Meaning of the category of sets

Question

There are several set theories, is there a category of sets for each of them? For example, ZF-Set, ZFC-Set, NBG-Set, etc…

Answer 1

When your definition of a category starts like this

A category consists of a set $C_0$ of objects and for each $x,y\in C_0$ a set $C_1(x,y)$ together with...

and your meta-theory is one of the standard set theories like ZFC, then you will not be able to form a category $Set$ of all sets. This is because famously there can be no set $Set_0$ of all sets. There are several work-arounds.

• You can conservatively extend your meta-set-theory (for example ZFC) to one which can speak about classes and sets (for example NBG). You can then change your definition to "A category consists of a class $C_0$ of objects together with...". Since there is a class of all sets, you are able to form $Set$.

• You can form the small category $Set_\kappa$ of sets which are smaller than a certain cardinal number. The underlying set of that category can not consist of all sets whose cardinality is less than $\kappa$, since this is still a proper class, but it can be for example the set of all cardinals which are less than $\kappa$. The properties of the category $Set_\kappa$ dependend on the closure properties of $\kappa$ and the axioms of your meta-theory. For example when your meta-theory has choice, then all epimorphisms will be split in $Set_\kappa$. If you like to have a category $Set_\kappa$ which behaves as if it were the category of sets of ZFC, then you can add an extra axiom, namely the existence of Grothendieck universes, and let $\kappa$ be such a universe.

But if you want to study directly how the properties of a set theory influence the properties of "its category of sets", then you can also follow the following approach. You study the various material set theories (first-order single sorted theories whith a single relation symbol $\in$) and their models. You need a meta-theory for that, but this can be anything (although it need be sufficiently strong so that you can show the existence of some models). Then you can associate to a model $(V,\in)$ of say ZF a category $Set_{(V,\in)}$ of sets whose underlying set of objects is the underlying set $V$ of the model. The properties which all the categories $Set_{(V,\in)}$ share, where $(V,\in)$ ranges over the models of a particular set theory, will depend on the set theory in question.

The most detailed paper which compares material and structural set theories and the properties of their associated categories is the paper Stack Semantics by Mike Shulman. It is a tough read though.