounds instead of 100 pounds for that you're getting all this but do they compare to Copic markers that's the question I've got some pastel markers that I would like to sort of compare them to you I mean always I believe that you get what you pay for so you know even if these are a little bit not as good as Copics it's still worth it maybe let's well I can't say that yet but let's trust them out so they have a nib point they don't come with a brush pen which is kind of a shame because I prefer using the brush nibs on these alcohol markers there's also a chisel tip which I always avoid using but lots of people do like them I think they are really good if you have that kind of style that you use them with yeah I want to test these out I don't know what I'm gonna draw but I think we'll just do like a chatty video today I keep you guys company on your your drawing drawing day I don't know so I've got my cup of instant coffee I'm afraid to say because desperate times guys and also I've got my lovely I love these they're very very like I have these four beautiful I love these Copic sketch markers that come with brush and chisel tip and they're all like pastel colours and I chose them when I was in Japan I think you guys saw that in my Japan haul video back in the day and yeah so I don't know if I'm gonna compare them because I think that's a little bit harsh because these are obviously top tier and these are affordable versions of an alcohol marker so we'll just we'll see how it goes I've got my sketchbook here sorry if you can hear these seagulls but I have moved locations slightly and I thought I would just draw whatever I want today when we're just gonna test stuff out so okay I'm I want to try this one because it looks like a really nice pastel-y pastel-y so what I mean like a light color yeah I gotta tell you I ain't a huge fan of the nib tip pen but you know I think they blend pretty nicely let's see how they blend together let's use a chisel let's just go for it chisel tip boom yeah these are very nice so I think that is really pretty impressive for the price point if you guys can't afford Copic markers or you know a fan of alcohol markers but you want to chest them out then this is a pretty good way to go maybe you don't want that many though so I don't know I don't know your story guys I know your life you know I don't know what to draw hmm do you know what I do want to draw maybe like some houses because I really enjoyed doing that with my previous horror comic is what I did was I drew loads of lutes of I draw lots I cannot speak I drew lots of kind of haunted houses to get inspiration and I used alcohol markers I used alcohol markers for that whole comic so I think it would be nice to revisit that idea because I know for my horror comic I have been saying sorry about cars guys this is gonna be a very relaxed video though so yeah that's just gonna be the case what I was saying was that I haven't really explored using anything else other than gouache for the horror comic style and I know that I did all my other comic with alcohol markers and pen and like a black pencil and I think that could be something nice to revisit I know I said I wanted a completely different style but if I have experience in something then maybe it shouldn't I shouldn't totally disregard trying that I mean it worked for me last time and I really enjoyed it so maybe it would be a wise option even though I am taken with the idea of doing it all in gouache paints I'm not sure it's the best approach so let's see maybe now that I have a lot of alcohol markers given to me would it be a good idea to use them for my horror comic let's find out yeah so I want to do I want to blend them together because I think that is a great thing about alcohol markers basically if you haven't heard of alcohol markers before they are made with believe it or not alcohol so when you put them down they mix together and you don't have those lines like you do with felt tip you remember in school when you'd have cheap felt tips and you'd kind of do it like this to get it not to have so many lines like that but with alcohol markers you can just blend them together and they should not have like such harsh lines I mean they can do but you it depends how you kind of work it and you can layer them which is also a very nice thing they also come with like a blender marker which they have provided here so I'm gonna test that out so the blender marker I believe is just more alcohol I am not 100% sure on that so don't quote me but yeah it's just a way to mix colors together and sort of ease them out because but I'm not a pro at it so you can see here that I'm not very I'm not the best at alcohol market but yeah you can go in and use like lighter colors to sort of blend out more nicely I wonder how everyone's doing I hope you're all staying safe and in your houses and just chilling was this whole horrible horrible situation passes us I'm currently working from home which is a very very weird feeling because it's just like totally different I actually went into the office and got all my equipment from there like my iMac that I use and yeah since doing that it's sped up the process but it's just been a bit of a panic it works been pretty rough week but we'll get through it guys do you know yeah so I'm just drawing this house I've seen on Pinterest basically whenever I get a bit stuck on what to draw I really genuinely just go on Pinterest and make like kind of a mood board and save lots of things together I'm gonna check this is going through the page it is I'm gonna use this because I don't swatch stuff mix stuff together and take inspiration and I think it used to really work for me on Instagram but since they changed the algorithm it kind of took away that aspect so now all I see is pictures of people's babies and things and I'm like mmm I like you guys but I want to use Instagram for inspiration not like your holiday trips and I mean I do like that aspect of it but when I need to get some inspiration for drawing or you know just want to see what everyone's up to art wise any who thought it was a ramble yeah like I said this is gonna be a rambly video and it's kind of different to the videos that I would usually do so I hope that you just stick on stick on what stick me on in the background of whatever you're doing and just draw along with me maybe or I don't know whatever your life presents really really really do not like this color combination that I've picked out and usually I feel like I have a good sense of color if that is okay to say like but this time I'm like what the heck is this ugly green and blue combo in gray I don't know but yeah this is this is life you know sometimes you pick your pick bag color choices so I really like these markers so far I think they're pretty close to Copics considering the price point yeah I like them I like them just to show you the Copic markers so these are the ones I like because they have this brush pen and they are very very smooth but yeah there's not that much difference from this to this so that's kind of nice the only thing is the brush tip what am I saying the brush tip is something that I enjoy but we can get used to these nips also the nips are like handy if you want to do little details which can't really just kind of hard to do with the Copic markers with the brush so there we go that's there's a plus obviously when you have alcohol markers whatever you lay down does bleed out slightly and these are bleeding out a little bit but that's normal so you've just got to bear that in mind when you're using them like when you place down a line if you want it to be a detailed line then just bear in mind that it's gonna bleed out slightly but that can be kind of nice for the style if you work with it don't go against it just kind of use it for your advantage advantage why am I speaking like that which I haven't done that with the road oh well done now but yeah the good thing about alcohol markers is that you can layer them up so if you think like one bit is a little bit weak you can just go back in and it will absorb into the page but it does give like a nice little effect of like a darker area and usually in mind do you mind and usually this just works out if you want to add a little bit of depth or you know a little bit more definition to a certain area to draw the eye that's really handy so there you go guys that's just the pro tip of the day I apologize