$Inj$ (or $Set_inj$) is the wide subcategory of the topos Set with its morphisms restricted to monomorphisms (injective functions).
The arrow category $Inj^\to$ of $Inj$ is, in turn, a wide subcategory of the quasitopos#mono? Mono? of set monomorphisms. The two categories have the same objects, namely injective functions, but in Mono the morphism are arbitrary commutative squares from one injection to another, whereas in $Inj^\to$ the other two arrows in the commutative square must also be injective. Note that $Mono$ is a reflective subcategory of the presheaf topos $Set^{\to}$ (the Sierpinski topos), and the category of separated presheaves for the double negation topology? on $Set^\to$.
For a set $S$, its power set ($\mathcal{P}S$ or $2^S$) is the slice category $Inj/S$ which has objects that are injections to $S$ and morphisms that are injections of injections that form commuting triangles. $2^S$ is a quasitopos because it is a Heyting algebra.
Last revised on September 8, 2017 at 15:48:36. See the history of this page for a list of all contributions to it.