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

This leads to some questions.

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

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

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 1) 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 $\mathfrak{C}[\Lambda^n_i]$.

So my question is: what's the general formula for the hom's of a (co)limit of $\mathscr{V}$-enriched categories?

Thanks in advance.

  • 1
    $\begingroup$ There is no general formula. Even colimits of ordinary categories are hard. Incidentally, (a) is not always the cartesian product. $\endgroup$ – Zhen Lin Jul 31 '15 at 12:56
  • $\begingroup$ @ZhenLin Well, if I can compute some simple cases, I think it's possible to write down a general formula. Anyway, do you know how to compute a) and c) ? Is 1 true? Any further ideas? $\endgroup$ – user40276 Jul 31 '15 at 13:05
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    $\begingroup$ Coproducts are easy, of course. But as you say, coequalisers are hard, and if you can't figure that out (even for $\mathcal{V} = \mathbf{Set}$), you don't have a general formula. $\endgroup$ – Zhen Lin Jul 31 '15 at 14:11
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    $\begingroup$ As for whether $\mathcal{V}\textbf{-Cat}$ is cocomplete – that's true, but it's hard. See [Wolff, $\mathcal{V}\textbf{-Cat}$ and $\mathcal{V}\textbf{-Graph}$]. $\endgroup$ – Zhen Lin Jul 31 '15 at 14:15
  • $\begingroup$ @ZhenLin Thanks for the reference, but what about products? Is 1 true? Why the naive product category (a) is not the product (what's the correct product)? $\endgroup$ – user40276 Aug 1 '15 at 5:41

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