Enrichments for higher categories David Kern Laboratoire Angevin de REcherche en MAthématiques — Séminaire doctoral 12th October 2018 David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 1 / 22 Sommaire - Section 1: The algebraic structure of free objects 1 The algebraic structure of free objects 2 Monoids and enrichments Monoidal and enriched categories Monads 3 Categories up to homotopy David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 1 / 22 I Free category FQ with objects the vertices of Q morphisms homFQ (V0; V1) = fstrings of composable arrows joining V0 to V1g Composition given by concatenation (and get identities as empty string) Adjunction property Let C be a category. Any functor FQ C is determined by its action on Q, and any graph morphism Q C extends uniquely (by concatenation): ! (F ; C) ' ( ; UC) U homCat! Q homQuiver Q (with the “underlying quiver” functor) The free category functor Let Q = (fverticesg; farrowsg) be a quiver. What is the “best” way to turn it into a category? I Need to freely add a composition operation, respecting associativity (and units) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 Adjunction property Let C be a category. Any functor FQ C is determined by its action on Q, and any graph morphism Q C extends uniquely (by concatenation): ! (F ; C) ' ( ; UC) U homCat! Q homQuiver Q (with the “underlying quiver” functor) The free category functor Let Q = (fverticesg; farrowsg) be a quiver. What is the “best” way to turn it into a category? I Need to freely add a composition operation, respecting associativity (and units) I Free category FQ with objects the vertices of Q morphisms homFQ (V0; V1) = fstrings of composable arrows joining V0 to V1g Composition given by concatenation (and get identities as empty string) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 The free category functor Let Q = (fverticesg; farrowsg) be a quiver. What is the “best” way to turn it into a category? I Need to freely add a composition operation, respecting associativity (and units) I Free category FQ with objects the vertices of Q morphisms homFQ (V0; V1) = fstrings of composable arrows joining V0 to V1g Composition given by concatenation (and get identities as empty string) Adjunction property Let C be a category. Any functor FQ C is determined by its action on Q, and any graph morphism Q C extends uniquely (by concatenation): ! (F ; C) ' ( ; UC) U homCat! Q homQuiver Q (with the “underlying quiver” functor) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 Examples I U : Vectk Set the forgetful “underlying set” functor; the free vector space k[S] on a set S is the k-linear span of its elements I U : CAlgk ! Vectk ; the free commutative k-algebra on V is the symmetric tensor • ∼ ⊗2 ⊗3 algebra: U (Sym V ) = k ⊕ V ⊕ (V )S2 ⊕ (V )S3 ⊕ · · · ! Remark: FX is functorial in X, giving F :C D ! Free objects in general Let U :D C be a “forgetful” functor. For any object X 2 C, the free object of D on X is FX 2 D with ηX : X UFX (in C) such that: ! ηX for any Y 2 D and any morphism f : X UY in X UFX! C, there is a unique f : FX Y (in D) making the Uf f diagram commute (in C) ! UY ! David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 3 / 22 Free objects in general Let U :D C be a “forgetful” functor. For any object X 2 C, the free object of D on X is FX 2 D with ηX : X UFX (in C) such that: ! ηX for any Y 2 D and any morphism f : X UY in X UFX! C, there is a unique f : FX Y (in D) making the Uf f diagram commute (in C) ! UY ! Examples I U : Vectk Set the forgetful “underlying set” functor; the free vector space k[S] on a set S is the k-linear span of its elements I U : CAlgk ! Vectk ; the free commutative k-algebra on V is the symmetric tensor • ∼ ⊗2 ⊗3 algebra: U (Sym V ) = k ⊕ V ⊕ (V )S2 ⊕ (V )S3 ⊕ · · · ! Remark: FX is functorial in X, giving F :C D David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 3 / 22 ! Adjoint functors Definition An adjunction F a G between categories C and D is F :C D and G :D C with bijections φX;Y : homD(FX; Y ) ' homC(X!; GY ) ! natural in X 2 C and Y 2 D. U :D C forgetful functor !