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Let $\mathcal C$ be a sufficiently-complete-and-cocomplete category. Let $C \in \mathcal C$, and let $A \rightarrowtail C \leftarrowtail B$ be subobjects. Let $A \cap B = A \times_C B$ be the intersection of these subobjects. Let $AB = A \amalg_{A \cap B} B$ be the pushout of $A,B$ over this intersection. There is an induced map $AB \to C$.

Question: What conditions on $\mathcal C$ ensure that $AB \to C$ is a monomorphism?

It suffices to suppose that $\mathcal C$ is either abelian or a topos, I think. Does it suffice to assume that $\mathcal C$ is exact?

Does it suffice for the category to be adhesive? (and incidentally is every abelian category adhesive?)

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    $\begingroup$ It suffices that $\mathcal{C}$ is a coherent category. The proof is mainly due to Joyal, and it appears e.g. in "Reyes, Gonzalo: From sheaves to logic - Studies in algebraic logic, M.A.A. studies in Math., vol. 9 (1974). The Elephant has also two different proofs, one using the internal language of the category. $\endgroup$
    – godelian
    Commented Dec 24, 2022 at 19:39
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    $\begingroup$ This comments seems to be an answer. $\endgroup$
    – Martin Brandenburg
    Commented Dec 24, 2022 at 20:14
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    $\begingroup$ I asked this myself, including the directed case, but got no answer. From some other question I learned that Michael Barr had written about this in Springer LNM 1348. My application was to show when extensional well founded coalgebras have "overlapping unions" like sets ($\epsilon$-structures). $\endgroup$
    – Paul Taylor
    Commented Dec 24, 2022 at 21:02
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    $\begingroup$ Exactness isn’t enough: you can have pairs of subgroups with trivial intersection whose union is not the free product. $\endgroup$
    – Kevin Carlson
    Commented Dec 24, 2022 at 21:30
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    $\begingroup$ Regularity isn't enough, eg ${\mathbf{Set}}^{op}$. There is a pushout in $\mathbf{Set}$ using sets with 1, 2, 2 and 3 elements where the pullback has 4 elements, so $3\to 4$ is not epi in $\mathbf{Set}$ (mono in ${\mathbf{Set}}^{op}$). $\endgroup$
    – Paul Taylor
    Commented Dec 29, 2022 at 9:20

2 Answers 2

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As per the comments, I'm giving one possible answer here.

It suffices that $\mathcal{C}$ is a coherent category. The proof is mainly due to Joyal, and it appears e.g. in "Reyes, Gonzalo: From sheaves to logic - Studies in algebraic logic, M.A.A. studies in Math., vol. 9 (1974). The Elephant has also two different proofs, one using the internal language of the category.

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  • $\begingroup$ Thanks! I'll hold off on accepting for now -- I'd really like a condition which includes the abelian case as well. $\endgroup$
    – Tim Campion
    Commented Dec 24, 2022 at 21:04
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    $\begingroup$ That's a long paper by Reyes, on many topics. Where in it is this particular result? $\endgroup$
    – Paul Taylor
    Commented Dec 24, 2022 at 21:36
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    $\begingroup$ @TimCampion: Barr's paper is applicable to Abelian categories as well as pretoposes. $\endgroup$
    – Paul Taylor
    Commented Dec 24, 2022 at 21:38
  • $\begingroup$ @PaulTaylor I don't have Reyes' paper at hand, but I reproduced the proof in my master thesis, page 13 Lemma 3.2.12 at cms.dm.uba.ar/Members/cespindo/tesis.pdf (there's a change of terminology though, but the category is coherent). $\endgroup$
    – godelian
    Commented Dec 24, 2022 at 21:44
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    $\begingroup$ Dowloadable book, not sure if it's legit. Vengo a Buenos Aires 5 de enero, donde encontraré Claudio Hermida. $\endgroup$
    – Paul Taylor
    Commented Dec 24, 2022 at 22:11
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Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:

Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • I think this still doesn't cover the abelian case.
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    $\begingroup$ It appears the proofs of those statements are omitted from the paper entirely... $\endgroup$
    – varkor
    Commented Jan 2, 2023 at 4:12
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    $\begingroup$ @varkor Here is what seems to be another version of the same paper (BRICS RS-03-31). Here, Theorem 4.7 and Corollary 4.8 have proofs. Thm 4.7 says that every adhesive category has effective unions, while Cor 4.8 shows that if unions are effective and pushouts of monos are stable under pullback along monos, the subobject lattices are distributive. So it seems the issue which prevents an abelian category from being adhesive is that pushouts of monos are not stable under pullback along monos. $\endgroup$
    – Tim Campion
    Commented Jan 2, 2023 at 18:18
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    $\begingroup$ I think the proof of Thm 4.7 only uses that pushouts of monos are stable under pullback, not the full strength of adhesiveness. $\endgroup$
    – Tim Campion
    Commented Jan 2, 2023 at 18:22

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