From: (John Baez) Newsgroups: sci.math.research Subject: This Week's Finds in Mathematical Physics (Week 115) Date: 1 Feb 1998 22:18:36 -0800 This Week's Finds in Mathematical Physics - Week 115 John Baez These days I've been trying to learn more homotopy theory. James Dolan got me interested in it by explaining how it offers many important clues to n-category theory. Ever since, we've been trying to understand what the homotopy theorists have been up to for the last few decades. Since trying to explain something is often the best way to learn it, I'll talk about this stuff for several Weeks to come. Before plunging in, though, I'd like mention yet another novel by Greg Egan: 1) Greg Egan, Diaspora, Orion Books, 1997. The main character of this book, Yatima, is a piece of software... and a mathematician. The tale begins in 2975 with ver birth as an "orphan", a citizen of the polis born of no parents, its mind seed chosen randomly by the conceptory. Yatima learns mathematics in a virtual landscape called the Truth Mines. To quote (with a few small modifications): The luminous object buried in the cavern floor broadcast the definition of a topological space: a set of points, grouped into `open subsets' which specified how the points were connected to one another - without appealing to notions like `distance' or `dimension'. Short of a raw set with no structure at all, this was about as basic as you could get: the common ancestor of virtually every entity worth of the name `space', however exotic. A single tunnel led into the cavern, providing the link to the necessary prior concepts, and half a dozen tunnels led out, slanting gently `down' into the bedrock, pursuing various implications of the definition. Suppose T is a topological space... then what follows? These routes were paved with small gemstones, each one broadcasting an intermediate result on the way to a theorem. Every tunnel in the Mines was built from the steps of a watertight proof; every theorem, however deeply buried, could be traced back to every one of its assumptions. And to pin down exactly what was meant by a `proof', every field of mathematics used its own collection of formal systems: sets of axioms, definitions, and rules of deduction, along with the specialised vocabulary needed to state theorems and conjectures precisely. [....] The library was fully of the ways past miners had fleshed out the theorems, and Yatima could have had those details grafted in alongside the raw data, granting ver the archived understanding of thousands of Konishi citizens who'd travelled this route before. The right mind-grafts would have enabled ver effortlessly to catch up with all the living miners who were pushing the coal face ever deeper in their own inspired directions... at the cost of making ver, mathematically speaking, little more than a patchwork clone of them, capable only of following in their shadow. If ve ever wanted to be a miner in vis own right - making and testing vis own conjectures at the coal face, like Gauss and Euler, Riemann and Levi-Civita, deRham and Cartan, Radiya and Blanca - then Yatima knew there were no shortcuts, no alternatives to exploring the Mines first hand. Ve couldn't hope to strike out in a fresh direction, a route no one had ever chosen before, without a new take on the old results. Only once ve'd constructed vis own map of the Mines - idiosyncratically crumpled and stained, adorned and annotated like no one else's - could ve begin to guess where the next rich vein of undiscovered truths lay buried. The tale ends in a universe 267,904,176,383,054 duality transformations away from ours, at the end of a long quest. What does Yatima do then? Keep studying math! "It would be a long, hard journey to the coal face, but this time there would be no distractions." I won't give away any more of the plot. Suffice it to say that this is hard science fiction - readers in search of carefully drawn characters may be disappointed, but those who enjoy virtual reality, wormholes, and philosophy should have a rollicking good ride. I must admit to being biased in its favor, since it refers to a textbook I wrote. A science fiction writer who actually knows the Gauss-Bonnet theorem! We should be very grateful. Okay, enough fun --- it's time for homotopy theory. Actually homotopy theory is *tremendously* fun, but it takes quite a bit of persistence to come anywhere close to the coal face. The original problems motivating the subject are easy to state. Let's call a topological space simply a "space", and call a continuous function between these simply a "map". Two maps f,g: X -> Y are "homotopic" if one can be continuously deformed to the other, or in other words, if there is a "homotopy" between them: a continuous function F: [0,1] x X -> Y with F(0,x) = f(x) and F(1,x) = g(x). Also, two spaces X and Y are "homotopy