it's not the best tip okay right I want to draw some trees in the background here because I think like it will add a little bit of this is a very very light green this is why people do swatches you know I want a darker green than that so let's get number 50 which looks like a very dark green these also don't I mean they do have names but they don't have them on the pen which would have been nice so 50 is they got the names here that you can reference them if you want to forest green let's go with forest green then I want to just do a little bit of like a darker grass grassy knoll over here just to make sure that it's in the background you know maybe it'd be a good idea to do some like haunted houses and sketch those right now because I want some inspiration for my horror comic and obviously the inspiration right now has been like motels creepy old motels and stuff but maybe I want to just explore ideas a little bit more I have been writing the story out a little bit and what I know now is that the characters there's gonna be two characters so there's gonna be the main character and there's gonna be like a little girl traveling with her I haven't really decided her name yet but her name her age but I want her to be like maybe 11 12 kind of mature but enough to know like what's going on but not mature enough to be able to kind of deal with stuff herself if that makes sense without an adult around but yeah I'm still thinking about names for these characters so let's see I really want you guys to be involved in this process a little bit more so if you guys want to name a character then just comment down below and give me your name suggestions and ideas so what it is is that one of the characters is in the aunt of this little girl this young lady I don't know her age yet probably a young teenager and they are traveling together because she's sort of rescuing her from not a very nice father and they're going somewhere else and that is my new idea and I think they can have a really good back and forth conversation wise whereas before I felt like just having that character by herself it was a little bit boring and I think it works have two characters together so I have decided that they will be two characters and if you would like to name them both or name one of them then let me know this takes place in roughly the 1950s in America again we're going to have that sort of like timeless quality but give me names down below and I will you know tell you if I use them and thank you very much so yeah that's the story I can't really tell you more about the characters to help name them because I haven't got that far yet I mean the aunt is kind of a little bit bitter but not bitter but just kind of wise and yeah I guess bitter and a smoker and just sharp and then the young girl is full of zest and life and she's just just full of optimism about certain situations so if you would like to name the characters and then comment below and I will look them up if you got that far into this video because this video is a logo yeah I just want to keep guys company whilst I draw because I haven't really done a draw along with me video since forever it's just gonna be very very relaxed and so yeah with my old comic I used a pencil for the lines and then I colored them like this and I think I do really genuinely like that style and I do kind of miss drawing with it so they can just be like that fun little example of that yeah my first impression of these is that they're quite nice and I enjoy them they're very good for the price point and yeah I like them I'm just gonna draw more more houses I think that's okay with you guys okay so I'm not a huge fan of the way that this drawing turned out but that is no fault of the alcohol markers I would say that is just me warming up so I'm going to draw some more spooky houses and let's try it out I mean this is just my sketchbook and usually I've found that I've been very been more sketchy in my sketchbook if that makes sense more rough because usually like before I would just I would have finalized pieces in my sketchbook because that's just the way I like drawing but recently I've been drawing more on separate sheets of water color paper and stuff so for me it just made sense to I'm gonna use the chisel tip a little bit more because I don't tend to use that and I think it will be nice to try and explore using that feels tip is great if you want like straight lines very easy to get like a perfect straight line with them which is very nice I think that's why a lot of like architects and people who design products use use the chisel tip a lot which makes sense probably makes sense to have swatch these out before just going for it because as you know not always the the colors aren't necessarily match the the lid but in this case I think I think they're pretty good they're pretty on point so that's that's good usually with cheap alcohol markers they don't really tend to bother with the lid or the color of this indicator matching what the actual color is so it's nice to know that at this price point they they have put four into it and they do care I also think that this whole idea of it being a carry case that you can just place down is really nice too I'm very impressed with the whole design and concept here very impressed I'm trying I want to get a little bit experimental here with these because I think that there is room to be a little bit a little bit weird with these which you know I'm a little bit weird with my artwork so that makes sense doesn't it I hope that this will be a little bit of a longer video than usually because like I said it's gonna be a very chilled video and we're just gonna chat so I'm hoping that you guys are enjoying this type of video if you don't let me know it's fine I don't need to be like this all the time I'm just doing a one-off thing of this because I've had a very long week and I think we all have really to be fair I think we've all been a little bit not down but just like a bit stressed out with everything going on so I think it makes sense for me to just it was either gonna be no video today or just this really chilled basic editing video that I'm doing right now and I hope you guys don't mind because I wanted to speak with you sort of you know I'm not really speaking with you but I am a little bit and I hope you don't mind that this is the yeah I keep saying the same thing I think I'm like just repeating this um it's like I said it's a little bit weird as you wouldn't see this in real life but it just works I think behind this I want to use my colored pencils because I I just love like colored pencils and and alcohol markers together I just I just love that kind of aesthetic and I love using them so I was gonna go back to Japan at the end of April and obviously that is not happening anymore because of everything going on it's a bit of a downer but that's life sometimes and I'm just trying to stay positive and keep my head up I think everybody is at this stage and then you know I'm gonna focus more on trying to do my own projects even though work has been crazy and especially trying to work from home has been crazy but you know yeah like this idea of like slightly crooked lines within it's not because I can't draw I just like crooked lines in my drawings no I think it just adds like a little bit of charm to have like these windows that look a little bit wonky wonk there's the correct term guys wonky wonk so yeah I am I really like these markers I'm so glad that I'll text them to me I'm really lucky it's part of the perks of you know working two years on YouTube um okay so I'm gonna do something weird now I'm gonna do the background in a really dark gray it looks really ominous but I do like the way that it looks anybody just joining this channel and have actually made it to this point in the video I just want to say that I am quite experimental with stuff I do and like quite rough with things I don't really think things out I just kind of go for it so if that's something you like then um welcome that's just how I like to do things because then it kind of takes the pressure away a little bit and also I get more into like the design of all the process of doing some of this stuff so I need to use a lighter color maybe I use like this pale sort of skin white skin color see how that looks showing definitely like a lot of texture in my work I feel like it adds visual interest when I'm doing