Abstract Homotopy Theory Switching Gears Today Will Be Almost All Classical Mathematics, in Set Theory Or Whatever Foundation You Prefer

Abstract homotopy theory Switching gears Today will be almost all classical mathematics, in set theory or whatever foundation you prefer. Michael Shulman

Slogan March 6, 2012 Homotopy theory is the study of 1-categories whose objects are not just “set-like” but contain paths and higher paths.

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Homotopies and equivalences Examples

Question What structure on a category C describes a “homotopy theory”?

We expect to have: • C = topological spaces, homotopies A × [0, 1] → B. 1 A notion of homotopy between morphisms, written f ∼ g. • C = chain complexes, with chain homotopies. This indicates we have paths f (x) g(x), varying nicely • C = categories, with natural isomorphisms. with x. • C = ∞-groupoids, with “natural equivalences” Given this, we can “homotopify” bijections: Deﬁnition A homotopy equivalence is f : A → B such that there exists g : B → A with fg ∼ 1B and gf ∼ 1A.

4 / 52 5 / 52 ∞-groupoids ∞-functors

Definition There are two ways to define morphisms of ∞-groupoids: Today, an ∞-groupoid means an algebraic structure: 1 strict functors, which preserve all composition operations 1 Sets and “source, target” functions: on the nose. 2 weak functors, which preserve operations only up to ··· ⇒ Xn ⇒ ··· ⇒ X2 ⇒ X1 ⇒ X0 specified coherent equivalences. Which should we use? X0 = objects, X1 = paths or morphisms, X2 = 2-paths or 2-morphisms, . . . • We want to include weak functors in the theory. 2 Composition/concatenation operations e.g. p : x y and q : y z yield p@q : x z. • But the category of weak functors is ill-behaved: it lacks limits and colimits. 3 These operations are coherent up to all higher paths. • The category of strict functors is well-behaved, but seems to miss important information. A topological space Z gives rise to an ∞-groupoid Π∞(Z).

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Coﬁbrant objects Coﬁbrant replacement

Theorem Theorem If A is a free ∞-groupoid, then any weak functor f : A → B is Any ∞-groupoid is equivalent to a free one. equivalent to a strict one. Proof. Proof. Given A, deﬁne QA as follows: Deﬁne ef : A → B as follows: • The objects of QA are those of A. • ef acts as f on the points of A. • The paths of QA are freely generated by those of A. • ef acts as f on the generating paths in A. Strictness of ef • The 2-paths of QA are freely generated by those of A. then uniquely determines it on the rest. • ... • f acts as f on the generating 2-paths in A. Strictness then e We have q : QA → A, with a homotopy inverse A → QA uniquely determines it on the rest. obtained by sending each path to itself. • ... However: QA → A is a strict functor, but A → QA is not!

8 / 52 9 / 52 Weak equivalences Some left derivable categories

1 ∞-groupoids with strict functors Definition • cofibrant = free A left derivable category C is one equipped with: • weak equivalences = strict functors that have homotopy inverse weak functors. • A class of objects called cofibrant. 2 chain complexes • A class of morphisms called weak equivalences such that • cofibrant = complex of projectives • if two of f , g, and gf are weak equivalences, so is the third. • ∼ weak equivalence = homology isomorphism • Every object A admits a weak equivalence QA −→ A from 3 topological spaces a cofibrant one. • cofibrant = CW complex • weak equivalence = isomorphism on all higher homotopy Remarks groups πn • A “weak morphism” A B in C is a morphism QA → B. 4 topological spaces • In good cases, a weak equivalence between cofibrant • cofibrant = everything • objects is a homotopy equivalence. weak equivalence = homotopy equivalence The homotopy theories of ∞-groupoids and topological spaces (CW complexes) are equivalent, via Π∞.

