Small categories and completeness

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(1) Can a small category be cocomplete? Meaning, have all small colimits? I'd be glad to see an example.

(2) Suppose $\mathcal C$ is a small category, with $Ob(\mathcal C)$ being of cardinality $\kappa$. May $\mathcal C$ have all small limits of cardinality $\leq \kappa$ ? Does this now allow examples which ware outruled in (1)?

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2 Answers

Small (co)complete categories are posets by a theorem of Freyd. If $C$ has all small coproducts and its class of morphisms $C_1$ is small, then $C(x,y)^{C_1}\simeq C(\coprod_{f\in C_1} x, y)\subseteq C_1$. If $C(x,y)>1$, then $C_1$ has a subset of strictly greater cardinality: contradiction.

A poset that has suprema and infima of all of its subsets is a complete category.

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I didn't know this theorem, it's really a beautiful piece of math. Probably, the same argument shows that you can only get (2) with posets. – Fernando Muro Oct 3 at 19:57

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You have to be careful about (2), since the OP seems to be measuring the size of a small category by the cardinality of its set of objects, not its set of morphisms. That measure might be a bad idea, by the way. – Todd Trimble Oct 3 at 20:44

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@Todd Trimble: since every object has an identity morphism, the set of morphisms has a greater cardinality than the set of objects. I fotgot to mention that. – Wouter Stekelenburg Oct 4 at 8:06
@Todd Trimble: What is OP? In addition, I see what you mean that measuring a category's size only by the cardinality of it's objects isn't such a good idea, as it might have a much greater cardinality of morphisms. Is there any other reason? – Shlomi A Oct 4 at 9:24

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I think that the best definition of cardinal for categories (and the one I've seen most used) is the cadinal of all morphisms in a skeleton (i.e. a subcategory with one object for each isomorphism class). – Fernando Muro Oct 4 at 11:14

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(1) Yes, the trivial (final) category, with only one object and one morphism (the identity in the unique object).

(2) If you only asked for colimits of cardinality $<\kappa$ then you would find many examples, e.g. finite abelian groups ($\kappa = \aleph_0$ here). If you insists in $\leq\kappa$ I think you face the same problem as above. Just think of vector spaces of dimension $<\kappa$. This category, up to isomorphism, has $\leq \kappa$ objects but it doesn't have colimits of sice $\leq \kappa$ since the coproduct of $\kappa$ copies of the ground field has dimension exactly $\kappa$.

I've been speaking about colimits in (2) instead of limits, which is what you ask for, so take opposite categories.

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Colimits is fine for me. Your example of vector spaces of dimension $< \kappa$ is clear. However, is there some more general argument that shows that any small category with $Ob(\mathcal C)$ of cardinality $\kappa$ cannot have all colimits of size $\kappa$? – Shlomi A Oct 3 at 22:14
@Shlomi, probably not, now I'd go to prove that any category with your conditions in (2) is a poset, have a try. – Fernando Muro Oct 4 at 6:00
@Fernando, thanks. It seems that a poset with $\kappa$ objects and a maximal element works if one requires all colimits of cardinality $\leq \kappa$. However, I haven't managed to convince myself that such a category (2) must be a poset. – Shlomi A Oct 4 at 10:37
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