Comparing Category Theory and Model Theory (with examples from Group Theory).

The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple:

(Q1)If someone write "the category of finite groups" what are the objects of this category? Surely, $\mathbb{Z}_6$ is in this category. What about the other instances of $\mathbb{Z}_6?$ What prohibits to add extra copies of $\mathbb{Z}_6$ into objects? There is no "equality" of objects on "objects" other than the "standard equality class" of the object which models its theory. But the theory of an "object" is not included in the category. Let us formalize the theory of $\mathbb{Z}_6$ groups by adding extra conditions to the axioms of groups so that if a usual group satisfies these extra conditions then its isomorphic to $\mathbb{Z}_6.$ Now, is every model of these abstract conditions in our category? Or not? Do you include model theoretic semantic into category or not?

What does category theory give different than the model theory then?

(Q2) If someone chooses objects up to isomorphisms then why a "functor" (should be morphism) is called isomorphism? I saw somewhere that there is a mono which is not a monomorphism in the usual sense. So are there , for instance finite groups, which are isomorphic in the categoric sense but not isomorphic in the normal sense? Please note that finite group category is just an example, you are welcome to add interesting examples.

Thank you.

Edit: "functor" of Q2 should be "morphism".

Solution 1:

The natural notion of isomorphism of categories is not isomorphism (the existence of two functors which are inverses) but equivalence. A good intuition for equivalence is that it behaves like homotopy equivalence of spaces. In particular, just as many different spaces with different sets of points (in particular, sets of points with different cardinalities) can be homotopy equivalent, many categories with different sets of objects can be equivalent. Thus in the category of categories, "the set of objects" is not well-defined as it is not invariant under equivalence, just as in the homotopy category, "the set of points" is not well-defined as it is not invariant under homotopy equivalence.

When someone says "the category of groups," they are refraining from specifying a particular set of objects and morphisms because any reasonable choice gives the same category up to equivalence. For example, you can take

• The category whose objects are sets $G$ (say in ZFC) equipped with maps such that etc. and whose morphisms are group homomorphisms, or
• The category whose objects are, roughly speaking, isomorphism classes of groups and whose morphisms are group homomorphisms

and the corresponding categories are equivalent; the latter is just the skeleton of the former.

Solution 2:

Necessary context: In my opinion, it is not obligatory to base mathematics on set theory, although, of course, comparisons and discussions are interesting. Similarly, I do not think it is obligatory to compare categorical "foundations" with set-theoretic "foundations". At one end, Grothendieck found it necessary to postulate the existence of very many large cardinals (see Weibel's "Homological..."). At another end, in fact, by this point I definitely do not think that set theory reflects the practice of mathematics. Exaggerated category theory does not, either. I do not worry about "the category of sets", either.

The category of finite groups includes all finite groups... :) Yes, it includes many different copies of the dihedral group $D_4$, including different versions painted blue, red, or yellow. Or copies which are the same color, but are "distinct".

After some years/decades, I worry much less about the alleged set of sets that don't include themselves. I do not want a list of prohibitions that also prevents me from "forming" this set. I also do not want a prohibition against sharp knives when I am cutting up vegetables, even tho' I may cut myself. I do not want a prohibition against water, even tho' I may drown myself in my own bathtub by lying face down and breathing it in. Perversities.

So, yes, there are all those different copies of the same isomorphy-class of a group. Yes, if one thinks of ways to squeeze out a paradoxical-seeming something, probably one can. But why should one? :)