nLab
universe

Much of ordinary mathematics can be thought of as taking place inside the archetypical category Set of sets. In as far as an arbitrary category may be thought of as a generalization of $\mathrm{Set}$, a category is a universe inside which mathematics may take place.

Of course, without further assumptions on the category, there is in general very little math that can be formulated inside it, but a few extra properties and structures are usually necessary to provide something interesting. On the other hand, perhaps too much can be formulated; in the terminal category everything is trivial; the attitude to take is that any specific category is merely one model, while a general class of categories is a theory

For instance:

• If the category is equipped with the structure of a site, then geometrical notions, such as defining arrows locally on the domain with patching conditions (or more generally descent theory), exist inside it.
• If it is a finitely complete category, the existence of (finite) products and terminal objects means that varieties of algebras can be defined.
• If the category is a topos with a natural numbers object, then one can do a certain sort of “ordinary” mathematics inside it, specifically impredicative constructive mathematics without the fancier tools of model theory? or (ironically) category theory.
• Further extra conditions on the category, such as being a Boolean topos or being a superextensive site, bring the internal mathematics closer to that of $\mathrm{Set}$.

For more information, write it here or see foundations and internalization.