nLab
category of V-enriched categories

Contents

Idea

For 𝒱\mathcal{V} a suitable context of enrichment there is a 22-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 a cosmos with underlying category 𝒱 0\mathcal{V}_0

Examples

Structure of the category of 𝒱\mathcal{V}-enriched categories for various contexts

  • If 𝒱\mathcal{V} is a category V 0V_0 equipped with a monoidal structure, then 𝒱\mathcal{V}Cat has a unit object ℐ\mathcal{I}, and an association sending every 11-morphism 𝒜→Fℬ\mathcal{A}\stackrel{F}{\to}\mathcal{B} to the a 11-morphism in the lax slice 2-category Cat//V 0\mathbf{Cat}//V_0:
    [ℐ,𝒜] op×[ℐ,𝒜] →[ℐ,F] op×[ℐ,F] [ℐ,ℬ] op×[ℐ,ℬ] 𝒜(−,−)↘ F¯⇗ ↙ℬ(−,−) V 0 \array{ [\mathcal{I},\mathcal{A}]^{op}\times[\mathcal{I},\mathcal{A}]&\stackrel{[\mathcal{I},F]^{op}\times[\mathcal{I},F]}{\to}&[\mathcal{I},\mathcal{B}]^{op}\times[\mathcal{I},\mathcal{B}]\\ \mathcal{A}(-,-)\searrow&\bar{F}\neArrow&\swarrow\mathcal{B}(-,-)\\ &V_0 }

    where

  1. [ℐ,𝒜][\mathcal{I},\mathcal{A}] is really just the underlying category with hom-collections given by A 0(A,B)=V 0(I,𝒜(A,B))A_0(A,B)=V_0(I,\mathcal{A}(A,B)).

  2. 𝒜(−,−)\mathcal{A}(-,-) is the fully faithful two-variable hom-functor from A 0 op×A 0→V 0A_0^{op}\times A_0\to V_0, with 𝒜(f,g)\mathcal{A}(f,g) defined as the composite 𝒜(B,C)→l −1r −1I⊗𝒜(B,C)⊗I→f⊗id⊗g𝒜(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 0V_0

  3. [ℐ,F][\mathcal{I},F] is the functor F 0F_0 from A 0A_0 to B 0B_0 underlying the enriched functor FF. This is defined by letting F 0fF_0f be the composite I→f𝒜(A,B)→F A,Bℬ(FA,FB)I\stackrel{f}{\to}\mathcal{A}(A,B)\stackrel{F_{A,B}}{\to}\mathcal{B}(FA,FB) where F A,BF_{A,B} is the family of morphisms in V 0V_0 defining the enriched functor FF.

  4. The natural transformation F¯:catA(−,−)→catB(F 0−,F 0−)\bar F\colon\cat A(-,-)\to\cat B(F_0-,F_0-) has for its components exactly the maps F A,BF_{A,B} above: i.e. F¯ A,B=F A,B\bar F_{A,B}=F_{A,B}.

  5. In particular, we recover j A:I→𝒜(A,A)j_A\colon I\to\mathcal{A}(A,A) as A¯\bar{A} when we consider AA an enriched functor of [ℐ,𝒜][\mathcal{I},\mathcal{A}]. The unit identities F A,A∘j A=j FAF_{A,A}\circ j_A=j_{FA} then hold automatically.

This is in fact a 11-functor from 𝒱\mathcal{V}Cat to Cat//V 0\mathbf{Cat}//V_0. It is not a 22-functor since if α\alpha is an enriched natural transformations from an enriched functor FF to an enriched functor GG, then α 0\alpha_0, the underlying natural transformation from F 0F_0 to G 0G_0 satisfies ℬ(id,α 0)F¯=ℬ(α 0,id)G¯\mathcal{B}(id,\alpha_0)\bar F=\mathcal{B}(\alpha_0,id)\bar G instead of ℬ(α 0 op,α 0)∘F¯=G¯\mathcal{B}(\alpha_0^{op},\alpha_0)\circ\bar F=\bar G that the definition of Cat//V 0\mathbf{Cat}//V_0 requires. Nevertheless, if we do define the codomain of the association to have only the objects and 11-morphisms as in the above triangles, and define 22-morphisms to satisfy the correct naturality condition (rather than the lax slice 2-category condition), then we still have a 22-category and the association is a 22-functor that is fully faithful on 22-morphisms, and faithful on 11-morphisms.

Two remarks are in order. First, something is deeply wrong here since this 22-category we have had to define arises out of the desire to make things easy and understandable, as opposed to natural. Second, the failure of the association to be faithful on 11-morphisms means that an enrichment of a functor F 0:A 0→B 0F_0\colon A_0\to B_0 consists of more than just giving a natural transformation from 𝒜(−,−)\mathcal{A}(-,-) to ℬ(F 0−,F 0−)\mathcal{B}(F_0-,F_0-). But more significantly, this seems to suggest that the monoidal structure on V 0V_0 is not captured by any structure on the 22-category 𝒱\mathcal{V}Cat. This state of affairs is remedied in any closed category context.

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

category: category

Revised on April 15, 2014 04:02:45 by Toby Bartels (98.16.173.18)