nLab category of V-enriched categories




For 𝒱\mathcal{V} a closed and complete cosmos for enrichment there is a 2 2 -category 𝒱Cat\mathcal{V} Cat whose

Sometimes one also considers 𝒱Cat\mathcal{V} Cat as a mere category by dropping the 22-morphisms (and using enriched strict categories).

Possible Contexts

  • 𝒱\mathcal{V} can be a monoidal category with underlying category 𝒱 0\mathcal{V}_0

  • 𝒱\mathcal{V} can be a closed category with underlying category 𝒱 0\mathcal{V}_0

  • 𝒱\mathcal{V} can be a multicategory with underlying category 𝒱 0\mathcal{V}_0

  • 𝒱\mathcal{V} can be a cosmos with underlying category 𝒱 0\mathcal{V}_0


Extra structure

Monoidal structure

If 𝒱\mathcal{V} is symmetric monoidal then 𝒱Cat\mathcal{V}Cat becomes itself a (very large) monoidal category with tensor product given by forming enriched product categories.

If 𝒱\mathcal{V} is in addition closed and complete, then 𝒱Cat\mathcal{V}Cat becomes itself a closed monoidal category with internal hom given by forming enriched functor categories.

[Kelly (1982), §2.3]


  • If 𝒱\mathcal{V} is a category 𝒱 0\mathcal{V}_0 equipped with a monoidal structure, then 𝒱\mathcal{V}Cat has a unit object \mathcal{I}, and a designated lax natural transformation [,] op 0 L[[,],𝒱 0]:𝒱[\mathcal{I},-]^{op}\stackrel{{}^-_0L}{\Rightarrow}[[\mathcal{I},-],\mathcal{V}_0]\colon\mathcal{V}Cat\toCat, where the former is a 22-functor flipping 22-morphisms, and the latter is a 22-functor flipping 11-morphisms (c.f. contravariant functor). For the sake of simplicity, we note that [,][\mathcal{I},-] is simply the forgetful 22-functor from 𝒱\mathcal{V}Cat to Cat, and hence abbreviate it as () 0(-)_0. Then the above lax natural transformation is given by the following data:
  1. For every object (i.e. 𝒱\mathcal{V}-enriched category) 𝒜\mathcal{A} of 𝒱\mathcal{V}Cat, we have to give a functor 𝒜 0 op 0 𝒜L[𝒜 0,𝒱 0]\mathcal{A}_0^{op}\stackrel{{}^{\mathcal{A}}_0L}{\rightarrow}[\mathcal{A}_0,\mathcal{V}_0]. By the cartesian closed structure of the 11-category Cat, we define these to be the hom-functors 𝒜 0 op×𝒜𝒜(,)𝒱 0\mathcal{A}_0^{op}\times\mathcal{A}\stackrel{\mathcal{A}(-,-)}{\rightarrow}\mathcal{V}_0 which are defined in terms of the monoidal structure 𝒱\mathcal{V} and the 𝒱\mathcal{V}-enrichment data of 𝒜\mathcal{A} by setting 𝒜(B,C)𝒜(f,g)𝒜(A,D)\mathcal{A}(B,C)\stackrel{\mathcal{A}(f,g)}{\rightarrow}\mathcal{A}(A,D) to be the composites 𝒜(B,C)l 1r 1I𝒜(B,C)Ifidg𝒜(C,D)𝒜(B,C)𝒜(A,B)( 𝒜) 2𝒜(A,D)\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}{\to}\mathcal{A}(A,D) in 𝒱 0\mathcal{V}_0.

  2. For every 11-morphism (i.e. 𝒱\mathcal{V}-enriched functor}) 𝒜F\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}, we have to give a natural transformation 0 FL{}^F_0L:

    𝒜 0 op F 0 op 0 op 0 𝒜L 0 FL L [𝒜 0,𝒱 0] [F 0,𝒱 0] [ 0,𝒱 0] \array{ \mathcal{A}_0^{op}&\stackrel{F_0^{op}}{\rightarrow}&\mathcal{B}_0^{op}\\ {}_0^{\mathcal{A}}L\downarrow&\stackrel{{}^F_0L}{\Rightarrow}&\downarrow{}^{\mathcal{B}}L\\ [\mathcal{A}_0,\mathcal{V}_0]&\stackrel{[F_0,\mathcal{V}_0]}{\leftarrow}&[\mathcal{B}_0,\mathcal{V}_0] }