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Cylinder objects Left homotopies What happened to our “notion of homotopy”? Deﬁnition Deﬁnition A cylinder object for A is a diagram A left homotopy f ∼ g is a diagram

A 1A F A F f FF FF FF FF i FF F 0 # ! i0 F# ! ∼ / Cyl(A) = A Cyl(A) / B i ; ; = 1 x i1 x xx xx xx xx xx xx A 1A A g

Examples for some cylinder object of A. • In topological spaces, A × [0, 1]. Remark • In chain complexes, A ⊗ (Z → Z ⊕ Z). For the previous cylinders, this gives the usual notions. • In ∞-groupoids, A × I, where I has two isomorphic objects.

12 / 52 13 / 52 Duality Path objects

A path object for A is a diagram

/ Definition 1A : A vv A right derivable category C is one equipped with: vv vvev1 ∼ v • A class of objects called fibrant. A / Path(A) HH ev0 • A class of morphisms called weak equivalences, satisfying HH HH the 2-out-of-3 property. H$ 1A / A ∼ • Every object A admits a weak equivalence A −→ RA to a fibrant one. Example [0,1] In topological spaces, Path(A) = A , with ev0, ev1 evaluation at the endpoints.

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Right homotopies Simplicial sets Deﬁnition A simplicial set X is a combinatorial structure of: A right homotopy f ∼ g is a diagram 1 A set X0 of vertices or 0-simplices f / B 2 A set X of paths or 1-simplices, each with assigned v: 1 vv source and target vertices vv vv p1 A / Path(B) 3 A set X2 of 2-simplices with assigned boundaries H HHp0 HH HH$ g / B 4 ... Example • A simplicial set X has a geometric realization |X|, a For the previous path object in topological spaces, this is again n the usual notion. topological space built out of topological simplices ∆ according to the data of X. • A topological spaces Z has a singular simplicial set S∗(Z ) whose n-simplices are maps ∆n → Z .

16 / 52 17 / 52 Kan complexes Homotopy theory of simplicial sets

We can also think of a simplicial set as a model for an ∞-groupoid via

n-simplices = n-paths • Fibrant objects = Kan complexes but it doesn’t have composition operations. • Weak equivalences = maps that induce equivalences of Deﬁnition geometric realization A simplicial set is ﬁbrant (or: a Kan complex) if

This is also equivalent to ∞-groupoids. 1 Every “horn” can be “ﬁlled” to a 2-simplex. 2 etc. . . . If Y is not ﬁbrant, then there may not be enough maps X → Y ; some of the composite simplices that “should” be there in Y are missing.

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Diagram categories Homotopy natural transformations

Question Question If α: F → G has each component αx a homotopy equivalence, is α a homotopy equivalence? If C has a homotopy theory, does a functor category CD have one? Let βx : Gx → Fx be a homotopy inverse to αx . Then for f : x → y, Fact A natural transformation α: F → G is a natural isomorphism iff βy ◦ G(f ) ∼ βy ◦ G(f ) ◦ αx ◦ βx each component αx is an isomorphism. = βy ◦ αy ◦ F(f ) ◦ βx Deﬁnition ∼ F(f ) ◦ βx A weak equivalence in CD is a natural transformation such that So β is only a natural transformation “up to homotopy”. each component is a weak equivalence in C. Conclusion: the “weak morphisms” of functors should include “homotopy-natural transformations”.

21 / 52 22 / 52 Fibrant diagrams Fibrations Theorem For g : G0 → G1 in spaces, the following are equivalent. Question 1 Every homotopy commutative square into g is homotopic For what sort of functor G is every homotopy-natural to a commutative one with the bottom map ﬁxed: transformation F → G equivalent to a strictly natural one? h h0 0 / ( D F0 G0 F ' Consider D = (0 → 1), so C is the category of arrows in C. 0 6 G0 f ' g = g f / F0 G0 F1 / G1 / h1 F1 G1 h1 ' 2 g is a ﬁbration: F1 / G1 / X 9 G0 0 g X × [0, 1] / G1

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Proof: 1 ⇒ 2 Proof: 2 ⇒ 1

h 0 / h0 F0 G0 X / G0 X / G0 F0 / G0 0 g g g f ' g ←→ 1 0 ←→ ' F / F × [0, 1] / G 0 0 9 1 F1 / G1 X × [0, 1] / G1 X / G1 h1 h1f