equivalent" if there are functions f: X -> Y and g: Y -> X for which fg and gf are homotopic to the identity. Thus, for example, a circle, an annulus, and a solid torus are all homotopy equivalent. Homotopy theorists want to classify spaces up to homotopy equivalence. And given two spaces X and Y, they want to understand the set [X,Y] of homotopy classes of maps from X to Y. However, these are very hard problems! To solve them, one needs high-powered machinery. There are two roughly two sides to homotopy theory: building machines, and using them to do computations. Of course these are fundamentally inseparable, but people usually tend to prefer either one or the other activity. Since I am a mathematical physicist, always on the lookout for more tools for my own work, I'm more interested in the nice shiny machines homotopy theorists have built than in the terrifying uses to which they are put. What follows will strongly reflect this bias: I'll concentrate on a bunch of elegant concepts lying on the interface between homotopy theory and category theory. This realm could be called "homotopical algebra". Ideas from this realm can be applied, not only to topology, but to many other realms. Indeed, two of its most famous practitioners, James Stasheff and Graeme Segal, have spent the last decade or so using it in string theory! I'll eventually try to say a bit about how that works, too. Okay.... now I'll start listing concepts and tools, starting with the more fundamental ones and then working my way up. This will probably only make sense if you've got plenty of that commodity known as "mathematical sophistication". So put on some Coltrane, make yourself a cafe macchiato, lean back, and read on. If at any point you feel a certain lack of sophistication, you might want to reread "The Tale of n-Categories", starting with "week73", where a bunch of the basic terms are defined. A. Presheaf Categories. Given a category C, a "presheaf" on C is a contravariant functor F: C -> Sets. The original example of this is where C is the category whose objects are open subsets of a topological space X, with a single morphism f: U -> V whenever the open set U is contained in the open set V. For example, there is the presheaf of continuous real-valued functions, for which F(U) is the set of all continuous real functions on U, and for any inclusion f: U -> V, F(f): F(V) -> F(U) is the "restriction" map which assigns to any continuous function on V its restriction to U. This is a great way of studying functions in all neighborhoods of X at once. However, I'm bringing up this subject for a different reason, related to a different kind of example. Suppose that C is a category whose objects are "shapes" of some kind, with morphisms f: x -> y corresponding to ways the shape x can be included as a "piece" of the shape y. Then a presheaf on C can be thought of as a geometrical structure built by gluing together these shapes along their common pieces. For example, suppose we want to describe directed graphs as presheaves. A directed graph is a bunch of vertices and edges, where the edges have a direction specified. Since they are made of two "shapes", the vertex and the edge, we'll cook up a little category C with two object, V and E. There are two ways a vertex can be included as a piece of an edge, either as its "source" or its "target". Our category C, therefore, has two morphisms, S: V -> E and T: V -> E. These are the only morphisms except for identity morphisms --- which correspond to how the edge is part of itself, and the vertex is part of itself! Omitting identity morphisms, our little category C looks like this: S --------> V E --------> T Now let's work out what a presheaf on C is. It's a contravariant functor F: C -> Set. What does this amount to? Well, it amounts to a set F(V) called the "set of vertices", a set F(E) called the "set of edges", a function F(S): F(E) -> F(V) assigning to each edge its source, and a function F(T): F(E) -> F(V) assigning to each edge its target. That's just a directed graph! Note the role played by contravariance here: if a little shape V is included as a piece of a big shape E, our category gets a morphism S: V -> E, and then in our presheaf we get a function F(S): F(E) -> F(V) going the *other way*, which describes how each big shape has a bunch of little shapes as pieces. Given any category C there is actually a *category* of presheaves on C. Given presheaves F,G: C -> Sets, a morphism M from F to G is just a natural transformation M: F => G. This is beautifully efficient way of saying quite a lot. For example, if C is the little category described above, so that F and G are directed graphs, a natural transformation M: F => G is the same as: a map M(V) sending each vertex of the graph F to a vertex of the graph G, and a map M(E) sending each edge of the graph F to a edge