[0025] Definition 2·a (Topos).

Let $E$ be a geometric universe. A $E$-topos $X$ or a topos in $E$ is defined to be a geometric universe $\Sh{X}$ equipped with an algebraic morphism $X:E\to\Sh{X}$ whose gluing fibration $\GL{X}$ has a small separator (Definition 4.3·d [002H]). A morphism of $E$-topoi $f:{X}\to{Y}$ is defined by an algebraic morphism $\Sh{f}:\Sh{Y}\to\Sh{X}$ of geometric universes equipped with a 2-isomorphism $\varphi_{f}:X \cong Y;\Sh{f}$ in $[E,\Sh{X}]$.

It will often be convenient to display pasting diagrams as string diagrams, which we read in diagrammatic order from top left to bottom right. A string diagram views 2-cell as a “transformer” through which information passes along a wire; the input/output interface of such a transformer is written on the wires. Regions of whitespace denote objects of the ambient 2-category. Our pasting diagram above is depicted as a string diagram in $\AlgGU$ below:

A 2-morphism $\alpha:{f}\to{g}$ in $[X,Y]$ is defined to be a 2-morphism $\Sh{\alpha}:\Sh{f}\to{\Sh{g}}$ compatible with $\varphi_{f},\varphi_{g}$ in the sense that the following pasting diagrams are equal:

We may likewise represent the equation above in terms of string diagrams.