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I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be combinatorial, the category of small categories must be locally presentable. Is this true or false?

Intuitively, every small category should be $\lambda$-small, for $\lambda$ an upper bound of the set of objects and all morphism sets. This leads me to think that the answer is 'yes', but I'm not sure since I don't really know how are colimits in the category of small categories.

PS If your answer has a nice description of how to compute colimits in the category of small categories (or just push-outs and filtered colimits) I will appreciate it very much.

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As Mike says, it's locally finitely presentable because $Cat$ is (equivalent to) the category of models of a finite limit sketch. A quick way of describing this is to say that a category is (or is the nerve of) a simplicial set such that certain squares in the combinatorial definition of simplicial set (via faces, degeneracies) have to be pullbacks, as stated more precisely in the sixth proposition of this section of the nLab article on nerves. This clearly gives the notion of category in terms of a finite limit sketch. (In fact, this will work with truncated simplicial sets $C$, involving only $C_j$ up to dimension $j = 3$ if I'm not mistaken.)

In other language, every category is a filtered colimit of finitely presented categories (meaning categories presented as coequalizers of pairs

$$R \rightrightarrows S$$

where $R, S$ are free categories on finite graphs). Colimits in $Cat$ can be a little bit unpleasant; coproducts are of course unproblematic, but coequalizers are another matter. An explicit construction of coequalizers can be found in this paper. (I mean to come back to this answer and improve it if possible.)

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    $\begingroup$ I don't think that colimits of categories are really complicated when one has understood colimits of monoids (since categories are just monoidoids). On the other hand, 2-colimits are more complicated. $\endgroup$
    – Martin Brandenburg
    Commented Dec 31, 2011 at 11:29
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    $\begingroup$ @Martin: The main difference to monoids is that a coequalizer in $Cat$ which identifies different objects can introduce new loops. One can witness this behavior by considering the coequalizer of the two object inclusions $1 \to 2$ into the generic arrow; the coequalizer is then $\mathbb{N}$ seen as a monoid. This doesn't come up in $Mon$. It's not that coequalizers in $Cat$ are ferociously complicated, but they are a little fiddly because of looping, and the sort of niceties available for $Mon$ (e.g., $U: Mon \to Set$ preserves reflexive coequalizers) are not quite as nice here. $\endgroup$
    – Todd Trimble
    Commented Dec 31, 2011 at 13:41
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Yes. The theory of categories is defined by a finite limit sketch, i.e. an essentially algebraic theory, and the category of models of any finite limit sketch is locally finitely presentable. See, for instance, section 3.D of Adamek & Rosicky.

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  • $\begingroup$ @Mike Shulman. Is interesting also the characterization of JW Gray n the article "The existence and construction of Lax limits" (see rows 4,5 p.280): numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_3_277_0 $\endgroup$
    – Buschi Sergio
    Commented Dec 28, 2011 at 19:46
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    $\begingroup$ filtred colimits is just the category with object set the filtred colimits os object sets, and the some for the morphisms (just because filtred colimits commuting by finite limits) $\endgroup$
    – Buschi Sergio
    Commented Dec 28, 2011 at 21:22
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    $\begingroup$ Yes, indeed: filtered colimits in any locally finitely presentable category are just colimits of the underlying sets. $\endgroup$
    – Mike Shulman
    Commented Dec 30, 2011 at 4:07
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About the construction of coequalizer in $Cat$: I call a graph unitary if for each objet $X$ is assigned the unit arrow $1_X: X\to X$) and unitary-graph morphism is a units-preserving morphims.

Let $F, G: A\to B$ two functors, consider these as morphisms of unitary graphs, its (unitary graphs) coker $B\to C$ has for object (resp. arrows) the coker of the objects (arrows) maps: $|F|_0, |G|_0$ (resp. $|F|_1, |G|_1$).

Let $F(C)$ the free category on $C$ (we can take it with the same units of $C$), and $Q:=C/\sim$ the quotient of $F(C)$ with the congruence generated by $g\cdot f\sim g\circ f$ (where "$\circ$" is the composition of $B$ and "$\cdot$" the free composition of $C$), then the composition of graph morphisms $q: B\to C\to F(C)\to Q$ is also a funtor $q: B\to Q$, and represent the coker of $F, G$.

I seem it work well (take with caution).

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