of the graph G, such that M(V) of the source of any edge e of F equals the source of M(E) of e, and M(V) of the target of any edge e of F equals the target of M(E) of e. Whew! Easier just to say M is a natural transformation between functors! For more on presheaves, try: 2) Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer-Verlag, New York, 1992. B. The Category of Simplices. This is a very important example of a category whose objects are shapes --- namely, simplices --- and whose morphisms correspond to the ways one shape is a piece of another. The objects of Delta are called 1, 2, 3, etc., corresponding to the simplex with 1 vertex (the point), the simplex with 2 vertices (the interval), the simplex with 3 vertices (the triangle), and so on. There are a bunch of ways for an lower-dimensional simplex to be a face of a higher- dimensional simplex, which give morphisms in Delta. More subtly, there are also a bunch of ways to map a higher-dimensional simplex down into a lower-dimensional simplex, called "degeneracies". For example, we can map a tetrahedron down into a triangle in a way that carries the vertices {0,1,2,3} of the tetrahedron into the vertices {0,1,2} of the triangle as follows: 0 -> 0 1 -> 0 2 -> 1 3 -> 2 These degeneracies also give morphisms in Delta. We could list all the morphisms and the rules for composing them explicitly, but there is a much slicker way to describe them. Let's use the old trick of thinking of the natural number n as being the totally ordered n-element set {0,1,2,...,n-1} of all natural numbers less than n. Thus for example we think of the object 4 in Delta, the tetrahedron, as the totally ordered set {0,1,2,3}. These correspond to the 4 vertices of the tetrahedron. Then the morphisms in Delta are just all order-preserving maps between these totally ordered sets. So for example there is a morphism f: {0,1,2,3} -> {0,1,2} given by the order-preserving map with f(0) = 0 f(1) = 0 f(2) = 1 f(3) = 2 The rule for composing morphisms is obvious: just compose the maps! Slick, eh? We can be slicker if we are willing to work with a category *equivalent* to Delta (in the technical sense described in "week76"), namely, the category of *all* nonempty totally ordered sets, with order-preserving maps as morphisms. This has a lot more objects than just {0}, {0,1}, {0,1,2}, etc., but all of its objects are isomorphic to one of these. In category theory, equivalent categories are the same for all practical purposes --- so we brazenly call this category Delta, too. If we do so, we have following *incredibly* slick description of the category of simplices: it's just the category of finite nonempty totally ordered sets! If you are a true mathematician, you will wonder "why not use the empty set, too?" Generally it's bad to leave out the empty set. It may seem like "nothing", but "nothing" is usually very important. Here it corresponds to the "empty simplex", with no vertices! Topologists often leave this one out, but sometimes regret it later and put it back in (the buzzword is "augmentation"). True category theorists, like Mac Lane, never leave it out. They define Delta to be the category of *all* nonempty totally ordered sets. For a beautiful introduction to this approach, try: 3) Saunders Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1988. C. Simplicial sets. Now we put together the previous two ideas: a "simplicial set" is a presheaf on the category of simplices! In other words, it's a contravariant functor F: Delta -> Set. Geometrically, it's basically just a bunch of simplices stuck together along their faces in an arbitrary way. We can think of it as a kind of purely combinatorial version of a "space". That's one reason simplicial sets are so popular in topology: they let us study spaces in a truly elegant algebraic context. We can define all the things topologists love --- homology groups, homotopy groups (see "week102"), and so on --- while never soiling our hands with open sets, continuous functions and the like. To see how it's done, try: 4) J. Peter May, Simplicial Objects in Algebraic Topology, Van Nostrand, Princeton, 1968. Of course, not everyone prefers the austere joys of algebra to the earthy pleasures of geometry. Algebraic topologists thrill to categories, functors and natural transformations, while geometric topologists like drawing pictures of hideously deformed multi-holed doughnuts in 4 dimensional space. It's all a matter of taste. Personally, I like both! D. Simplicial objects. We can generalize the heck out of the notion of "simplicial set" by replacing the category Set with any other category C. A "simplical object in C" is defined to be a contravariant functor F: Delta -> C. There's a category whose objects are such functors and whose morphisms are natural transformations between them. So, for example, a "simplicial abelian