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The Formal Theory of Adjunctions, Monads, Algebras, and Descent

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The Formal Theory of Adjunctions, Monads, Algebras, and Descent Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent The formal theory of adjunctions, monads, algebras, and descent Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl Reimagining the Foundations of Algebraic Topology Mathematical Sciences Research Institute Tuesday, April 8, 2014 Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Formal theory of adjunctions, monads, algebras, descent Joint with Dominic Verity. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Plan Part I. adjunctions and monads htpy u ? context: 2-CAT / ( ; 2)-CAT • incl 1 Theorem. Any adjunction in a homotopy 2-category extends to a• homotopy coherent adjunction in the ( ; 2)-category. 1 (Interlude on weighted limits.) Part II. algebras and descent definitions of algebras, descent objects: via weighted limits • proofs of monadicity, descent theorems: all in the weights! • Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Some basic shapes categories with a monad weight (\shape") for ∆+ = / o / underlying object • • / •··· / ∆ = o / o / descent object • / • o / •··· / ∆1 = o / o / object of algebras • • o •··· Moreover: ∆ o ? _∆ B / dscB + > Q D K aCC ~ D vv + CC ~~ , DD vv D ` C ~~` D v a CC ~ ` D vv C 0 P / ~ ~ D ! {v v X . ∆1 q g X algB e Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent ( ; 2)-categories 1 An ( ; 2)-category is a simplicially enriched category whose hom-spaces1 are quasi-categories. ho htpy 2-cat u ? u ? Cat / qCat 2-CAT / ( ; 2)-CAT N incl 1 Examples. 2-categories: categories, monoidal categories, accessible categories, algebras for any 2-monad, . ( ; 2)-categories: quasi-categories, complete Segal spaces, Rezk1 objects, . Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent The free adjunction Adj := the free adjunction, a 2-category with objects + and − op hom(+; +) = hom( ; ) := ∆+ − − op hom( ; +) = hom(+; ) := ∆1 − − Theorem (Schanuel-Street). Adjunctions in a 2-category K correspond to 2-functors Adj K. ! η / η / o u id η / uf o u ufuf ufη / ufufuf ufη / o ufu ··· ufufη / η / η / o u u η / ufu o u ufufu ufη / ufufufu o u ufη / o ufu ··· o ufu ufufη / o ufufu Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent The free homotopy coherent adjunction Theorem (Schanuel-Street). Adjunctions in a 2-category K correspond to 2-functors Adj K. ! A homotopy coherent adjunction in an ( ; 2)-category K is a simplicial functor Adj K. 1 ! data in Adj: , + ; ; ; ; ;... − n-arrows are strictly undulating squiggles on n + 1 lines − 0 1 1 − 0 − 0 2 1 1 2 1 1 3 2 2 3 2 2 4 3 3 4 3 3 + + + Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Homotopy coherent adjunctions A homotopy coherent adjunction in an ( ; 2)-category K is a simplicial functor Adj K. 1 ! Theorem. Any adjunction in the homotopy 2-category of an ( ; 2)-category extends to a homotopy coherent adjunction. 1 Theorem. Moreover, the spaces of extensions are contractible Kan complexes. Upshot: there is a good supply of adjunctions in ( ; 2)-categories. 1 Proposition. Adj is a simplicial computad (cellularly cofibrant). Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Homotopy coherent monads Mnd := full subcategory of Adj on +. A homotopy coherent monad in an ( ; 2)-category K is a simplicial functor Mnd K., i.e., 1 ! + B K 7! 2 ∆+ hom(B; B) =: the monad resolution ! η / η µ / 2 o 3 idB η / t o µ t tη / t tη / o tµ ··· ttη / t2 and higher data: ηt ¡@ == µ ¡¡ ∼ == ! t ¡ t Warning: A monad in the homotopy 2-category need not lift to a homotopy coherent monad. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Interlude on weighted limits Let K be a 2-category or an ( ; 2)-category, A a small 2-category. 1 weightop diagram limit × 7! 2 2 2 {−;−} (CatA)op KA A K × −−−−−! Facts: W; D A := some limit formula using cotensors f g a representable weight evaluates at the representing object colimits of weights give rise to limits of weighted limits Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Cellular weighted limits f g Example (A := b a c). Define W sSetA by: −! − 2 0 0 homb @∆ homc @∆ / × t × ; 0 0 p ftg 1 homb ∆ homc ∆ / homb homc o homa @∆ × t × t × q 1 W; A =: comma object W o homa ∆ f −} × A cellular weight is a cell complex in the projective model structure on CatA or sSetA. Example: Bousfield-Kan homotopy limits. Completeness hypothesis: K admits cellular weighted limits. