Question

I need a name for a regular category where the inverse image maps have right adjoints.

If $\mathcal C$ is a regular category, then the poset of subobjects $\mathsf{Sub}(X)$ of any object $X$ is a semilattice and the inverse image map of any arrow $f:X\to Y$ has a left adjoint $\exists_f:\mathsf{Sub}(X) \to \mathsf{Sub}(Y)$. If $\mathcal C$ is a Heyting category, then the inverse image map $f$ also has a right adjoint $\forall_f:\mathsf{Sub}(X) \to \mathsf{Sub}(Y)$. But Heyting categories also have all finite coproducts and I want a name for regular categories that just have those right adjoints.

Do you know if this category of categories already has a name? Can you suggest a name?

Update: Heyting categories or logoses need not have all finite coproducts, but posets of subobjects are lattices, where I only need semilattices.

Answer 1

From Freyd and Scedrov's book Categories, Allegories: a logos is a regular category in which $Sub(A)$ is a lattice for each object $A$, and in which the inverse-image operation $f^*: Sub(B) \to Sub(A)$ has a right adjoint for each morphism $f: A \to B$ (page 117).