Topos

The category \(\mathbf{Set}\) can be studied from the categorical point of view (see Categorical approach). It has finite limits, exponential objects, and a way to represent subobjects by characteristic functions. A topos is a category with analogous structure. Thus a topos should not be thought of as a single set; it is a category that behaves in many ways like the category of sets.

This point of view allows one to do set-like constructions in categories other than \(\mathbf{Set}\). Different toposes can therefore be treated as different mathematical universes, each with its own internal logic.