## name my cat: regular categories where inverse images also have right adjoint

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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.

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Is there a name for the fragment of logic containing only $\wedge$, $\exists$, $\Rightarrow$, and $\forall$?  If so, you could borrow the same name for your categories, since it would probably be their internal logic. – Mike Shulman Oct 15 2010 at 20:29
+1 for the title. – Dave Penneys Nov 23 2010 at 19:12

## 1 Answer

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).

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I don't want a lattice structure on my subobject posets. But maybe I could call my categories "hemilogoses" or something like that. – Wouter Stekelenburg Oct 18 2010 at 9:14

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Sorry I don't know a name for precisely what you want. It seems like a natural-enough notion. – Todd Trimble Oct 18 2010 at 10:50
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