stuff and just something I enjoy like a lot of people probably don't enjoy that in the stuff like they probably like smooth clean lines and smoothness but honestly I prefer it I like not when looking at other people's stuff like I like when I like other people's stuff I like it but um for me I like to use a lot of texture okay house number two done you know what like maybe you can add a little bit of pop of color here hmm maybe that's too dark let's go with the light a one first and then we can build it up if it's too dark so just getting bigger and bigger with this piece I'm gonna run out of paper I don't know why I've done it like that like yeah lock out this section they're a massive house for no reason just because I am dumb okay now it's time to go darker let's see how this works so yeah I'm just gonna do some trees in the background and I think I want to add like pops of color to all of these because I think like that's more my style than this kind of gray stuff that I've been doing so I'm just gonna be a bit quirky with it because I think that's kind of cool that style um yeah I want to do like a darker blue in the background of this one that definitely adds something to it like this style a bit more blue here yeah I like that so I'm gonna do some more pops of color around that's a great thing about alcohol markers is you can just definitely layer them like whatever and it works quite nicely okay guys so I think that is a really cool page and yeah I think I'm done here I really like the way that I did in the end it got a little bit more experimental which is more my style and more my thing and I really love these markers I'm definitely gonna use more of them and sketch out some more stuff if you like to see some random sketches and things like that then please follow me on instagram because I like to post up lots of images that I've done from videos and also other stuff and I like to do different stories and things I'm hoping to post more on there so if you follow me there then you can get some inspiration from me maybe and we can chat and stuff and I yeah I don't know about you guys but if you like alcohol markers then maybe they're worth testing out I think they're a good cheap alternative to copy markers and they work really nicely and they look pretty damn good too so that's it from me I hope you guys are staying safe and well during this time and I really look forward to our time together or my talking to the camera to you together if that makes sense so I will see you next time hopefully and yeah good luck everyone bye


Felly ydych chi'n mynd i gyd yw'r ffordd o'i gilydd gwybod chi? Felly mae'n iawn chi'n edrych yma. Rwy'n ei ddweud o'r ffordd a'i adeiladol. Roedd yn cychwyn i fynd wedi gyda'r tîm. Mae'r ffordd o'i gydag ffordd, a'r hyn wedi'i gydag, mae'n ddod i amser y gydag. A'r ffordd o'i gydag. Mae'r cyffredin ni'n gilydd o'i gilydd. Mae'r cyffredin ni'n cyffredin i'r cyffredin i'r cwm. Mae'r meddwl, ond mae'n modd i ddweud. Felly mae'r meddwl yn ei wneud yn gweithio, a'r cyfleaeth yn cwmwylliant y gallwn amser, felly mae'n gyfreidio'r gweithio, oherwydd mae'r gweithio'r gweithio'r gweithio'r gweithio'r gweithio. Mae'r gweithio'r gweithio ar y ddesun. A'n eu chwrs ac mae'n gweithio'r gweithio, ac mae'r gweithio'r gweithio ac mae'r gweithio'r gweithio. Mae'n credu'r gweithio? Ie ddim yn ymwybod ar ddatblygu'r ymdîm mae'n gofyniad i'n bwrdd yn gweithio'r gwaith a'u cyfnod ar y casg yma? Mae'n gwybod, mae'n ddweud... Mae'n gweithio'n ddigon i'r ffymdd, gyda'n ddigon, yn ymwysig. Dwi'n gweithio'n gweithio'n gweithio. Mae'n gweithio'n dweud o'r mwy o Gymru. Mae'n gweithio'n gweithio'n dweud o'u gweithio'n dweud o'r gwaith. Mae oherwydd mae'n llagl ceisio hynny i amddangos i'r pwrdd yn cael ei wneud mae'r ddefnyddio y dragell, ti'n trwy'n bwysig fel ymgyrch hyn. Mae'n cael ei gael bod yn cyst wordsdag gyda'r ddau a'u cychwynllus, efo'i un nôl o eu llei, fel ei wedi ei gael i'u clywiau. Yn y c nhw i chi ei gael, mae yna eftan yn ei sgol yn gweinol i ddweithio ac yn gweinolahod ac yn gweinol a'u gwneud. Ond mae'n cyfgrifedd o'r ridw i'r ysgol i'r byd, Mae gennym i ddim yn rhaid o'r siŵr, ac mae gennym i ddim yn rhaid o'r neid. Mae'r gwybod i chi'n cael gweithio'r hyn yn ymlaen i'r teimlo yn y Llyfrgellol Cymru? Ymgyrchau y Llyfrgell. Mae'r Llyfrgell yn ymgyrchau, ac mae'n hoffodd yma o'r agorau. Mae gennym i ddim yn gwybod i'n gweithio'r agorau, A os ydych chi'n teimlo i'r clyweddach yn yich gallu hwn yn ei gysylltu gwybod ym mwynhau? A os ydych chi'n ei gael'r penedig a rydych chi'n ei ddenion ddigonio nad oedd yn rhoi gofyn o'r gofyn. Mynd i rydyn nhw'n ei gael ei nosion eich godiadau. Oni advise i chi yw bethau gyda'r du yn batech chi ei wneud unrhyw oherwydd dyda chi'n ei gael. Byddai'n rhoi'r straiker ac nid yw'r unwyddu'r un i'r witnesses. If you could tell me where I can get one from, I might be interested in another straiker. We've searched for a long time for the club before I arrived. As you well know, as everybody well knows. This one we've focused on as quickly as we can. gyda'r number of strikers that we looked at since I arrived, I suppose. There are other players that you're being linked with. Midfielder, Jean-Michel Ceri at Nice. Theo Wolcott is another name that's been linked as well. I think one at a time, you know, one at a time would be the order of the day. I think that we are at weird message yesterday that we're going to allow Janik Balasi to go on loan, you know, and his agent is wondering why he's heard that. And I think to the agent and to everybody else that sat here and everybody, this is January for you, that speculation creates massive problems for all managers because of the tip for tap that goes on, the whispers, the Chinese whispers that get rumoured. And it's very disconcerting to particularly a player if he hears that message. And having only got Janik back after such a long injury period, it's ludicrous to think we're actually going to leave on loan. But the question was asked by his agent and it can be very disruptive to the players, but that's what we have to deal with. You know, hopefully by the end of January we'll have got some business done and we'll be a stronger squad for it. I'll take it, then you're happy with the progress Janik is making then as well because you look lively against United the other night. Well, I think he's way ahead of where he should be. I mean, I think that he shouldn't be starting games yet, but we need some pace in the side. So we're trying. We're trying to balance off that problem that we have that we've got to be able to do the transition from defending well to attacking. And then when attacking, we've got to be able to defend well. And that's what led us down against Manchester United because we were attacking. We attacked more. We had more shots on target against one of the top boys that we did against Liverpool or Chelsea. But then we lost 2-0. And for me, two boys, two goals that could have been avoided, two great finishes, but could really easily have been avoided up to that. So I still got, well, I haven't had no time to work on that. That's the problem. So until we get into the end of this month and next month, we won't be able to establish the basics better and improve on that department because we can't get on the coaching field and do the coaching that we want at the moment. Just with the squad that you've got and possible outgoings, I suppose, obviously Kevin Morales was linked with Olympiacos in the summer. What's likely to happen with him this month? Do you feel that you're looking to move him on? Depends on many factors. It depends on one, does Kevin want to go? Does he fancy the club that comes in for him? Do we agree with the fee that's being offered? Is it right for us? And then if it is for us, we ask him, is it right for him? And then that transfer may well occur. Many, many, many players in the January who knows. You never say never to what goes on in the January window when people are in a position where they want to either move players on or they want to strengthen the squad. I think that it's the hardest window to work in. But we're trying at the moment to cure one of the areas where we feel we have a problem and everybody's been aware of that problem even before I came. And hopefully this is a huge pressure on the lad. So I hope you can live with it. We all do. And we can't guarantee when you come from a road you can hit the ground running in January. You saw that, obviously, with Sandro as well. Seville have been linked with him. Has there been any contact, anything in that at all? There's been no bids for Sandro at this moment in time. There'll be no loans as far as my recommendation would be. I don't make the final decision, the decision is made by everybody. But I think that personally if anybody's going to leave, it would have to be the right fee for us. And we consider we could use that money somewhere else to strengthen our squad. And still the same for us. No contact will be of God to his future. I would expect there would be in the next or sometime in this window if there's going to be any interest from one of the big boys. Until then, though, available for selection, possibly? Is he getting close to that? Well, he's back in training full time.  