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)


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.

Thanks in advance.

  • 3
    $\begingroup$ It seems to me that Zhen Lin provided a highly relevant reference, and somehow the discussion got sidetracked by a discussion of products, not colimits. Did you work through the paper by Wolff? Here's a direct link: sciencedirect.com/science/article/pii/0022404974900188 In practice, one thing that's not easy to compute is coequalizers (as you note). $\endgroup$
    – Todd Trimble
    Aug 24, 2015 at 0:08
  • 2
    $\begingroup$ There is a paper Rigidification of Quasi-categories by Dugger and Spivak devoted to the problem of computing $\mathfrak{C}[X]$. Depending on what $X$s you are interested in, you may find fairly explicit answers there. $\endgroup$ Aug 24, 2015 at 3:26
  • 2
    $\begingroup$ @user40276 for future reference, anything four years old or older on ScienceDirect in most if not all pure mathematics journals is free to read. Part of Elsevier trying to win back the hearts of mathematicians (you'll notice it's not true for as many journal in other fields, like theoretical physics, that have similar half-life for their papers). $\endgroup$
    – David Roberts
    Aug 24, 2015 at 7:30
  • 2
    $\begingroup$ There is a paper by Kelly and Lack called "Cat$_V$ is locally presentable or locally bounded if $V$ is so" This gives you a condition under which the answer to (2) is yes. Note that asking $V$ to be locally presentable is more than asking it to be cocomplete, but often satisfied in practice. See the book of Adamek and Rosicky for more information. $\endgroup$ Aug 24, 2015 at 12:09
  • 1
    $\begingroup$ Giovanni Caviglia has a paper on arxiv where he computes a bunch of colimits in Cat$_V$ as part of putting a model structure on it. See also Berger-Moerdijk "On the homotopy theory of enriched categories" which Giovanni was generalizing. You ask in your last paragraph for examples and this contains some. The goal was to compute a pushout in $Cat_V$ as a transfinite composition of pushouts of things in $V$ (well, technically several copies of $V$). I think this would be a good example to work through. $\endgroup$ Aug 24, 2015 at 12:15


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy