# Class models in set theory and category theory

Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, groups as of models of group theory, topologies as of models of topology, and so on?

There is one big difference, and maybe the essence lies within: all "normal" models have to been based on sets, while categories may be based on proper classes.

Why would it be a problem for "normal" models of "normal" theories to be based on proper classes, but not for categories as models of category theory?

Side remark: Forgive me for talking at large: "Normal" models of "normal" theories are required to be based on sets, because only then one can apply the machinery of set theory. For categories one definitely wants to apply another machinery, the machinery of category theory. But if this machinery can handle class models in principle, why not apply this machinery on class models of "normal" theories, too - and in a direct way?

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@Qiaochu: I'll take your advice and will accept some answers. Thanks for the hint. – Hans Stricker Feb 21 '11 at 19:38
I love your trick of using <sup>, but may I suggest you use some other way of marking side remarks? – Mariano Suárez-Alvarez Feb 22 '11 at 3:00

I'll take the liberty of interpreting your question as follows. What are the benefits and/or disadvantages of considering categories as models for the first order theory $T=T_{Cat}$ of categories?
Considering categories as models of $T$ can be done. While it is not the commonest definition of category and it is certainly not emphasized that it is being done it certainly can be done. The article "Enlargement of categories" by Brunjes and Serpe is an example where the advantage of doing just that is clear. Namely, they extend the familiar notion of enlargement of sets in logic to enlargement of categories. Thus the evident advantage is that one can use all of the tools of logic to the study of categories.
The disadvantage with this approach is that it is limited to the study of categories whose objects form a set and thus many important categories (e.g., $Top$, $Set$, $Grp$, ...) are excluded from such a study.
Another important point to be made here is that while categories can be defined in terms of sets it is also possible to use categories instead of sets as foundations of mathematics. This is a very fruitful approach that would be somewhat limited if we demand to base category theory on sets. The name for a category that can be used instead of $Set$ for the purposed of logic is Topos (there are plenty of introductory texts about topos theory).