The touch won't be quite as good as it used to be. The match fitness won't be quite as good as it used to be. And we'd have to, with like James and like Yannick, if he's the case, when he comes back in, like Sheamus, like Leighton, we'll have to suffer that for a while till we get them back up to speed. So I think we've had a stat where we've suffered more injuries than anybody else. You know, I think there's a... Unfortunately, we're the top of the league for the wrong reason, where too many players have missed too many games and we need to address that one. I can't address it just yet, but eventually I'll address it in terms of our prevention of injury policy. And how are you looking squad wise?  Yes, I think that from a selection point of view, we've probably got looking at what happened on the Manchester United game. Michael Keane had to have five or six stitches in his foot right on the top of the crown of his foot, which I think will be too dangerous to risk him. But everybody else that's in the squad, I think, is OK. And some rotation has been happening, sadly for Rossi, it hasn't produced the sort of points that we wanted, but we have to accept that in this period, if you don't rotate, you can't keep the same time side and then up. And for me, I'd kept the same side building up to Christmas, which was doing a great job in getting results and points. But unfortunately, it wouldn't last. And I saw that most at Bournemouth, like you mean, where, you know, the fatigue kicked in and we ended up losing in the final minutes because of that fatigue. And like I said, Manchester United was OK, but defending wasn't what I expected. And we are a team that, if you look at the statistics, if we are to win a game, we need a clean sheet. We can win a game without a clean sheet, but that would be rare. So what importance do you put on the FA Cup for the season? Well, I think now we're in the position, while the security of the Premier League is always the first priority. I think we're in the position where we go for it, and it's not a league game. There's no points available. It's win or bust. Simple as that. It's a cup game. It's different to the Premier League. It's less pressure. It's a glory game, and you go out and seek the glory and try and win it. If we're to get through, we have to beat Liverpool. Simple as that. So effectively, a draw is not good enough. Liverpool can set a new and beaten record against Everton. So does that come into the thinking as well? Really, you know, my responsibility is trying to get this team winning. And the players, even if they knew that statistic, won't really make any difference to their mentality to go out and try and win anyway. You don't need a statistic like that to want to go out and beat Liverpool. Well, I just wonder if there's that burden of history, again, in this fixture? What affects the players? I don't think so. I don't think the professional enough today to know what the size of the game is and make sure they try and deliver the best, and that's what we've got to try and do Friday night. On usual game to play Friday night. When you look at the difference between the two clubs, how would you sum it up? Obviously, Liverpool could go out and spend £75 million on Van Dijk, yet there's still that speculation that's maintained over Coutinho's future as well. Well, I've only just arrived, so my ambitions to build the club will, based on me, continue to get results that keeps me in the job. And when it keeps me in the job, then I can build the club as big as I think with the backing of everybody else and the dreams of what the owners have as big as anything it possibly wants to go to. So there's a number of places we will all work together on and all improve on as a club in total in every area, and if we do that, we can grow and get to where we want to go, but that can't happen in 12 months, 18 months, two years. It's got to be three, two, minimum, three, five-year plan to have a sustainability to be as big a club as you want to be when you've seen the looks of Tottenham. If you look at Tottenham's history, it took them 10 years to get to become an established, Premier League club that gets in the Champions League on a regular basis. So it's not an easy task, and it can be achieved, but for me, if I'm to achieve it, I have to continue to get results to stay in the job first and then help the club build what it wants to be. In terms of planning for this game, obviously the club has rotated quite a bit in recent times as well. Does that make it difficult in terms of planning, or do you pretty much know what to expect from it? With all of our injuries, like everybody suffers at this period of time, there's no need for all the coaches from abroad to say what we need, because we've been saying it for 15 years, and certainly I've been saying it. The good thing is they've probably got a bigger voice than I have, and they may listen to that, but we do need to shut this league down after the third round of the FA Cup. Is there any question about that now, and I think that we all know that players need to recover from the serious fatigue that they're suffering in this period, and from our point of view, rotation either gets you results and you keep going, and you say, well, he's good at rotating on, you're rotating on, you don't get results, always bad at it, you know what I mean? So that's what we have to face, but you know. There we go. I was just wondering about it. Is there any news on addressing a great, gay contract set in October, when everybody came in, that the deal was very close, have you heard any kind of talk? Surprise baffled me. I don't know nothing about that at all, so it's not my department's contracts anyway, so if there is a contract being discussed, then I'm sure we'll get the information to you if and when that contract has been negotiated. Sorry. You said earlier that you would maybe expect in contact regards Ross from some clubs in the next few days a week. Does that mean you'll resign to the fact that he will move this month? I think that we are, and I think the club was resigned before I got here that they were going to lose Ross, so I can't see that changing. Okay, thank you. Cheers.


Now, let's talk about these reports. As you said, you folks keep do a lot of reports, but let's talk about the state of the internet report, talk about the idea, the driver. One of the things that we are asked by our customers is, what are my peers doing that I'm not that's doing well? What's happening in my industry or what's happening in my region? And is it unique from others? And so, that's why a lot of our reports were able since we're an international company, we have customers across every time zone, we have customers in every industry. And so, what we wanna do is we wanna say, these are best practices, these are lessons learned, and this is a trend we're seeing. We're seeing a trend of, in this case, let's say we went and we looked at 90 different ransomware groups. We studied victims across different groups and we studied a time period of almost two years. It was 20 months total. And as we looked at all that, we wanna come back and make recommendations. How should you be thinking about protecting it? Should you shift resources based on how the threat is shifting their attacks? And so, those are kind of the things that, as we protect against the edge and internal and we have a threat hunting team on our segmentation. We have shadow hunting team on APIs. We have a security operation center for DDOS. So, we see so many lessons learned, we just wanted to share those.


