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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 InjSet^{\to} \to Inj?

    I have yet to add some tags and links in quasitopos. See Sandbox/1054.

    InjInj 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.

    SurjSurj has been somewhat discussed in nForum:constant functor

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 5th 2017

    No, InjInj is not a quasitopos, and there is no reflector Set InjSet^\to \to Inj because InjInj 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 MonoMono whose objects are monomorphisms ABA \to B and whose morphisms are commutative squares? That is a quasitopos, and there is a reflector Set MonoSet^\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 InjInj had a slick definition.

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

    We have Sierpinski topos already for Set 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 MonoMono 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 injSet_{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 CC is abstractly determined by the category C C^\to without knowing which of its objects “are identities”. I.e. can inequivalent categories have equivalent arrow categories? I don’t know.