# How to compute (co)limits of enriched categories?

I’ve asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I’m posting it here.

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories enriched over $\mathscr{V}$. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$.

This leads to some questions.

1)When $\mathbf{Cat}_{\mathscr{V}}$ is (co)complete?

Or a more, specific question

2) If $\mathscr{V}$ is (co)complete, is it true that $\mathbf{Cat}_{\mathscr{V}}$ is (co)complete?

Now, let’s forget 1) and 2) for a moment and get back to the main question. To simplify, let’s start with simple “computations” (which I’m not sure whether they’re correct)

3)

a) The product of two $\mathscr{V}$-enriched categories $\mathscr{C} \times \mathscr{D}$ is given on objects by pairs $(c, d)$ and on morphisms by $\mathscr{C} \times \mathscr{D} ((c, d), (c’, d’)) = \mathscr{C}(c, c’) \otimes \mathscr{D} (d, d’)$

b) The coproduct of two $\mathscr{V}$-enriched categories $\mathscr{C} \coprod \mathscr{D}$ is given on objects by the disjoint union and the morphisms are the same ones.

c) The pushout of the category $\mathscr{C} = a \rightarrow b$ with the category $\mathscr{D} = c \rightarrow d$ along the inclusion of the terminal category $*$ (where $0 \in *$ goes to $b$ and $0$ goes to $c$ ) is the category $x \rightarrow y \rightarrow z$ such that $[x, y] = \mathscr{C}(a, b)$ and $[y, z] = \mathscr{D}(c, d)$. Furthermore $[x, z]$ should be intuitively a kind of pushout $\mathscr{C}(a, b) \coprod_X \mathscr{D}(c, d)$ (where $X$ is something that considers the endomorphisms of $b$ and $c$).

So even for the simple example (c) I could not compute the colimit. Clearly, the problem is how to compute the (co)equalizers. Maybe my failure is simply because 2) is not true.

The motivation for this question is that I was trying to compute $\mathfrak{C} [X] = \int^{[n] \in \Delta} X_n \otimes \mathfrak{C}[\Delta^n]$ explicitly for some common $X \in \mathbf{sSet}$, where $\mathfrak{C}[\Delta^n]$ is Cordier enrichment of $[n]$ by declaring $\mathfrak{C}[\Delta^n] (i, j) = (\Delta^1)^{j – i -1}$ (and empty when $i > j$). For instance, I could not compute $\mathfrak{C}[\Delta^2 \coprod_{\Delta^1} \Delta^2]$ (two triangles glued along an edge) and $\mathfrak{C}[\Lambda^n_i]$.

4) If $\mathbf{Cat}_{\mathscr{V}}$ is (co)complete, what’s the general formula for the hom objects in the (co)limit?

In my previous question, Zhen Lin said that writing down such formula is almost impossible, so I’m not expecting for a general formula. I just want to know how to compute some simple toy examples (so I can get the main idea and apply it in practice in more general cases). Furthermore I’m mainly interested in the case of topological and simplicial categories.