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I’ve finally created the page Inj to record some facts and give something to link to, though I don’t fully understand what I’ve written so it needs to be checked.
What is the reflector $Set^{\to} \to Inj$?
I have yet to add some tags and links in quasitopos. See Sandbox/1054.
$Inj$ had been discussed in nForum: power set & Inj though I haven’t reread all the comments there yet.
I haven’t yet made Surj which at least appears in partition.
$Surj$ has been somewhat discussed in nForum:constant functor
No, $Inj$ is not a quasitopos, and there is no reflector $Set^\to \to Inj$ because $Inj$ is not even a complete category (it lacks products for instance, and even lacks a terminal object).
Might you be thinking of the category $Mono$ whose objects are monomorphisms $A \to B$ and whose morphisms are commutative squares? That is a quasitopos, and there is a reflector $Set^\to \to Mono$ that can be described. But this category is (in the nLab) denoted $\mathcal{M}$; see M-category.
No, Inj is not a quasitopos,
Hmm. I was relying on the text in quasitopos#examples which says
The following examples are categories of separated presheaves for the $\neg\neg$-topology on various presheaf toposes:
- The category of monomorphisms between sets (as presheaves on the interval category).
I guess that section needs to be more specific.
I (wrongly) thought that $Inj$ had a slick definition.
We have Sierpinski topos already for $Set^\to$, but maybe we don’t want that now in view of #3.
Re #4: saying “presheaves on the interval category” is disambiguating, and makes clear that $Mono$ doesn’t mean the category of sets with injections as morphisms, but the category whose objects are injections, as described in #3. Anyway, I’m still not sure which of those two categories you want Inj to mean; I’m just noting that the version of the article of #1 isn’t right.
I’ve updated Inj so at least is is not wrong I hope this time, though still not particularly useful.
I also added a tags so that double negation#topology and M-category#mono can be directly linked.
Can I assume that there is no way to recover a category from its arrow category even for the special cases considered here? I haven’t been able to google up mention of this.
Can I assume that there is no way to recover a category from its arrow category
The original category is the full subcategory of its arrow category on the identity arrows. Is that what you are after?
I still don’t think it was quite right: the arrow category of $Set_{inj}$ has as objects monomorphisms, but as morphisms commutative squares of monomorphisms, whereas in Mono only the objects are monos, not the arrows. I fixed it.
Re: #8, it’s an interesting question, though, whether $C$ is abstractly determined by the category $C^\to$ without knowing which of its objects “are identities”. I.e. can inequivalent categories have equivalent arrow categories? I don’t know.
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