X / G ( h : 0 X ' G 0 uu 6 0 h0 F0 / G0 uu ( 9 0 uu g ←→ F ' G ss uu g 0 6 0 0 sss g uu ss / g ←→ 1 s X × [0, 1] G1 / f F / F × [0, 1] / G X G1 0 0 9 1 F1 / G1 h1 h1f

25 / 52 26 / 52 Fibrations and coﬁbrations Lifting properties

Definition Conclusions Given i : X → Y and q : B → A in a category, we say i q if any • Fibrations can be the fibrant objects in the category of commutative square / arrows. X ? B • Similarly, cofibrations (defined dually) can be the cofibrant i q objects. Y / A • A “category with homotopy theory” should have notions of admits a dotted filler. fibration and cofibration. • D And maybe more stuff, for diagrams C other than arrows? • I = { q | i q ∀i ∈ I } • This is starting to look like a mess! • Q = { i | i q ∀q ∈ Q } • fibrations = {X → X × [0, 1]}

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Closure properties of lifting properties Weak factorization systems

Deﬁnition Lemma A weak factorization system in a category is (I, Q) such that If i q, then i (any pullback of q). 1 I = Q and Q = I. Proof. 2 Every morphism factors as q ◦ i for some q ∈ Q and i ∈ I.

/ / X ? D 7 B Examples _ i q • in sets, I = surjections, Q = injections. / / Y C A • in sets, I = injections, Q = surjections. Note: (I), I always satisﬁes condition 1. • Similarly, (any pushout of i) q. • Q = ﬁbrations = {X → X × [0, 1]} • Also closed under retracts. I = Q Factorization?

30 / 52 31 / 52 The mapping path space Acyclic coﬁbrations

Definition Facts Given g : X → Y in topological spaces, its mapping path space • ρ : Ng → Y is a fibration. is the pullback g • The map λ : X → Ng defined by x 7→ (x, g(x), c ) is a Ng / X g g(x) cofibration and a homotopy equivalence.

λg ρg • The composite X −→ Ng −→ Y is g. ev0 ρg Paths(Y ) / Y • (fibrations) = (cofibrations) ∩ (homotopy equivalences) ev1 ) Y Definition An acyclic cofibration is a cofibration that is also a homotopy equivalence. Ng = { (x, y, α) | x ∈ X, y ∈ Y , α: g(x) y } Theorem and ρg(x, y, α) = y. (acyclic cofibrations, fibrations) is a weak factorization system.

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Mapping cylinders Acyclic ﬁbrations

Given g : X → Y , its mapping cylinder is the pushout Facts g / X Y • X → Mg is a coﬁbration 0 • Mg → Y is a ﬁbration and a homotopy equivalence. Cyl(X) / Mg • The composite X → Mg → Y is g.

• (coﬁbrations) = (ﬁbrations) ∩ (homotopy equivalences)

Definition An acyclic fibration is a fibration that is also a homotopy equivalence.

X Theorem (coﬁbrations, acyclic ﬁbrations) is a weak factorization system. Y

34 / 52 35 / 52 Model categories Homotopy theory in a model category

Definition (Quillen) • X is cofibrant if ∅ → X is a cofibration. A model category is a category C with limits and colimits and • Y is fibrant if Y → 1 is a fibration. three classes of maps: • For any X, we have ∅ → QX → X with QX cofibrant and • C = cofibrations QX → X an acyclic fibration (hence a weak equivalence). • F = fibrations • For any Y , we have Y → RY → 1 with RX fibrant and • W = weak equivalences Y → RY an acyclic cofibration. such that • Any X has a very good cylinder object

1 W has the 2-out-of-3 property. acyc. ﬁb. X + X −−→cof. Cyl(X) −−−−−→ X 2 (C ∩ W, F) and (C, F ∩ W) are weak factorization systems. • Any Y has a very good path object Not all “categories with homotopy” are model categories, but acyc. cof. ﬁb. very many are. When it exists, a model category is a very Y −−−−−→ Paths(Y ) −→ Y × Y convenient framework.

