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This question may be too general. I am interested in references or proofs for special cases. I follow the definition in Chapter I of Kelly's Enriched Category.


Let $V$ be a monoidal category theory. $V$-Cat be the $2$-category of small $V$-enriched categories with morphisms $V$-morphisms.

When can we say that colimits/limits exist in $V$-Cat?


For example, I am interested in knowing whether the category of categories enrihced in simplicial sets admits colimits.

It seems necessary that $V$ to admit such limits/colimits. I do not know if this is true nor if it is sufficient.

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  • $\begingroup$ if you are intrested in categories enriched in simplicial sets, you could try to look in Luries books, he heavily looks at those simplicial cats $\endgroup$
    – Enkidu
    Sep 2, 2019 at 8:33
  • $\begingroup$ I am aware of this book, but I could not find where he addresses this. Do you know which page? $\endgroup$
    – W. Zhan
    Sep 3, 2019 at 7:01
  • $\begingroup$ it is books, and no sorrz, i do not know those 2000 pages by hard $\endgroup$
    – Enkidu
    Sep 4, 2019 at 8:01

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It's basically necessary that $V$ be complete and cocomplete (for instance, it's definitely necessary if $V$ admits any absorbing object for its monoidal product), so let's assume that.

For limits of $V$-categories to exist, this is sufficient-in fact only completeness is necessary. The limit of a diagram $D:J\to V-\mathrm{Cat}$ is the $V$-category with objects the limit of $\mathrm{ob} D(j)$ taken in $\mathrm{Set}$, while the morphisms $\mathrm{lim} D((a_j),(b_j))$ are just given by the limit $\mathrm{lim} D(j)(a_j,b_j)$ taken in $V$. The composition operation uses the canonical map $$\mathrm{lim} D(j)(a_j,b_j)\otimes \mathrm{lim} D(j)(b_j,c_j)\to \mathrm{lim}\left(D(j)(a_j,b_j)\otimes D(j)(b_j,c_j)\right).$$ Since this canonical map does not exist in the case of colimits, the latter are harder.

I don't have either a proof or a counterexample available in the case that $V$ is merely cocomplete, but this is easy to show cocompleteness in the case of simplicially enriched categories, which you say is your main interest. Indeed, it's immediate that the category of simplicial objects in categories, that is, the functor category $\mathrm{Cat}^{\Delta^{\mathrm{op}}}$, is cocomplete, since $\mathrm{Cat}$ is. And simplicially enriched categories may be identified with the subcategory of $\mathrm{Cat}^{\Delta^{\mathrm{op}}}$ such that the simplicial set $\mathrm{ob} C_\bullet$ of objects is discrete. This property is preserved by colimits, which gives colimits for simplicial categories.

This argument can be generalized by replacing $\Delta^{\mathrm{op}}$ with any small category $A$ and the condition that $\mathrm{ob} C_\bullet$ be discrete with the condition that it be a constant presheaf (of sets) on $A$. Then the point is that the "constant presheaf" functor is fully faithful with a right adjoint. For most other $V$ of interest, we will not be able to embed $\mathrm{Set}$ fully faithfully via a left adjoint, so we cannot reduce the question of colimits of categories enriched in $V$ to the easier question of colimits of categories internal to $V$ like this. Cocompleteness should still hold at least if $V$ is monoidally locally presentable, which will make $V$-categories accessible monoidal over $V$-graphs and thus locally presentable. But I don't have a reference for this argument.

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