Everybody forced here up in 2022. I'm at collegiate first off the line five box case random team style Results on screen can be found anytime a link sports cards calm, and I dropped you a link in chat, too Here we go. I'm not photo. I thought something to self-destruct I'm at Christmas like the opting flawless. Yeah There's some of the big ones right there I love stunning luminance over here Still haven't got that shirt in all right first up is a Joe Montana 34 of 35 this goes to the 49ers for longest tenure There we go 49ers Immaculate ink auto Ozzy Newsom 86 of 99 that goes browns, right? I don't think he played anywhere else Ozzy Newsom never doesn't go to Ravens Ozzy Sorry wrong Ozzy again. I keep doing that Yeah, only played for Cleveland. There you go Shane.  Okay, you go falcons on the drake london bullpatch auto Devonte price nine out of ten Devonte price is packers, correct I'm not sure I believe he is Devonte price bullpatch auto nine of ten It's a free agent currently, but he was definitely on a team. Let's see. Oh, he's colt colts actually colts Yeah colts are a thing of different player Colts on the Devonte price patch out of nine of ten I think he went to the same high school to Jake. Really? I think so. That's cool. Jake told me that Redemption Justin Ross premium patch rookie auto emerald going to the chiefs Chiefs on that one Unless i'm gonna double check. I think he's just been on their practice squad or like the injury list for like the whole season, right? Something like that Who was it? Justin Ross That sounds right. Yeah. Yeah, it doesn't say that he was never he was on ir. Yeah, so he's never anywhere else So chiefs Justin Ross And jersey card on the back is brian robinson jr. Commanders to 42. I guess that jesson ross Yeah, the emerald is the first off the line by us. That's the first off the line right there The emerald is the first off the line. It's like 13. I think I think it's trying to cool down. Let's start close out streams and stuff. I guess. I don't know Fox two. We've got erin rogers number 26 of 99 Packers for now Signature moves rookie auto cardinals tray mc bride 91 of 99 I'm sure the microphone's big enough. That's that's that's a lot You know, that's why I'm trying to quiet it. I don't know about our laptop over here Immaculate, uh, two of 10 breeze hall auto jets Breeze the beast latent sports cards.com So we have more breaks on monday and personals tomorrow and post that i'm gonna close out and reopen this They just need to restart the laptop You need to do that, but i'm just restarting a restream There you go jets nice breeze to 10 Prime patch rookie auto jack cone. I believe he is all he was colts. I'm gonna double check Uh, he's on the uh, san antonio bromp bramas, which is a xfl team. But anyway nfl Only colts Colts on the jack cone emerald first off the line prime patch rookie auto bryan robinson jr. 5 of 13 commanders You go commanders. Where was jack at colts? In jumbo jersey giants cave on tibido 50 of 99 I was watching breaks today. Even kenzie was sleeping her the hits Performance review, I just uh recommend Look, okay, y'all you want to sleep everything and he just yeah, he would exactly Fine i'll sleep everything for you rogers stall back 85 and 99 for the cowboys No, taylor haven't even gotten the game yet. It's just it's been delayed for with fedex for however long So it's a it's a touchy subject It's been no cal if like three days now, dude I'm Going to drive up there and go get it Jalen widemire team slogan signatures die cuts gold jalen widemire is bills. I believe Going to royals, did he? Uh, it says colts currently. Um Um Yeah colts on the jalen widemire. I know he was he was bills. Yeah, it is colts. He says sign with them in october Don't say anything else after that Colts jalen widemire. I look that up Wondell robinson signature moves 35 of 49 giants What benefit about uh watching the collegiate breaks is you get to know where everyone is Yeah, I think this might do it for me too taylor might do it for me too signature moves alec pierce colts 50 of 99 You've been playing though Star Wars breaks this week. Uh, not on this side. There might be someone late in gaming But you want to check late in gaming and you're gonna want to check late in sports cards I I would I wouldn't think so, but I don't know for sure. You just have to check the schedules There you go. And then late in gaming is right here Jets emerald prime patch rookie auto garret wilson Three out of 13 Um, that is a great hit offensive rookie of the year Yeah Edward p congrats congrats I'm excited to start it when everybody's done playing already And jumbo patch karsten strong 17 and 25. Is he still eagles? No, he has cardinals All right, so we're gonna check that donnie says cardinals. I I believe him. I just Because lane was making fun of me for not knowing who my quarterbacks were because I didn't know karson strong He's a free agent currently Oh, is it really? It's gonna go eagles. He was only with the cardinals for a week.  So Yeah eagles on the strong and you know what we pulled a taylor heineke auto earlier today So good But he's actually playing right, but it took him a few years It's true Got jonathan taylor 77 of 99 for the colts Uh, next allman actually be in the next few weeks or so Two of ten auto brian robinson jr. Commanders a lot of brian robinson for washington But I don't know for sure morris. We don't know for sure yet It won't be this week though. A lot of big releases this week Immaculate ink michael vick auto 31 of 49. I believe he goes falcons for longest tenure I know he Yeah, he's definitely It's kind of I know it's kind of close though. He was only with philly Friday 3 or Atlanta did he was the philly for five years really his little atlanta for six years Yeah, I told you man. It's it's closer than people think. No, I didn't realize. Yeah, it's atlanta, but it was only by one year That's crazy. It is kind of crazy Redemption cj stroud immaculate ink ruby There we go. That's a big one right there. That's random to everybody Good luck on this one Wow, we can magnate. Oh, wow the next one's crazy too. Oh Oh, man, I don't like to show this on stream we got a uh What kind of leaf is that I'm just getting buckeye leaf helmet sticker garret wilson six of nine for the jets That's cool. They should uh Legalize buckeye leaves in florida And it's six nine. Oh my god. Come on now You can magnus. Yeah Six of nine on the on the on the leaf helmet sticker And karsten strung prime patrick. He auto emerald nine of 13 going to the eagles the first off the line parallel What a troll park. Let's go magic bench orland. Oh magic bench orland. Oh magic bench We gotta keep it up It's left Staw back again for the cowboys 24 of 99 james cook on the introductions auto 39 to 49 for the uh for the bills bills shrey lance immaculate ink auto 15 to 25 49ers shrey mcbride three color patch auto there we go cardinals 14 to 25 some of those uh Colorado like flag uniforms essentially pretty cool Nice mcbride to 25 good rookie tight end there John mechie emerald rookie premium patch auto 13 to 13 They go texans very nice Everybody's a john mechie fan. That's for sure and quad jersey. We've got laurence lance fields and mac So is that from this year? Yeah, it's from this year. So jags niners There's pats pats victor r Niners no e g Where's the jags at ever piece? Yeah, it's a random. So you'd have to have over 50 to have it. So random here Right now And we'll recap. Oh, what was it uh jaguars Who's next lance? Bears Patriots I'll show it again. All right, we're ending now for the uh, laurence lance fields and mac You got jags niners bears and pats good luck Five and four nine times whoever's up top gets it one Two three four five six seven eight and nine Bears up top bears Bears beats What are you doing? What is going on? Oh, we've got a rear randomized to stroud donny That's a total don you'll move right there Stroud redemption another random this one's to everybody. I'll re-roll since we just Here we go. Good luck big time card there six and four ten times Whoever's up top gets the card So all 30 spots are here. So they have like the combos and stuff too ten times one two three four five six seven eight Nine and ten going to the Texans.  There's a giant's hit colts jalen watermeyer a team slogan signatures die cuts gold gary wilson jets Magnet one for your jets. We had karson strong michael vick It was falcons brian robinson jr. John met you the third tray mcbryde again tray lance james cook And the uh six out of nine buckeye helmet sticker gary wilson for the jets. Congrats on that That's the well with some gary wilson's and breeze hall at the ten We got more football coming up. Thanks everybody