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Homotopy theory in a model category Some model categories

Example Topological spaces, with Let X and Y be fibrant-and-cofibrant in a model category C. • Fibrations as before • Left and right homotopy agree for maps X → Y . • Cofibrations defined dually • Homotopy is an equivalence relation on maps X → Y . • Weak equivalences = homotopy equivalences • A map X → Y is a weak equivalence iff it is a homotopy equivalence. Example • In good cases, every functor category CD is also a model Topological spaces, with category. • Fibrations as before • Cofibrations = homotopy equivalent to rel. cell complexes

• Weak equivalences = maps inducing isos on all πn The second one is equivalent to ∞-groupoids.

38 / 52 39 / 52 Some model categories Some model categories

Example Example Chain complexes, with Small categories (or groupoids), with • Fibrations = degreewise-split surjections • Fibrations = functors that lift isomorphisms • Coﬁbrations = degreewise-split injections • Coﬁbrations = injective on objects • Weak equivalences = chain homotopy equivalences • Weak equivalences = equivalences of categories

Example Example Chain complexes, with Any category, with • Fibrations = degreewise surjections • Fibrations = all maps • Coﬁbrations = degreewise-split injections with projective • Coﬁbrations = all maps cokernel • Weak equivalences = isomorphisms • Weak equivalences = maps inducing isos on all Hn

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Simplicial sets Display map categories

Recall Example A display map category is a category with Simplicial sets, with • A terminal object. • Fibrations = Kan ﬁbrations • A subclass of its morphisms called the display maps, • Coﬁbrations = monomorphisms denoted B A or B A. _ • Weak equivalences = geometric realization equivalences. • Any pullback of a display map exists and is a display map.

This is unreasonably well-behaved in many ways. Note: The right class of any weak factorization system can be a class of display maps.

42 / 52 44 / 52 Identity types in d.m. categories Identity types in d.m. categories The reﬂexivity constructor x : A ` reﬂ(x):(x = x) The dependent identity type must be a section ∆∗Id / Id x : A, y : A ` (x = y): Type ? A A must be a display map IdA / A × A A ∆ A × A or equivalently a lifting

IdA y< reﬂ yy yy yy yy / A × A A ∆

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Identity types in d.m. categories Identity types in d.m. categories

The eliminator says given a dependent type with a section In fact, refl all display maps. ∗ refl C / C C / ∗ / @ @ A f< C C there exists _ a compatible refl section IdA IdA / B f A / IdA refl IdA

In other words, we have the lifting property Conclusion Identity types factor ∆: A → A × A as / A = C refl q refl A −−→ IdA −− A × A Id Id A A where q is a display map and refl (display maps).

47 / 52 48 / 52 General factorizations Modeling identity types Theorem (Gambino–Garner) In a display map category that models identity types, any Theorem (Awodey–Warren,Garner–van den Berg) morphism g : A → B factors as In a display map category, if

i q A / Ng / / B (display maps), ((display maps)) where q is a display map, and i all display maps. is a well-behaved weak factorization system, then the category models identity types. Ng = y : B, x : A, p :(g(x) = y) J K products ←→ categorical products is the type-theoretic mapping path space. disjoint unions ←→ categorical coproducts Corollary . . • I = (display maps) identity types ←→ weak factorization systems • Q = I is a weak factorization system.

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Type theory of homotopy theory Other homotopy theories

The model category of simplicial sets is well-behaved. Conclusion To model homotopy type theory, we need a category that We can prove things about ordinary homotopy theory by 1 has ﬁnite limits and colimits (for ×, +, etc.) reasoning inside homotopy type theory. 2 has a well-behaved WFS (for identity types), (The complete theory of simplicial sets). 3 is compatibly locally cartesian closed (for Π), 4 has a univalent universe (for coherence and univalence) Interpreted in ordinary homotopy theory, These requirements basically restrict us to (∞, 1)-toposes. • Function extensionality holds. • The univalence axiom holds (Voevodsky). Unfortunately, no one has yet found sufﬁciently coherent

• A space A is n-truncated just when πk (A) = 0 for k > n. univalent universes in any (∞, 1)-topos other than simplicial • An equivalence is a classical (weak) homotopy sets (i.e. ∞-groupoids). equivalence.

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