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• CommentRowNumber1.
• CommentAuthorRodMcGuire
• CommentTimeSep 5th 2017

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

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeSep 5th 2017

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).

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeSep 5th 2017
• (edited Sep 5th 2017)

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.

• CommentRowNumber4.
• CommentAuthorRodMcGuire
• CommentTimeSep 5th 2017
• (edited Sep 5th 2017)

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:

I guess that section needs to be more specific.

I (wrongly) thought that $Inj$ had a slick definition.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeSep 5th 2017
• (edited Sep 5th 2017)

We have Sierpinski topos already for $Set^\to$, but maybe we don’t want that now in view of #3.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeSep 5th 2017

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.

• CommentRowNumber7.
• CommentAuthorRodMcGuire
• CommentTimeSep 7th 2017

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 7th 2017
• (edited Sep 7th 2017)

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?

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeSep 8th 2017

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.