Homotopy theory and classifying spaces Bill Dwyer Copenhagen (June, 2008) Contents 1. Homotopy theories and model categories . 3 What is a homotopy theory T? – Examples of homotopy theories (Tpair presentation) – The homotopy category Ho(T) – Examples of homotopy categories – Model cat- egory: (C, E) with extras – Examples of model categories (C, E) – Dividends from a model category structure on (C, E) – Equivalences between homotopy theories – 0 0 T(C, E) ∼ T(C , E ) for model categories – Examples of equivalences between ho- motopy theories 2. Homotopy limits and colimits . 17 Colimits and related constructions – Aside: notation for special coends – Homo- topy colimits and related constructions – Model category dividend – Construction of hocolim, version I – Construction of hocolim, version II – Homotopy limits and related constructions – Another model category dividend – Construction of holim, version I – Construction of holim, Version II – Properties of homotopy (co)limits – Mysteries of (I) and (II) revealed 3. Spaces from categories . 29 Categories vs. Spaces – The Grothendieck Construction – Thomason’s Theorem – Variations on CnF (all ∼ on N) – Extension to homotopy coends – The parallel universe of Hom and ⊗ – Properties of Kan extensions – Does pulling back preserve hocolim? – Terminal functors – Does pulling back preserve holim? – Initial functors 4. Homology decompositions . 40 Approximation data for BG – Approximation data from G-orbits – Obtaining alterna- tive approximation data – Six Z/p-homology decompositions – How to obtain the six decompositions – Identify hocolim for subgroup diagram – Identify hocolim for cen- tralizer diagram – Relate posets KC2 ∼ KC1 – Relate posets KC3 ∼ KC2 1 2 5. Localizations . 46 Left Bousfield localization – Example with discrete categories – Another model cat- egory dividend – Examples of model category localizations (I) – Examples of model category localizations (II) – Localization with respect to R-homology, R ⊂ Q – Localization with respect to Z/p-homology – Mixing LQ(X) with the Lp(X)’s to recover X – An approximation to Lp – The Bousfield-Kan p-completion Cp – The Bousfield-Kan p-completion Cp: good & bad 6. Cohomology of function spaces . 56 The Steenrod algebra Ap – Modules and algebras over Ap – The functor T – The functor T ↔ function spaces out of BV – Lannes works his magic – Outlining the proof – The case X = K = K(Z/p, n) – The case in which X is p-finite. – Tower {Us} = {· · · → Un → Un−1 → · · · → U0} – T and maps from BV – In the presence of a volunteer... 7. Maps between classifying spaces . 68 Cohomology of homotopy fixed point sets – X a finite complex: setting the scene – X a finite complex (II) – Aside: sighting of Cp(p-bad space) – Maps BQ → BG: individual components – Maps BQ → BG: how many components? – Maps into CpBG for G finite – Fusion functors – Fusion systems 8. Linking systems and p-local classifying spaces . 78 Does F determine CpBG? – Switch to the p-centric collection – The p-centric cen- tralizer diagram – The categorical p-centric centralizer model – The linking system – Lc vs. Fc – the orbit picture – Lifting from Ho(Sp) to Sp – Fusion relations suffice! – p-local finite group X 9. p-compact groups . 89 Definition of a p-compact group – Some basic properties of p-compact groups – More cohomology of homotopy fixed point sets – The centralizer of V in X is a p-compact group. – The centralizer of V in X is a subgroup – ∃ non-trivial V → X – Existence of a Sylow p-subgroup for finite G – Existence of a maximal torus in X 10. Wrapping up . 99 Another commutative diagram? – Homotopy theories – Localizations & completions – The functor T and H∗ Homh(BV, −) – Homology decompositions and maps to BG – p-compact groups and p-local finite groups – Conclusion Lecture 1: Homotopy theories and model categories 3 Lecture 1. Homotopy theories and model categories These are rough notes. Read at your own risk! The presentation is not necessarily linear, complete, compact, locally connected, orthographically defensible, grammatical, or, least of all, logically watertight. The author doesn’t always tell the whole truth, sometimes even on purpose. This first lecture is deep background: before getting to classifying spaces, I’d like to describe some homotopy theoretic machinery. The first question to ask before trying to understand this machinery is a very basic one. Slide 1-1 Slide 1-1 What is a homotopy theory T? Equivalent answers • Tpair category C with a subcategory E of equivalences • Ten category R enriched over spaces (simplicial sets) • Tsc Segal category • Tcss complete Segal space • Tqc quasi-category • T∞ ∞-category (or (∞, 1) category) Just like categories h hT1 Internal function objects T3 = Cat (T1, T2) = T2 The main message here is that there has been a remarkable convergence of opinion over the last few years about