η φX;Y : hom(FX; Y ) 3 g 7 Ug ◦ ηX 2 hom(X; UY ) X X UFX 9! Uf bijective by free property f ! UY = F : C ? D : U David Kern (LAREMA — SemDoc) Enrichments for higher categories) 12th October 2018 4 / 22 Notation: η and " are called the unit and counit of the adjunction " and η determine the adjunction φ, that is: F and G are adjoint iff there are ": FG idD and η: idC GF with the compatibility condition F ) )G G Fη F ηG C FGF D = C idF D and D GFG C = D idG C "F G" F G F G The algebra of adjunctions ' φX;FX : homD(FX; FX) − homC(X; GFX) maps idFX 7 ηX : X GFX I By functoriality of φ = natural transformation η: idC GF -1 ! ' ! ! Similarly, by φGY ;Y : hom(GY ; GY ) − hom(FGY ; Y ), get ": FG idD ) ) ! ) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 5 / 22 The algebra of adjunctions ' φX;FX : homD(FX; FX) − homC(X; GFX) maps idFX 7 ηX : X GFX I By functoriality of φ = natural transformation η: idC GF -1 ! ' ! ! Similarly, by φGY ;Y : hom(GY ; GY ) − hom(FGY ; Y ), get ": FG idD Notation: η and " are called) the unit and counit of the adjunction) ! ) " and η determine the adjunction φ, that is: F and G are adjoint iff there are ": FG idD and η: idC GF with the compatibility condition F ) )G G Fη F ηG C FGF D = C idF D and D GFG C = D idG C "F G" F G F G David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 5 / 22 Similarly, FηG is a comultiplication FG (FG)◦2 on the counital FG on D = GF and FG are an algebra and a cogebra for the composition products on ) EndoFun(C) and EndoFun(D) ) From adjunctions to monads Using the counit, we get a natural transformation µ := G"F :(GF) ◦ (GF) = G ◦ (FG) ◦ F GF. ) I The endofunctor GF on C has a “multiplication” µ:(GF)◦2 GF and a “unit” η: idC GF (associative and unital by compatibility condition) ) U :D C Example: ) forgetful functor UF creates a free object and forgets the added structure µ remembers reduction! along this structure (e.g. F = k[-]; µ is linear dependence) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 6 / 22 From adjunctions to monads Using the counit, we get a natural transformation µ := G"F :(GF) ◦ (GF) = G ◦ (FG) ◦ F GF. ) I The endofunctor GF on C has a “multiplication” µ:(GF)◦2 GF and a “unit” η: idC GF (associative and unital by compatibility condition) ) U :D C Example: ) forgetful functor UF creates a free object and forgets the added structure µ remembers reduction! along this structure (e.g. F = k[-]; µ is linear dependence) Similarly, FηG is a comultiplication FG (FG)◦2 on the counital FG on D = GF and FG are an algebra and a cogebra for the composition products on ) EndoFun(C) and EndoFun(D) David Kern)(LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 6 / 22 Sommaire - Section 2: Monoids and enrichments 1 The algebraic structure of free objects 2 Monoids and enrichments Monoidal and enriched categories Monads 3 Categories up to homotopy David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 7 / 22 Sommaire - Section 2: Monoids and enrichments 1 The algebraic structure of free objects 2 Monoids and enrichments Monoidal and enriched categories Monads 3 Categories up to homotopy David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 8 / 22 Examples I (ModR ; ⊗R ; R) I (EndoFun(C); ◦; idC) is strict monoidal (strictly associative) I Any category with finite products1 ( is the final object) Monoidal categories Definition A monoidal structure on a category V is a bifunctor ⊗:V × V V and a choice of object 1 2 V, with natural transformations expressing associativity and unitality ! ' I A braiding is a natural isomorphism τX;Y : X ⊗ Y − Y ⊗ X -1 I Symmetric structure if τX;Y = τY ;X ! ' A monoidal functor (V; ⊗; 1) (N; ; I) is F :V N with F(X ⊗ Y ) − FX FY ' (natural) and F(1) − I ! ! ! ! David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 I (EndoFun(C); ◦; idC) is strict monoidal (strictly associative) I Any category with finite products1 ( is the final object) Monoidal categories Definition A monoidal structure on a category V is a bifunctor ⊗:V × V V and a choice of object 1 2 V, with natural transformations expressing associativity and unitality ! ' I A braiding is a natural isomorphism τX;Y : X ⊗ Y − Y ⊗ X -1 I Symmetric structure if τX;Y = τY ;X ! ' A monoidal functor (V; ⊗; 1) (N; ; I) is F :V N with F(X ⊗ Y ) − FX FY ' (natural) and F(1) − I Examples ! ! ! ! I (ModR ; ⊗R ; R) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 I Any category with finite products1 ( is the final object) Monoidal categories Definition A monoidal structure on a category V is a bifunctor ⊗:V × V V and a choice of object 1 2 V, with natural transformations expressing associativity and unitality ! ' I A braiding is a natural isomorphism τX;Y : X ⊗ Y − Y ⊗ X -1 I Symmetric structure if τX;Y = τY ;X ! ' A monoidal functor (V; ⊗; 1) (N; ; I) is F :V N with F(X ⊗ Y ) − FX FY ' (natural) and F(1) − I Examples ! ! ! ! I (ModR ; ⊗R ; R) I (EndoFun(C); ◦; idC) is strict monoidal (strictly associative) David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 Monoidal categories