Before we get into the video people, I am giving away 11,500 Apex coins every single month to be in the chance of winning them, make sure you are subbed to this channel, drop a like on the video and leave a comment down below, more details are in the video description. So what is going on guys, welcome back to the channel and I appreciate you being here. Today I bring you another Apex Legends video and today I've decided to change things up a little, you guys know what I do top 5s for various different games, but I am getting so many clips to submit now I literally can't keep on top of them, nor can I post all the best clips as I'll be posting like 20 top 5 videos a day, but what I can do is instead of posting 20 top 5 videos a day, I can do something which works better for me and better for your viewing pleasure, and that is make highlight videos of all the best clips submitted, while also doing maybe 1 or 2 top 5 videos a week or month, and that's what you are seeing today guys. So guys if you have any clips to submit, do so to the email just linked in the video description, if you guys do enjoy the video leaving a like really helps out, as always credit for every clip can be found in the video description also, but guys enjoy this highlights video, expect more to come and hopefully I will see you on that next one. Thank you guys for watching this video, I hope you guys enjoyed it, see you in the next one. Oh come on dude, they're one, they're one. I know, I'm not. One over here dude. I know, I can't do anything to set dude, I'm almost dead. Dude. I'm almost dead dude. You got this bruh. Dude. There he is.     You're correct dude, that's mine, that's mine coming in. Mine coming in. Got in front of me. One's melted. Melted. You got a third pilot? Oh my god. I'm the only one that died, I'm done dude. There's a supply drop right there, pockets.  They're up there. Yo pockets, what is this?  Big strats man. Big strats. Big strats.


the talk is going to be about elementary higher toposes having a natural number of please all right and so thank you to organizers for giving me a chance to speak and thank you all for being here so this is a talk on like an ongoing project on studying so toposteoretic and specific like elementary toposteoric phenomena in like the higher categorical setting and there's something I specifically like about this title because unlike many foundation stocks I give this one already has like a specific sentence like the title already tells you what the talk is going to be about there's an object and has a certain property so the talk is obviously not going to be very long it's 20-25 minutes and so I have to skip some of the details of the proofs that I would have liked to give and so if there's anything that's interesting to you then please feel free to reach out to me and can talk me whatever I would love to talk more about this but I also want to respect the time here of this conference I know it's like a long day all right so let's go into the toposteoretic part so this should be familiar more or less to most of you can like break down toposes in various ways and this conference has been a great example of like the different kind of perspectives on the topos but like so one way to think about toposes is like there's like a gluttonyck perspective that arose like in the study of like algebraic geometry in approval key setting and then there's like an elementary perspective so the elementary toposes which is like maybe more connected to like type theory other stuff and obviously connected but also have like the differences and so gluttonyck toposes are in some sense like very nice they have like lots and lots of amazing properties they're like locally cottage and close then particularly presentable and they have subalgae classifiers and so we can use them to do lots of math unfortunately elementary toposes are not as nice so they did lose some of these properties so in particular elementary topos is by default not locally presentable and we still have some limiting elements like only the finite ones we don't have infinite elements and we only preserve some of the like the like axioms that you have in Gero's theorem so for example um co-products are still disjoint but not all the properties survive and so the ones that I have bolded here are the ones that we take as a definition right so we want an elementary topos to be a category that has finite limits locally cottage and close and has subalgae classifier and so with any generalization the benefit of it is that we gain more examples usually which for example the category of finite sets is an elementary topos but it is certainly not good indeed because it does not have infinite co-limits and there are other kind of interesting examples such as filter products or realizability toposes that come up in kind of more foundations and logical frameworks but then the drawback of the more examples is that it's like expressive right so there's bunch of stuff that we cannot prove anyone and in some sense the way I think about it is there's like this intuitive sense of like something infinite and if something is infinite then it's harder to like construct it in the in an elementary setting whereas in good and equal possessive payoff we can use local presentability to get this and so one thing we want is something that intermediates so we want to strengthen the notion of elementary topos without going all the way like without actually going all the way to a good and equal which has local presentability and one way of doing that is by the notion of a natural number object which is an object we add to elementary topos to strengthen the theory so and because the talk is a lot about national object that may actually give you like definitions and there will be several and each one of them kind of hinges on focusing on certain aspects of natural numbers in the category of sets so one way I think about natural numbers is that okay it's an internet set which means it has like a an injective but not surjective self-injection which cannot happen for a finite set and so if you try to use that to form some kind of axiomatic definition you end up with a freight natural number object which is just an object along with the point and endomorphism so that we have these two following two column diagrams then another aspect of natural numbers is well induction so we know that natural numbers have like the inductive property from a piano axiom or we can make that into like some kind of axiomatic definition i.e. we have a natural numbers we have an object we have the point and we have this endomorphism which is like a successor map and now we say okay whenever we have a sub-object that is closed under zero and the successor then it's just the object itself which is precisely the category statement of induction if one is in there and if n implies n plus one then you're hoping and then the final version which is I think the one that most people might not seen before so it's the one that if you just google natural number object it's probably going to like show up first is the Navier natural number object and that just says okay it's an object along with a point and endomorphism that is initial with that data so whenever I have another object with some endomorphism u and a point b then I get a unique map and that unique map just defines a kind of recursive function which just applies u n times 3d object and so I have now three notions and all of these three capture a certain aspect of n and fortunately all of these three coincide inside the elementary topos and that is why we usually don't actually distinguish so if you're just reading like a paper they will not distinguish between these notions they will just say you have an elementary topos plus natural number object because whichever notion you pick is equivalent to the other two so that's what we usually don't really matter don't care about this distinction and now let me come back to my original promise so as soon as you have a natural number object in our elementary topos we can now prove new results that we couldn't do before and there are many examples of that I want to focus on one here as soon as our natural object we get free monoids and the equation for the free monoid is explicitly given by by this formula here where like n1 is called the universal finite corner and it's just constructed out of the product of the natural number object itself and so notice this here the existence of the natural number object is necessary right it's like actually it's the vision and necessary but let's not focus on that it is actually necessary in fact if it doesn't exist this would fail and an easy example is finite sets inside the category of finite sets I don't have free monoids even a free monoid on the point or something doesn't exist because it wants to be n itself and n is not a finite set so assuming an natural number object is necessary to get such