The formal theory of homotopy coherent monads

Transcription

1 The formal theory of homotopy coherent monads Emily Riehl Harvard University 23 July, 2013 Samuel Eilenberg Centenary Conference Warsaw, Poland

2 Joint with Dominic Verity. Slogan: It s all in the weights!

3 Plan 1 Homotopy coherent adjunctions 2 Homotopy coherent monads and the monadic adjunction 3 Codescent in the Eilenberg-Moore quasi-category 4 Monadicity theorem

4 Quasi-categories A quasi-category is a simplicial set A in which any inner horn Λ n,k A 4 0 < k < n has a filler n The homotopy category ha has objects = vertices morphisms = homotopy classes of 1-simplices Via the adjunction h Cat qcat quasi-category theory extends category theory.

5 Adjunctions of quasi-categories qcat 2 := the 2-category of quasi-categories, consisting of quasi-categories A, B functors (maps of simplicial sets) g : A B natural transformations (homotopy classes of 1-simplices) A 0 g i g 0 α B 1 α B A k i 1 4 k 0 An adjunction of quasi-categories is an adjunction in qcat 2. A f u B : id B uf ɛ: fu id A

6 Some theorems and examples Theorems. f u induces adjunctions f X u X and C u C f for any simplicial set X and quasi-category C. Any equivalence can be promoted to an adjoint equivalence. Right adjoints preserve limits. f : B A has a left adjoint iff f a has a terminal object for each a A. Examples. ordinary adjunctions, topological adjunctions simplicial Quillen adjunctions colim const lim loops suspension

7 A coherence question qcat := the simplicial category of quasi-categories. Given A f u id B B in qcat 2, what adjunction data exists in qcat? ufu u uɛ 2222 α u u id u uf in B B ɛ fu id A in A A in B A fuf f ɛf β f f id f uf uf uf uf id B uf id B uf ufuf uɛf ufuf uɛf But do there exist fillers with the same bottom face? in A B filling Λ 3,1 B B

8 The free adjunction Adj := the free adjunction, a 2-category with objects + and Adj(+, +) = Adj(, ) op := + Adj(, +) = Adj(+, ) op := Theorem (Schanuel-Street). 2-functors Adj qcat 2 correspond to adjunctions in qcat 2. id uf u uɛ ufu uɛ uf uɛ uf ufuɛ ufuf ufufu uɛ uf ufuɛ ufuf uɛ uf ufuɛ ufuf ufufuɛ ufufuf ufufufu

9 The free homotopy coherent adjunction Conjecture. The free homotopy coherent adjunction is Adj, regarded as a simplicial category under 2-Cat sset-cat. n-arrows are strictly undulating squiggles on n + 1 lines Proposition. Adj is a simplicial computad (i.e., cofibrant).

10 Homotopy coherent adjunctions u = f = = ɛ = s = Theorem. Any adjunction Adj qcat 2 lifts to a homotopy coherent adjunction Adj qcat. Theorem. Such extensions are homotopically unique: the spaces of extensions are contractible Kan complexes.

11 Homotopy coherent monads Mnd := full subcategory of Adj on +. Definition. A homotopy coherent monad is a simplicial functor T : Mnd qcat., i.e., + B qcat t B B =: the monad resolution + id B t µ t t 2 µ t tµ tt t 3 and higher data, e.g., t 2 t µ t t

12 Weighted limits Fix a simplicial functor T, a diagram of shape A. A weight is a simplicial functor W : A sset. The weighted limit {W, T } represents the simplicial set of cones of shape W over T. Key facts: The limit weighted by hom a evaluates at a. The weighted limit bifunctor is cocontinuous in the weights. Upshot: Weights built by gluing representables will define cones of the expected shape.

13 Weighted limits in the quasi-categorical context Proposition. qcat has all limits weighted by projective cofibrant simplicial functors. Mnd Adj hom + W + hom sset 1 W Mnd Adj a simplicial computad W + and W projective cofibrant.

14 The Eilenberg-Moore quasi-category Fix a homotopy coherent monad T : Mnd qcat {W +, T } = B {W, T } =: B[t], the Eilenberg-Moore quasi-category By definition B[t] = eq ( B B + ) so a vertex is a map B of the form: b β tb µ t tβ t 2 b µ t tµ tt ttβ t 3 b and higher data, e.g., tb β b b

15 The monadic homotopy coherent adjunction... is all in the weights! Adj op hom sset Adj res sset Mnd {,T } qcat op f u hom W B[t] + hom + W + B f t u t Proposition. If V W is identity-on-0-cells, then {W, T } {V, T } is conservative. E.g., W + W. Corollary. The monadic forgetful functor u t : B[t] B is conservative.

16 Codescent in the Eilenberg-Moore category Suppose (b, β) is an algebra for a monad t on a category B. Fact. There is a canonical colimit diagram in B[t] ttβ tt t 3 b tµ t 2 b t µ tβ t µ 6 tb β b which is a u t -split reflexive coequalizer diagram, and preserved by u t. (b, β) a u t -split (augmented) simplicial object op + B[t] B u t

17 Codescent in the Eilenberg-Moore quasi-category Theorem. Every vertex in B[t] is the colimit of a canonical u t -split simplicial object that is preserved by u t. Proof. By cocontinuity, op + W + W + {W, T } B[t] op + op + W W B res u t B op + {W, T } B[t] op + B[t] ev 1 const in qcat 2. res B[t] op

18 Codescent in the Eilenberg-Moore quasi-category Theorem. {W, T } B[t] ev 1 B[t] op + B[t] op res const defines an absolute left lifting diagram in qcat 2 that u t preserves. Proof. Similar to: Theorem. B ev 1 const B res B op defines an absolute left lifting diagram in qcat 2 that is preserved by any functor. Proof. See The 2-category theory of quasi-categories.

19 The classical monadicity theorem Let t be the monad induced by an adjunction f u. Theorem (Beck). There is a comparison functor commuting with the adjunctions. L A B[t] R f u f t B u t If A has u-split coequalizers, then R has a left adjoint. If u preserves them, then L is fully faithful. If u is conservative, then L R is an adjoint equivalence. Goal. Prove the analogous theorem for the homotopy coherent monad of a homotopy coherent adjunction.

20 Defining the comparison map Adj H qcat Mnd B[t] = {W, res H} = {lan W, H} T Weights for the monadic adjunction, revisited. weight for the Eilenberg-Moore quasi-category: lan res hom weight for the monadic adjunction: lan res hom lan res The counit of sset Adj sset Mnd defines a map of weights lan res hom hom and hence a natural transformation R A B[t] u f t f u t B between homotopy coherent adjunctions.

21 The weight for u-split simplicial objects Define a weight op hom + hom + {W, H} A op op hom W B res u B op Definition. The quasi-category A admits colimits of u-split simplicial objects if there is an absolute left lifting diagram {W, H} A colim const in qcat 2. A op

22 The proof of the monadicity theorem Proof. The obvious map W lan W induces B[t] {W, H}. If A has colimits of u-split simplicial objects, define L := B[t] {W, H} colim A. From the universal property of absolute left liftings, L R. If u preserves these colimits, then u t carries the unit of L R to an isomorphism. As u t is conservative, the unit is an isomorphism. If u is conservative, it follows that the counit is also an isomorphism, and A B[t] is an adjoint equivalence of quasi-categories.

23 Further reading The 2-category theory of quasi-categories arxiv: A weighted limits proof of monadicity on the n-category café. Homotopy coherent adjunctions and the formal theory of monads coming soon!