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 ∞ Theorem. Any adjunction in a homotopy 2-category extends to •a homotopy coherent adjunction in the ( , 2)-category. ∞
(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 / •··· / ∆∞ = 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 . ∆∞ q g X algB e Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent ( , 2)-categories ∞
An ( , 2)-category is a simplicially enriched category whose hom-spaces∞ are quasi-categories.
ho htpy 2-cat u ⊥ u ⊥ Cat / qCat 2-CAT / ( , 2)-CAT N incl ∞
Examples. 2-categories: categories, monoidal categories, accessible categories, algebras for any 2-monad, . . . ( , 2)-categories: quasi-categories, complete Segal spaces, Rezk∞ 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(+, ) := ∆∞ − − 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. ∞ →
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. ∞ →
Theorem. Any adjunction in the homotopy 2-category of an ( , 2)-category extends to a homotopy coherent adjunction. ∞
Theorem. Moreover, the spaces of extensions are contractible Kan complexes.
Upshot: there is a good supply of adjunctions in ( , 2)-categories. ∞ 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., ∞ → + B K 7→ ∈ ∆+ 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. ∞ weightop diagram limit × 7→ ∈ ∈ ∈
{−,−} (CatA)op KA A K × −−−−−→
Facts:
W, D A := some limit formula using cotensors { } 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: −→ ←− ∈
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 ∆ { −} ×
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 f∈ o homotopy coherent adjunction algB ⊥ B u /
Adj Mnd Mnd u: E B K ∆∞ Cat uuu E ∈ ∈ , uu E" Mnd / K + 7→ B