group" is a simplicial object in the category of abelian groups. Just as we may associate to any set X the free abelian group on X, we may associate to any simplicial set X the free simplicial abelian group on X. In fact, it's more than analogy: the latter construction is a spinoff of the former! There is a functor L: Set -> Ab assigning to any set the free abelian group on that set (see "week77"). Given a simplicial set X: Delta -> Set we may compose with L to obtain a simplicial abelian group XL: Delta -> Ab (where I'm writing composition in the funny order that I like to use). This is the free simplicial abelian group on the simplicial set X! Later I'll talk about how to compute the homology groups of a simplicial abelian group. Combined with the above trick, this will give a very elegant way to define the homology groups of a simplicial set. Homology groups are a very popular sort of invariant in algebraic topology; we will get them with an absolute minimum of sweat. Just as a good firework show ends with lots of explosions going off simultaneously, leaving the audience stunned, deafened, and content, I should end with a blast of abstraction, just for the hell of it. Those of you who remember my discussion of "theories" in "week53" can easily check that there is a category called the "theory of abelian groups". This allows us to define an "abelian group object" in any category with finite limits. In particular, since the category of simplicial sets has finite limits (any presheaf category has all limits), we can define an abelian group object in the category of simplicial sets. And now for a very pretty result: abelian group objects in the category of simplicial sets are the same as simplicial abelian groups! In other words, an abstract "abelian group" living in the world of simplicial sets is the same as an abstract "simplicial set" living in the world of abelian groups. I'm very fond of this kind of "commutativity of abstraction". Finally, I should emphasize that all of this stuff was first explained to me by James Dolan. I just want to make these explanations available to everyone. [deletia. The picks up with the following post. -- djr] ============================================================================== From: (John Baez) Newsgroups: sci.math.research Subject: This Week's Finds in Mathematical Physics (Week 116) Date: 7 Feb 1998 21:42:11 -0800 [deletia -- djr] Now let me continue the tour of homotopy theory I began last week. I was talking about simplices. Simplices are amphibious creatures, easily capable of living in two different worlds. On the one hand, we can think of them as topological spaces, and on the other hand, as purely algebraic gadgets: objects in the category of finite totally ordered sets, which we call Delta. This gives simplices a special role as a bridge between topology and algebra. This week I'll begin describing how this works. Next time we'll get into some of the cool spinoffs. I'll keep up the format of listing tools one by one: D. Geometric realization. In "week115" I talked about simplicial sets. A simplicial set is a presheaf on the category Delta. Intuitively, it's a purely combinatorial way of describing a bunch of abstract simplices glued together along their faces. We want a process that turns such things into actual topological spaces, and also a process that turns topological spaces back into simplicial sets. Let's start with the first one. Given a simplicial set X, we can form a space |X| called the "geometric realization" of X by gluing spaces shaped like simplices together in the pattern given by X. Given a morphism between simplicial sets there's an obvious continuous map between their geometric realizations, so geometric realization is actually a functor | |: SimpSet -> Top from the category of simplicial sets, SimpSet, to the category of topological space, Top. It's straightforward to fill in the details. But if we want to be slick, we can define geometric realization using the magic of adjoint functors --- see below. E. Singular Simplicial Set. The basic idea here is that given a topological space X, its "singular simplicial set" Sing(X) consists of all possible ways of mapping simplices into X. This gives a functor Sing: Top -> SimpSet. We make this precise as follows. By thinking of simplices as spaces in the obvious way, we can associate a space to any object of Delta, and also a continuous map to any morphism in Delta. Thus there's a functor i: Delta -> Top. For any space X we define Sing(X): Delta -> Set by Sing(X)(-) = hom(i(-),X) where the blank slot indicates how Sing(X) is waiting to eat a simplex and spit out the set of all ways of mapping it --- thought of as a space! --- into the space X. The blank slot also indicates how Sing(X) is waiting to eat a *morphism* between simplices and spit out a *function* between