results okay so before I move on to the more general high category stuff let me summarize what we did so we can define an elementary topos we want to prove some stuff like we want to construct free monoids or other similar stuff for that we have several notions of natural number object that all coincide assuming the existence of such an object we can prove that we have free monoids for example but if you don't assume it we cannot so it's it's independent of the axioms of the elementary topos all right so this was in something like a because of you so depending on who you are just might or might not have been very familiar to you so now the next step is I want to move into the world of like phoenix category stuff and once you okay how does this how does this picture look like in phoenix categories do we have do we have natural number objects or not and what do they imply and because I'm now using the the word in phoenix category then I should like say something about what I mean by that uh and so let me just give you like a like a one slide summary so if if you're already familiar or if you did attend the first lecture of the the Hayatopos school last week by by Charles Resk then you should probably be already familiar with the phoenix categories some extent if not then just take this one I will just give this one slide summary so for the purpose of this talk and in phoenix categories just like there's a collection of objects or x y whatever and for any two objects there's a mapping space and that is really important I really want to fix that I have a space of maps from x to y where space can either mean topological space or con complex that doesn't matter and then that those spaces should give me some sort of weak enrichment meaning that whenever I have like two maps so that the codomain domain are appropriate I have some kind of composition but the composition only chosen up to some contractable ambiguity and then that composition is also associative but again associated only after some homotopies and and so on and all the classical categorical terms all the classical definitions of category theory limits adjunctions, Cartesian closure, adjoint functor theorem, presentability all all of those notions appropriately generalized to this infini category setting and that can be done either using like quasi categories or an infinity cosmoi or complete single spaces but for the purpose of the talk this is all I need like I will use some terms like high category terms but just believe that they exist in a primary category so now but just one slide we all have mastered infini category theory so we can move on to what infini categories you want to focus on so the main goal is to apply this to an elementary infini intervals and as the title suggests and so how you want to define elementary infini intervals we will certainly want the conditions that you can already find in an elementary intervals i.e. finite limits and co-limits, local collision close and sub-object classifier and then you additionally want some kind of object classifier universe the the issue is the exact notion of like universe you want is a little bit up to debate so there's like you can kind of decide on how strict the universe should be or how functorial how close it is on a certain construction and the claim I want to make is that it really doesn't matter for the purposes of this talk whichever notion of universe you think of the result I claim holds so in order to kind of make this stronger claim I would just not use the universe and we use some weaker condition that just follows from any notion of universe that is out so instead of using universe I just use finite descent and a finite descent is is like a is like the descent condition which was introduced by also by Charles Reckon in the second lecture and it's about like the interaction between pullbacks and pushouts the finite I'm using here is only because well we don't have infinite co-limits like the definition here is finite co-limits and so for finite co-limits we have the descent and it's the same condition follows from any notion of universe that is out there and that also suffices for the result I want to state so I don't even need to go to the strength of the whole universe I just need the descent condition that's enough and before I move on let me note that as soon as you have an infinity category which has these properties I can take the subcategory of zero truncated objects and that subcategory will be elementary topos because it will still have finite limits, peak localization closed and have a subcategory as I can make a nice bonus and notice this definition does have like lots of examples the most difficult example is the infinity of spaces but then you also get pre-sheeps you also get all higher toposes in the sense of like lyrical risk or something but in addition to those you also get like not presentable examples such as filter product between toposes so there's like a bunch of examples of this condition I stated okay so now we have a certain infinity category we want to study what do we want to study about it natural number objects so what is an appropriate notion of a natural number object in the infinity setting and because I've already defined them in the one setting I'm just like a bullet summary how they would look like in the infinity setting and for freight and piano the generalization is pretty straightforward right so the freight natural number object is just you have two certain diagrams of co-limit diagrams and I can just make exactly the state statement I still have exactly the same diagram and just say put those on a co-limit diagrams but inside some infinity category and for a piano it's the same I mean I just want a certain sub-object that is closed under os to be the same as the original object again you can just phrase that and the fact that you know it's a kind of high category setting has no influence on on the definition. Lavian natural number objects are not like that so for Lavian natural number object now I need to actually strengthen the condition I need to say that whenever I have a triple of an object x and some endomorphism u with a map on the point b then I have a space of maps from the natural number object to x and I want that space to be contractable so that is very different from the more classical categorical setting where we have a set of morphisms and we want that to be unique so there's no uniqueness anymore there's only contractability so it's like a much weaker form um but yeah so piano and freight are pretty much similar I would say so we are defined our infinity categories and the natural number object so I'm finding a position where I can state the the main result and the main result is that if e satisfies the condition I stated before so it's looking like a region closed finality by complete and satisfies the standard has a salvage classifier then all three notions of natural object coincide and exist and so that last point is very important but unlike the classical setting I can prove the existence from these accidents and also that they coincide okay so let me give like a summary of the proof so in an ideal world I would like to spend the next like I guess 25 minutes on this proof but I don't have 25 minutes so that means I have the next five minutes on this proof and the fun fact about the proof is that it breaks down nicely into like three almost completely independent steps and so the first step is motivated by results from I guess any introduction to algebraic topology course and in school right so so if you go like any intro to algebraic topology course the first result they will probably teach you is that pi one of s one is that and so I want to categorify that result I'm going to take a more general view of that and so how can I do that I just think of okay that is the free group on one generator so really what I'm saying is that pi one of s one is the free group on one generator and then even more generally okay if I don't want really pi one of s one I just want the loop loops on s one so pi one is like the the homotopy class of loops but I can just ignore the homotopy part and just say I want loops on s one and then loops of s one can be expressed in any category with finite limits and columns so that is not familiar then you just have to test beyond that that I can always write a circle as a certain column and I can write the loops on the circle as a certain limit as a certain pullback and now the statement then becomes if you have a category with like these settings as a matter of fact all I need is finite limits finite co-limits and descent then I can form loops on s one and loops on s one will be the free group on one generator it will satisfy the universal property here and so that's like the first step and it's motivated clearly