what a homotopy theory is [11]. All formulations give notions which are equivalent (in a homotopy theoretic sense, see slide 1–10 below), although the objects involved look very different in detail. A pair (C, E) is a relative category, and from the point of view of Tpair homotopy theory is relative category theory. This is the form under which homotopy theories usually show up in nature; E is usually some collection of morphisms in C which are not isomorphisms but have some claim to be considered honorary isomorphisms (for instance if C = Top is the category of topological spaces, E might be the collection of homotopy equivalences). But any category C gives a homotopy theory: take E to the identity maps or (it turns out equivalently) the isomorphisms in C. Another possibility is to take E = C. A category R enriched over topological spaces is an ordinary category furnished with a topology on each morphism space [9]. From the point of view of Ten homotopy theory is a continuous form of category theory. (Not too continuous: notice that we don’t worry about topologies on sets of objects.) The transition Tpair =⇒ Ten involves inverting the arrows in E in a derived sense [38]. Alternatively, the function spaces Lecture 1: Homotopy theories and model categories 4 in the simplicial category can be view as spaces of zigzags in the original category C, where the backwards-point arrows lie in E [36]. A Segal category is a simplicial space which is discrete at level 0 and satisfies some homotopical product conditions [89] [10]. The transition Ten =⇒ Tsc amounts to taking the nerve, and treating it as a simplicial space. A complete Segal space is simplicial space which is not necessarily discrete at level 0 and satisfies some homotopy fibre product conditions and some other homotopical conditions [89] [11]. The transition Tsc =⇒ Tcss requires repackaging group-like topological monoids of equivalences into their associated classifying spaces, but leaving the non-invertible morphisms alone. This is an unusually transparent model for a homotopy theory: an equivalence is just a map between simplicial spaces which is a weak equivalence at each level. Internal function objects are also easy to come by here. A quasi-category (∞-category) is a simplicial set, treated from what classically would be a very peculiar point of view [62] [63]. In some sense this is the most economical model for a homotopy theory. 1.1 Exercise. Let C be a category and E its subcategory of isomorphisms. In this particular case, describe in detail each of the various models for the homotopy theory of T(C, E). h hT1 The notation Cat (T1, T2) or T2 denotes the homotopy theory of functors from the first homotopy theory to the second, but taken in the correct homotopy theoretic way. hT1 The notation T2 is very similar to a notation for homotopy fixed point sets that will come up later on (2.28), but I’ll use it anyway. This same ambiguity comes up without the “h”: if X and Y are spaces, Y X is the space of maps from X to Y , but if Y is a space and G is a group Y G is the fixed-point set of the action of G on Y . The above internal function objects for homotopy theories are tricky to define correctly in some models, but can have a very familiar feel to them. If (C, E) and (C0, E0) are two category pairs in which all of the morphisms in E and E0 are invertible, then the function h object Cat (T(C, E), T(C0, E0)) is equivalent in the sense of homotopy theories to the category in which the objects are functors C → C0 and the morphisms are natural transformations. (To promote this to a homotopy theory, pick natural isomorphisms between functors as equivalences.) ? 1.2 Exercise. Let G and H be two discrete groups, treated as one-object categories or as homotopy theories (the latter by designating all morphisms as equivalences). De- scribe the groupoid of functors G → H in terms of the group structures of G and H. How many components are there to the groupoid? What are the vertex groups? 1.3 Exercise. Let C be a category, and let E = C. Let Top be the homotopy the- ory of topological spaces, where the equivalences are taken to be weak homotopy equivalences. What geometric structures do you think are described by the homotopy h theory Cat (T(C, E), Top)? Lecture 1: Homotopy theories and model categories 5 Slide 1-2 Slide 1-2 Examples of homotopy theories (Tpair presentation) Geometry Notation CE Tophe Top homotopy equivalences Top Top weak homotopy equivalences TopR Top R-homology isomorphisms Top≤n Top iso on πi for i ≤ n Algebra Sp simplicial sets |f| an equivalence in Top sGrp simplicial groups |f| an equivalence in Top simp. rings, Lie alg.,etc (same) ChR chain complexes over R homology isomorphisms DG algebras homology isomorphisms The first four examples illustrate the fact that a category can be associated with many different homotopy theories.