nLab category of V-enriched categories

Idea

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

• objects are $V$-enriched categories;

• morphisms are $V$-enriched functors; and

• $2$-morphisms are $V$-enriched natural transformations.

Sometimes one also considers $\mathcal{V} Cat$ as a mere category by dropping the $2$-morphisms (and using enriched strict categories).

Possible Contexts

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

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

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

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

Examples

• For $\mathcal{V} = ($Set$, \times)$, $\mathcal{V}Cat \simeq$ Cat, the $2$-category of locally small categories.

• For $\mathcal{V} = ($Cat$, \times)$, $\mathcal{V}Cat \simeq$ Str2Cat, the $2$-category of strict 2-categories.

Extra structure

Monoidal structure

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

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

[Kelly (1982), §2.3]

Involutions

• If $\mathcal{V}$ is a category $\mathcal{V}_0$ equipped with a monoidal structure, then $\mathcal{V}$Cat has a unit object $\mathcal{I}$, and a designated lax natural transformation $[\mathcal{I},-]^{op}\stackrel{{}^-_0L}{\Rightarrow}[[\mathcal{I},-],\mathcal{V}_0]\colon\mathcal{V}$Cat$\to$Cat, where the former is a $2$-functor flipping $2$-morphisms, and the latter is a $2$-functor flipping $1$-morphisms (c.f. contravariant functor). For the sake of simplicity, we note that $[\mathcal{I},-]$ is simply the forgetful $2$-functor from $\mathcal{V}$Cat to Cat, and hence abbreviate it as $(-)_0$. Then the above lax natural transformation is given by the following data:

1. For every object (i.e. $\mathcal{V}$-enriched category) $\mathcal{A}$ of $\mathcal{V}$Cat, we have to give a functor $\mathcal{A}_0^{op}\stackrel{{}^{\mathcal{A}}_0L}{\rightarrow}[\mathcal{A}_0,\mathcal{V}_0]$. By the cartesian closed structure of the $1$-category Cat, we define these to be the hom-functors $\mathcal{A}_0^{op}\times\mathcal{A}\stackrel{\mathcal{A}(-,-)}{\rightarrow}\mathcal{V}_0$ which are defined in terms of the monoidal structure $\mathcal{V}$ and the $\mathcal{V}$-enrichment data of $\mathcal{A}$ by setting $\mathcal{A}(B,C)\stackrel{\mathcal{A}(f,g)}{\rightarrow}\mathcal{A}(A,D)$ to be the composites $\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}{\to}\mathcal{A}(A,D)$ in $\mathcal{V}_0$.

2. For every $1$-morphism (i.e. $\mathcal{V}$-enriched functor}) $\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}$, we have to give a natural transformation ${}^F_0L$:

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

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

3. The lax naturality of ${}^-_0L$ says that for every $2$-morphism (i.e. a $\mathcal{V}$-enriched natural transformation) $F\stackrel{\alpha}{\Rightarrow} G\colon\mathcal{A}\to\mathcal{B}$ in $\mathcal{V}$Cat, the natural transformations ${}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ F_0^{op}$ and ${}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ G_0$ have to satisfy a compatibility condition with the natural transformations $G_0^{op}\stackrel{\alpha_0^{op}}{\Rightarrow}F_0^{op}$ and $[F_0,\mathcal{V}_0]\stackrel{[\alpha_0,\mathcal{V}_0]}{\Rightarrow}[G_0,\mathcal{V}_0]$. Explicitly, the condition is that the composite natural transformation ${}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\stackrel{[\alpha_0,\mathcal{V}_0].({}^{\mathcal{B}}_0L\circ F_0)}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]$ is the same as the composite natural transformation ${}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ G_0^{op}\stackrel{([G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L).\alpha_0^{op}}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]$. Unraveling the condition leaves us with the requirement that for every pair of objects $A,B$ of $\mathcal{A}_0$ the following diagram in $\mathcal{V}_0$ must commute:

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

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

• Supposing that $\mathcal{V}_0$ was a self-enriched category, i.e. isomorphic to the underlying category $\mathcal{V}^e_0$ of a $\mathcal{V}$-enriched category $\mathcal{V}^e$, then it is natural to require that the above lax natural transformation $[\mathcal{I},-]\stackrel{{}^-_0L}{\Rightarrow}[(-)_0,\mathcal{V}_0]$ is in fact the whiskering of a lax natural transformation $[\mathcal{I},-]\stackrel{{}^-L}{\Rightarrow}[-,\mathcal{V}^e]$ with the forgetful $2$-functor $(-)_0=[\mathcal{I},-]$. Such a lax natural transformation should give us most (if not all) of the closed structure on $\mathcal{V}^e_0\cong\mathcal{V}_0$…

…

Related concepts

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

• Pos

• Set

  • Rel

• Grpd, ∞Grpd

• Cat

  • Ho(Cat)

  • AccCat

  • PrCat

  • LexCat

  • MonCat

  • VCat

  • CatAdj

  • Prof

  • Operad

• 2Cat

• ModCat, CombModCat

• (∞,1)Cat

  • Pr(∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

References

• 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