by like background knowledge and algebraic topology and then the next step is if you recall as soon as I have this e I have an underlying elementary topos and now you just use kind of very classic literature on this that proves that it's whenever you have elementary topos and you have the free group on one generator you have a natural number on you can just construct it right so you use this object classifier and you carve out the n out of the z basically so that's like a very kind of summary step of the pool right that when you have z and it's like sub-element like a sub-object of that and just get it out of and so after step two then you prove that the underlying elementary topos has a natural number object what you want then to prove is that okay this underlying elementary to this national object in the underlying elementary topos is also not an object in the whole infinity category so there's like a lifting kind of argument and that doesn't follow obviously at first the thing is as I explained so the freight and piano conditions are very similar so as soon as they're holding the underlying one category they will hold also in the infinity category there's not that much more effort like you have a certain column of diagram and you just want to check that the same column of diagram is also a column of diagram in the infinity category so there's there's some work but it's not too bad the the main challenge of the proof and something like the hardest step is to show that this if you assume that's a love your natural number object in the underlying elementary topos then it is also love your natural number object in the infinity category and that is quite challenging because now you're making it like you're strengthening the claim quite a lot you're going from a statement about uniqueness of maps to contractability of some mapping spaces and that's just not obvious and that is also the part where unfortunately you cannot just use elementary topos methods because elementary topos don't really work well with like homotopy stuff and so the trick then is is to like use the wisdom of homotopy technologists because they they have been digging a lot about how logic and homotopy theory cannot be tracked so it's not actually homotopy type of per state but just look at the work and like try to get some ideas from there and adapt it to like the categorical setting and using that you can then find an appropriate proof that if you have a love your natural object in the underlying elementary topos it will also be a love your natural object for the full thing all right cool i still have a lot of time so i want to move on to the who cares so now that you have a natural object we can use it to do what's the point of it and let me focus on like three examples and one future direction so the first fun fact is so let me give you some history um the the good nick topos is it can be lifted to a higher topos like an infinity topos and it is in fact that whenever you pick any good nick topos you can find an infinity topos which has that good nick topos as a zero truncation and that infinity topos is often called the enveloping infinity topos so this was already known 10 15 years ago and one thing one might have expected is that something similar holds in the elementary setting so whenever i have an elementary topos i can lift it to some elementary infinity topos in a similar way but this result implies that this cannot be true it's just impossible right so for example the category of finite sets because it doesn't have a natural number object can never be the zero truncated objects of any elementary infinity just not going to happen and in particular this also implies that the category of finite spaces cannot be elementary infinity topos because it doesn't have natural number object and so if you want some kind of like finitary version of spaces that has that is still like a topos you have to like aim bigger so you need to focus on like couple small spaces for some couple that's like satisfy some properties and this has been studied by Loma and Aqua and make a recent paper and like this is like a like a nice contrast and i don't know it is i found it surprising and another like fun example and fun implication of the existence of NNOs is that we can like study um infinite columnets and so there's two aspects that we can use NNOs to study them one is that okay when we have a natural number object we can kind of reduce and determine quite straightforwardly when our infinity category does have infinite columnets and kind of reduces down to having a couple of the terminal objects so it's like a useful fact on studying infinite columnets in this elementary infinity topos setting i think more importantly it allows us to like define so-called internal sequential limits and like unfortunately i don't have enough time to like break down what they are but we can define like internal sequences inside our infinity categories and then define like the internal sequential limit of that internal sequence as a certain co-equalizer which then exists that's nice and like using the internal sequential limit you can actually construct negative one truncations um this is again one of those results that in the ingredient infinity holds by default and that was also part of the lectures of the summer school i think that lecture number three um but and there you just use like local presentability to use it but here you don't have local presentability so you use like NNOs to kind of circumvent that and get around decision and this should remind us of like a similar result in the elementary setting of epi-mono factorization right so one of the classical results of elementary topos is that you can always factor every map into an epi followed by mono and so this is a straightforward generalization of that like negative one truncation is precisely like epi-mono factorization of the map so we cannot just do it without the NNO but with an NNO we can get the same result um as a last step so one key example of the application of the natural number object in the elementary topo setting was the existence of free monoids so one natural question that would be okay you have to do the existence of a natural number object so can you now use it to do the existence of free like monoids in your infinity category setting and the answer is kind of a little disappointing so i i guess the answer is i don't know i don't know yet at least and not only that i don't know but it turns out this question is kind of quite challenging and so let me end this talk by giving you a sense of what the challenge here is when you're trying to like approach this question um if unlike the the one category setting in order to define monoids in some kind of infinity category you use you need to use some kind of like like algebra gadget like so operas or the view theories or something that is because you need to have operations for all possible levels but then ironically it can operate itself the definition of an upright and sometimes depends on the natural numbers so you have an operation for every n and so now the natural question you can ask yourself is should you use the natural number object that exists inside the topos or should you use the external one that exists in spaces and now you might want to okay what's the difference but it turns out there is a difference like in in certain elementary toposes there are so-called non-standard natural number objects which have additional uh non-standard natural numbers and so the definition would be different like you would get a new and different definition of an upright which has like additional operations and yeah that would just not coincide today the first question that could you ask here is when you say monoid what do you mean like do you mean the monoid where you have just the standard operations or where you also have the additional ones and so like not only like is it challenging to construct it it's even challenging to define what the appropriate notion should be and this challenge doesn't arise in the in like in the one setting right so when you try to define a monoid in certain elementary topos you never face the challenge regardless of what the natural number object is you never face the challenge of having to define additional operations because it always suffices to like define operations for level like zero one and two you never need the higher levels because your associativity everything will be strict this is kind of the challenge that only arises when you try to study um homotopical algebra like so higher algebra setting so there seems to be like this um surprising interaction between non-standard natural number odds and higher algebra that is clearly not well understood and definitely merits for the study so with this um I end my talk thank you very much