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

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Abstract Homotopy Theory Switching Gears Today Will Be Almost All Classical Mathematics, in Set Theory Or Whatever Foundation You Prefer Homotopy theory 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. 1 / 52 3 / 52 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 = 1-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 1-groupoids 1-functors Definition There are two ways to define morphisms of 1-groupoids: Today, an 1-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 1-groupoid Π1(Z). 6 / 52 7 / 52 Cofibrant objects Cofibrant replacement Theorem Theorem If A is a free 1-groupoid, then any weak functor f : A ! B is Any 1-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 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 1-groupoids and topological spaces (CW complexes) are equivalent, via Π1. 10 / 52 11 / 52 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 1-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. 14 / 52 15 / 52 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 jXj, 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 1-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 1-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. 18 / 52 19 / 52 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 23 / 52 24 / 52 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 = f q j i q 8i 2 I g • This is starting to look like a mess! • Q = f i j i q 8q 2 Q g • fibrations = fX ! X × [0; 1]g 27 / 52 29 / 52 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 2 Q and i 2 I. / / X ? D 7 B Examples _ i q • in sets, I = surjections, Q = injections. / / Y C A • in sets, I = injections, Q = surjections.
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