name my cat: regular categories where inverse images also have right adjoint - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:18:37Z http://mathoverflow.net/feeds/question/42147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42147/name-my-cat-regular-categories-where-inverse-images-also-have-right-adjoint name my cat: regular categories where inverse images also have right adjoint Wouter Stekelenburg 2010-10-14T13:18:10Z 2010-10-15T12:36:45Z <p>I need a name for a regular category where the inverse image maps have right adjoints.</p> <p>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.</p> <p>Do you know if this category of categories already has a name? Can you suggest a name?</p> <p>Update: Heyting categories or logoses need not have all finite coproducts, but posets of subobjects are lattices, where I only need semilattices.</p> http://mathoverflow.net/questions/42147/name-my-cat-regular-categories-where-inverse-images-also-have-right-adjoint/42164#42164 Answer by Todd Trimble for name my cat: regular categories where inverse images also have right adjoint Todd Trimble 2010-10-14T16:03:04Z 2010-10-14T16:03:04Z <p>From Freyd and Scedrov's book <i>Categories, Allegories</i>: a <b>logos</b> 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). </p>