    Since we have defined 0 𝒜L{}_0^{\mathcal{A}}L to be the hom-functor 𝒜(,)\mathcal{A}(-,-), to give a natural transformation 0 FL{}^F_0L is to give a natural transformation 𝒜(,)(F 0 op,F 0):𝒜 0 op×𝒜𝒱 0\mathcal{A}(-,-)\Rightarrow\mathcal{B}(F_0^{op}-,F_0-)\colon\mathcal{A}_0^{op}\times\mathcal{A}\to\mathcal{V}_0. We thus define the ob(𝒜 0)ob(\mathcal{A}_0)-indexed family of morphisms 𝒜(A,B) 0 FLA,B(F 0A,F 0B)\mathcal{A}(A,B)\stackrel{{}_0^FL_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B) in 𝒱 0\mathcal{V}_0 to be simply the family of morphisms 𝒜(A,B)F A,B(F 0A,F 0B)\mathcal{A}(A,B)\stackrel{F_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B) in 𝒱 0\mathcal{V}_0 defining the 𝒱\mathcal{V}-enriched functor 𝒜F\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}.

  3. The lax naturality of 0 L{}^-_0L says that for every 22-morphism (i.e. a 𝒱\mathcal{V}-enriched natural transformation) FαG:𝒜F\stackrel{\alpha}{\Rightarrow} G\colon\mathcal{A}\to\mathcal{B} in 𝒱\mathcal{V}Cat, the natural transformations 0 𝒜L 0 FL[F 0,𝒱 0] 0 LF 0 op{}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ F_0^{op} and 0 𝒜L 0 GL[G 0,𝒱 0] 0 LG 0{}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ G_0 have to satisfy a compatibility condition with the natural transformations G 0 opα 0 opF 0 opG_0^{op}\stackrel{\alpha_0^{op}}{\Rightarrow}F_0^{op} and [F 0,𝒱 0][α 0,𝒱 0][G 0,𝒱 0][F_0,\mathcal{V}_0]\stackrel{[\alpha_0,\mathcal{V}_0]}{\Rightarrow}[G_0,\mathcal{V}_0]. Explicitly, the condition is that the composite natural transformation 0 𝒜L 0 FL[F 0,𝒱 0] 0 LF 0 op[α 0,𝒱 0].( 0 LF 0)[G 0,V 0] 0 LF 0 op:𝒜 0 op[𝒜 0,𝒱 0]{}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\stackrel{[\alpha_0,\mathcal{V}_0].({}^{\mathcal{B}}_0L\circ F_0)}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0] is the same as the composite natural transformation 0 𝒜L 0 GL[G 0,𝒱 0] 0 LG 0 op([G 0,𝒱 0] 0 L).α 0 op[G 0,V 0] 0 LF 0 op:𝒜 0 op[𝒜 0,𝒱 0]{}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ G_0^{op}\stackrel{([G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L).\alpha_0^{op}}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]. Unraveling the condition leaves us with the requirement that for every pair of objects A,BA,B of 𝒜 0\mathcal{A}_0 the following diagram in 𝒱 0\mathcal{V}_0 must commute:

    𝒜(A,B) F A,B (F 0A,F 0B) G A,B (F 0A,(α 0) B) (G 0A,G 0B) ((α 0) A,G 0B) (F 0A,G 0B) \array{ \mathcal{A}(A,B)&\stackrel{F_{A,B}}{\rightarrow}&\mathcal{B}(F_0A,F_0B)\\ G_{A,B}\downarrow&&\downarrow\mathcal{B}(F_0A,(\alpha_0)_{B})\\ \mathcal{B}(G_0A,G_0B)&\stackrel{\mathcal{B}((\alpha_0)_A,G_0B)}{\rightarrow}&\mathcal{B}(F_0A,G_0B) }

    But a 𝒱\mathcal{V}-enriched natural transformation α\alpha is by definition a collection of morphisms α 0\alpha_0 in 0\mathcal{B}_0 such that the above diagram commutes.

  • Supposing that 𝒱 0\mathcal{V}_0 was a self-enriched category, i.e. isomorphic to the underlying category 𝒱 0 e\mathcal{V}^e_0 of a 𝒱\mathcal{V}-enriched category 𝒱 e\mathcal{V}^e, then it is natural to require that the above lax natural transformation [,] 0 L[() 0,𝒱 0][\mathcal{I},-]\stackrel{{}^-_0L}{\Rightarrow}[(-)_0,\mathcal{V}_0] is in fact the whiskering of a lax natural transformation [,] L[,𝒱 e][\mathcal{I},-]\stackrel{{}^-L}{\Rightarrow}[-,\mathcal{V}^e] with the forgetful 22-functor () 0=[,](-)_0=[\mathcal{I},-]. Such a lax natural transformation should give us most (if not all) of the closed structure on 𝒱 0 e𝒱 0\mathcal{V}^e_0\cong\mathcal{V}_0

(n+1,r+1)(n+1,r+1)-categories of (n,r)-categories


  • Max Kelly, Chapter 1 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]
category: category

Last revised on March 10, 2024 at 11:23:10. See the history of this page for a list of all contributions to it.