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A sequence $$E(\zeta) \stackrel{\theta}{\to} X^2 \rightrightarrows X \stackrel{\zeta}{\to} Y $$ where the unlabelled arrows are the two projection, is said to be exact iff

  • $\zeta$ is the coequalizer of $ p_1 \theta$ and $ p_2 \theta$
  • $\theta$ is equalizer of $\zeta p_1$ and $\zeta p_2$.

I do know that these kind of structures where studied by Barr, but probably they where investigated in many contexts. My feeling, which I hope will be confirmed, is that the right environment for the subject are regular categories.

Can you name some general reference about these structure?

To be more precise, I am interested in a kind of snake lemma for this kind of exact sequences.

$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llll} E(f) & \ra{p_i \circ \theta} & X & \ra{f} & Y \\ \da{g_1} & & \da{g_2} & & \da{g_3} & & \\ E(v) & \ra{q_i \circ \eta} & V & \ra{v} & Z & \\ \end{array} $$

Can I find a map $E(g_3) \to E(v)/E(f)$? Or, more in general, was such a question investigated?

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  • $\begingroup$ This sounds like the sort of thing that might be in the book of Borceux and Bourn, but perhaps they start by imposing further conditions on your category that are too restrictive for your purposes. (That said, one of their main aims is to work with good categories that are not additive; is this the kind of thing you want?) $\endgroup$
    – Yemon Choi
    Commented Jan 29, 2018 at 16:46
  • $\begingroup$ Yes it is, thanks for the reference. I will give a look. $\endgroup$
    – Ivan Di Liberti
    Commented Jan 29, 2018 at 16:47
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    $\begingroup$ I feared as much. In that case I do not know of a suitable reference. Perhaps you could ask on the category theory mailing list? $\endgroup$
    – Yemon Choi
    Commented Jan 29, 2018 at 18:22
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    $\begingroup$ How do you define $E(v)/E(f)$? $\endgroup$
    – Arnaud D.
    Commented Jan 30, 2018 at 10:06
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    $\begingroup$ I would go to the original reference for Barr-exact categories, with the warning that you shouldn't expect to do homology theory in an exact (let alone regular) category. There's a reason Borceux-Bourn called them homological categories. In particular, I don't see how one would hope to define $E(\nu)/E(f)$ without being in a pointed category, and realistically much more than that. Here's Barr's original, if you can pass Springer's paywall: link.springer.com/content/pdf/10.1007%2FBFb0058580.pdf $\endgroup$
    – Kevin Carlson
    Commented Feb 7, 2018 at 6:03

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