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Algebras for a homotopy coherent monad Mnd K Fix a homotopy coherent monad: + ! B 7! Goal: define the object of algebras algB K and the monadic f2 o homotopy coherent adjunction algB ? B u / Adj Mnd Mnd u: E B K ∆1 Cat uuu E 2 2 , uu E" Mnd / K + 7! B ∆1 ∆+×∆1 algB := ∆1;B Mnd = eq B B f g ⇒ Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Algebras, continued ∆1 ∆+×∆1 algB := ∆1;B Mnd = eq B B f g ⇒ Example: K = qCat. A vertex in algB is a map ∆1 B of the form: ! η / η / o µ b η / tb o µ t2b tη / t3b o β tη / o tµ ··· o tβ ttη / o ttβ and higher data, e.g., tb η @ >> β ∼ >> b b Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent The monadic homotopy coherent adjunction . is all in the weights! yoneda res {−;BgMnd Adjop / CatAdj / CatMnd / Kop 7! hom− 7! ∆1 7! algB −I T T J f a u ` ` f a u Ù + 7! hom+ 7! ∆+ 7! B Note: Adj a simplicial computad these weights are cellular. Q: Doesn't this imply that up-to-homotopy monads have monadic adjunctions and hence are homotopy coherent? A: No! Homotopy 2-categories don't admit cellular weighted limits. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Descent data for a homotopy coherent monad Fix B KMnd. A descent datum is a coalgebra for the induced monad2 on the object of algebras. dscB := coalg(alg(B)) ∆ CatMnd 2 ∆ ∆+×∆ dscB := ∆;B Mnd = eq B B f g ⇒ Example (K = qCat). A vertex in dscB is a map ∆ B: ! η / η / o µ b o β tb tη / t2b and higher data γ / o tβ ··· tγ / _ ∆+ o ? ∆ B / dscB aC ~> P DD u K ) CC ~~ * DD uuu C ~` D u ` a C0 P / ~~ ` D" zuu C ~ Mnd D Z / ∆1 p e in Cat in K Z algB c u Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Totalizations of cosimplicial objects in an object of K The monadicity and descent theorems require geometric realization of simplicial objects valued in an object of an ( ; 2)-category. 1 An object B K admits totalizations iff there is an absolute right lifting2 diagram in hoK: B = {{ tot {{ {{ const {{ +ν B∆ / B∆ id Equivalently: const ∆ o an adjunction B ? B in hoK. / 9 tot const ∆ o a homotopy coherent adjunction B ? B in K. / 9 tot Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Totalizations of split augmented cosimplicial objects Theorem. In any ( ; 2)-category with cotensors, the totalization of a split augmented1 cosimplicial object is its augmentation., i.e., B < ev xx 0 xx xx const xx +κ ∆1 ∆ B res / B is an absolute right lifting diagram for any object B. 1 O [0] }} }} ! Proof: is all in the weights! }} is an equationally ~}} +κ ∆1 o ∆ incl witnessed absolute right extension diagram in Cat. Apply B(−). Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Canonical presentations of algebras and descent data Theorem. Any descent datum is the totalization of a canonical dscB : uu id uu cosimplicial object of free descent data: uu const is an uu +κ u ∆ dscB can / dscB absolute right lifting diagram. Theorem. Any algebra is the geometric realization of a canonical algB : uu id uu simplicial object of free algebras.: uu const is an uu *κ u ∆op algB can / algB absolute left lifting diagram. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Monadic descent in an ( ; 2)-category 1 Theorem. For any homotopy coherent monad in an ( ; 2)-category with cellular weighted limits, there is a canonical map1 h c _? [ W B v l _______ / dscB Q E K EE uu EE uu E uu ` ` E" zuu algB that admits a right adjoint if B has totalizations that is full and faithful if elements of B are totalizations of their monad resolution that is an equivalence if comonadicity is satisfied The theory of comonadic codescent is dual: replace the weights by their opposites. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Monadicity theorem in an ( ; 2)-category 1 Theorem. For any homotopy coherent adjunction f u with homotopy coherent monad t, there is a canonical mapa d _ Z V wm h ? A ? _______ / algB ? ?? f t z< _ ? ?` u z z` ?? ?? zz zz ?? ?? zz zz f ?? z zz ut B |zz that admits a left adjoint if A has geometric realizations of u-split simplicial objects that is an adjoint equivalence if u creates these colimits. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Further reading \The 2-category theory of quasi-categories" arXiv:1306.5144 \Homotopy coherent adjunctions and the formal theory of monads" arXiv:1310.8279 \A weighted limits proof of monadicity" on the n-Category Caf´e.
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