sets. Having said what Sing does to *spaces*, what does it do to *maps*? The same formula works: for any map f: X -> Y between topological spaces, we define Sing(f)(-) = hom(i(-),f). It may take some headscratching to understand this, but if you work it out, you'll see it works out fine. If you feel like you are drowning under a tidal wave of objects, morphisms, categories, and functors, don't worry! Medical research has determined that people actually grown new neurons when learning category theory. In fact, even though it might not seem like it, I'm being incredibly pedagogical and nurturing. If I were really trying to show off, I would have compressed the last couple of paragraphs into the following one line: Sing(--)(-) = hom(i(-),--). where Sing becomes a functor using the fact that for any category C there's a functor hom: C x C -> Set. Or I could have said this: form the composite i x 1 hom Delta x Top ------> Top x Top -----> Set and dualize this to obtain Sing: Top -> SimpSet. These are all different ways of saying the same thing. Forming the singular simplical set of a space is not really an "inverse" to geometric realization, since if we take a simplicial set X, form its geometric realization, and then form the singular simplicial set of that, we get something much bigger than X. However, if you think about it, there's an obvious map from Sing(|X|) down to X. Similarly, if we start with a topological space X, there's an obvious map from X into |Sing(X)|. What this means is that Sing is the left adjoint of | |, or in other words, | | is the right adjoint of Sing. Thus if we want to be slick, we can just *define* geometric realization to be the right adjoint of Sing. (See "week77"-"week79" for an exposition of adjoint functors.) F. Chain Complexes. Now gird yourself for some utterly unmotivated definitions! If you've taken a basic course in algebraic topology, you have probably learned about chain complexes already, and if you haven't, you probably aren't reading this anymore - so I'll just plunge in. A "chain complex" C is a sequence of abelian groups and "boundary" homomorphisms like this: d_1 d_2 d_3 C_0 <--- C_1 <--- C_2 <--- C_3 <--- ... satisfying the magic equation d_i d_{i+1} x = 0 This equation says that the image of d_{i+1} is contained in the kernel of d_i, so we may define the "homology groups" to be the quotients H_i(C) = ker(d_i) / im(d_{i+1}) The study of this stuff is called "homological algebra". You can read about it in such magisterial tomes as: 2) Henri Cartan and Samuel Eilenberg, Homological Algebra, Princeton University Press, 1956. or 3) Saunders Mac Lane, Homology, Springer-Verlag, Berlin, 1995. But it you want something a bit more user-friendly, try: 4) Joseph J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979. The main reason chain complexes are interesting is that they are similar to topological spaces, but simpler. In "singular homology theory", we use a certain functor to convert topological spaces into chain complexes, thus reducing topology problems to simpler algebra problems. This is usually one of the first things people study when they study algebraic topology. In sections G. and H. below, I'll remind you how this goes. Though singular homology is very useful, not everybody gets around to learning the deep reason why! In fact, chain complexes are equivalent to a certain especially simple class of topological spaces, called "products of Eilenberg-MacLane spaces of abelian groups". In such spaces, topological phenomena in different dimensions interact in a particularly trivial way. Singular homology thus amounts to neglecting the subtler interactions between topology in different dimensions. This is what makes it so easy to work with --- yet ultimately so limited. Before I keep rambling on I should describe the category of chain complexes, which I'll call Chain. The objects are just chain complexes, and given two of them, say C and C', a morphism f: C -> C' is a sequence of group homomorphisms f_i : C_i -> C'_i making the following big diagram commute: d_1 d_2 d_3 C_0 <--- C_1 <--- C_2 <--- C_3 <--- ... | | | | f_0| f_1| f_2| f_3| | | | | V V V V C'_0 <-- C'_1 <-- C'_2 <-- C'_3 <--- ... d'_1 d'_2 d'_3 The reason Chain gets to be so much like the category Top of topological spaces is that we can define homotopies between morphisms of chain complexes by copying the definition of homotopies between continuous maps. First, there is a chain complex called I that's analogous to the unit interval. It looks like this: d_1 d_2 d_3 d_4 Z+Z <---- Z <---- 0 <----- 0 <----- ... The only nonzero boundary homomorphism is d_1, which is given by d_1(x) = (x,-x). (Why? We take I_1 = Z and I_0 = Z+Z because the interval is built out of one 1-dimensional thing, namely itself, and two 0-dimensional things, namely its endpoints. We define d_1 the way we do since