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1. • CommentRowNumber1.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 5th 2017
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI'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](https://ncatlab.org/nlab/revision/Sandbox/1054). $Inj$ had been discussed in [nForum: power set & Inj](https://nforum.ncatlab.org/discussion/5399/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 ](https://nforum.ncatlab.org/discussion/319/constant-functor/?Focus=63677#Comment_63677)

   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

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNo, $Inj$ 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).

   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).

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2017
   • (edited Sep 5th 2017)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexMight 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]].

   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.

4. • CommentRowNumber4.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 5th 2017
   • (edited Sep 5th 2017)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItex> 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 5th 2017
   • (edited Sep 5th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWe have [[Sierpinski topos]] already for $Set^\to$, but maybe we don't want that now in view of #3.

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

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorRodMcGuire
   • CommentTimeSep 7th 2017
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI'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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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?

   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?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeSep 8th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI 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.

   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.