the boundary of an oriented interval consists of two points: its initial endpoint, which is positively oriented, and its final endpoint, which is negatively oriented. This remark is bound to be obscure to anyone who hasn't already mastered the mystical analogies between algebra and topology that underlie homology theory!) There is a way to define a "tensor product" C x C' of chain complexes C and C', which is analogous to the product of topological spaces. And there are morphisms i,j : C -> I x C analogous to the two maps from a space into its product with the unit interval: i,j : X -> [0,1] x X i(x) = (0,x), j(x) = (1,x) Using these analogies we can define a "chain homotopy" between chain complex morphisms f,g: C -> C' in a way that's completely analogous to a homotopy between maps. Namely, it's a morphism F: I x C -> C' for which the composite i F C ----> I x C ----> C' equals f, and the composite j F C ----> I x C ----> C' equals g. Here we are using the basic principle of category theory: when you've got a good idea, write it out using commutative diagrams and then generalize the bejeezus out of it! The nice thing about all this is that a morphism of chain complexes f: C -> C' gives rise to homomorphisms of homology groups, H_n(f): H_n(C) -> H_n(C'). In fact, we've got a functor H_n: Chain -> Ab. And even better, if f: C -> C' and g: C -> C' are chain homotopic, then H_n(f) and H_n(g) are equal. So we say: "homology is homotopy-invariant". G. The Chain Complex of a Simplicial Abelian Group. Now let me explain a cool way of getting chain complexes, which goes a long way towards explaining why they're important. Recall from item D. in "week115" that a simplicial abelian group is a contravariant functor C: Delta -> Ab. In particular, it gives us an abelian group C_n for each object n of Delta, and also "face" homomorphisms partial_0, ...., partial_{n-1} : C_n -> C_{n-1} coming from all the ways the simplex with (n-1) vertices can be a face of the simplex with n vertices. We can thus can make C into a chain complex by defining d_n: C_n -> C_{n-1} as follows: d_n = sum (-1)^i partial_i The thing to check is that d_n d_{n+1} x = 0 for all x in C_{n+1}. The alternating signs make everything cancel out! In the immortal words of the physicist John Wheeler, "the boundary of a boundary is zero". Unsurprisingly, this gives a functor from simplicial abelian groups to chain complexes. Let's call it Ch: SimpAb -> Chain More surprisingly, this is an equivalence of categories! I leave you to show this --- if you give up, look at May's book cited in section C. of "week115". What this means is that simplicial abelian groups are just another way of thinking about chain complexes... or vice versa. Thus, if I were being ultra-sophisticated, I could have skipped the chain complexes and talked only about simplicial abelian groups! This would have saved time, but more people know about chain complexes, so I wanted to mention them. H. Singular Homology. Okay, now that we have lots of nice shiny machines, let's hook them up and see what happens! Take the "singular simplicial set" functor: Sing: Top -> SimpSet, the "free simplicial abelian group on a simplicial set" functor: L: SimpSet -> SimpAb, and the "chain complex of a simplicial abelian group" functor: Ch: SimpAb -> Chain, and compose them! We get the "singular chain complex" functor C: Top -> Chain that takes a topological space and distills a chain complex out of it. We can then take the homology groups of our chain complex and get the "singular homology" of our space. Better yet, the functor C: Top -> Chain takes homotopies between maps and sends them to homotopies between morphisms of chain complexes! It follows that homotopic maps between spaces give the same homomorphisms between the singular homology groups of these spaces. Thus homotopy-equivalent spaces will have isomorphic homology groups... so we have gotten our hands on a nice tool for studying spaces up to homotopy equivalence. Now that we've got our hands on singular homology, we could easily spend a long time using it to solve all sorts of interesting problems. I won't go into that here; you can read about it in all sorts of textbooks, like: 5) Marvin J. Greenberg, John R. Harper, Algebraic Topology: A First Course, Benjamin/Cummings, Reading, Massachusetts, 1981. or 6) William S. Massey, Singular Homology Theory, Springer-Verlag, New York, 1980. which uses cubes rather than simplices. What I'm trying to emphasize here is that singular homology is a composite of functors that are interesting in their own right. I'll explore their uses a bit more deeply next time. ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at For a table of contents of all the issues of This Week's Finds, try A simple jumping-off point to the old issues is available at If you just want the latest issue, go to