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Regular categories may equivalently defined as those with:

  • finite limits
  • coequalizers of kernel pairs
  • pulback stable regular epis

or

  • finite limits
  • pullback stable regular epi/mono factorization

When carefully proving the equivalence, the only limits required are pullbacks i.e. in a category with pullbacks:

coequalizers of kernel pairs & stable regular epis $\iff$ stable regular epi/mono factorization.

Is there a compelling reason to require all finite limits?

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    $\begingroup$ Possibly the link to the logic of the internal language suggests one might want a terminal object. $\endgroup$
    – David Roberts
    Feb 12, 2018 at 20:35
  • $\begingroup$ Perhaps one wants equalizers and products (finite ones) and so we might as well require all finite limits, since their existence follows from those ? $\endgroup$ Feb 12, 2018 at 20:38
  • $\begingroup$ David: could you elaborate? $\endgroup$ Feb 12, 2018 at 20:47
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    $\begingroup$ @Tyler Mike said it better in his answer. $\endgroup$
    – David Roberts
    Feb 13, 2018 at 0:44

1 Answer 1

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A category with pullbacks and equalizers that satisfies the rest of the definition of a regular category is called locally regular, since this is equivalent to saying that all of its slice categories (which of course have terminal objects) are regular in the usual sense. Locally regular categories share many other properties of regular ones, for instance one can construct a bicategory of relations and show that locally regular categories are essentially the same as "tabular allegories" (A3.2.7 in Sketches of an elephant).

There are many reasons one might give for why the notion of "regular category" includes a terminal object (and hence all finite products), but I think one fairly compelling one is that, as David said in a comment, one wants the internal logic of a regular category to be regular logic, and one needs all finite products in order to define a type theory and internal logic: a term $x:A, y:B \vdash t:C$ is a morphism $A\times B\to C$, and a term $\cdot \vdash t:C$ is a morphism $1\to C$. (One can make do with a cartesian multicategory instead, but a locally regular category doesn't have an underlying one of those either.)

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  • $\begingroup$ Thanks @mike-shulman. I see you had a hand in writing the relevant nlab locally regular category. Is the definition in your answer here equivalent to that given there? Specifically, are there equalizers in a category with stable reg. epi/mono factorization and pullbacks? Side note: pullbacks + reg. epi/mono factorization (not necessarily stable) $\implies$ reg. epi = strong epi = extremal epi. so the extremal epi/mono factorization will be reg. epi/mono already. $\endgroup$ Feb 12, 2018 at 22:48
  • $\begingroup$ Haha, I forgot that that page existed or I would have linked to it. It looks like I also misremembered the definition; I don't think equalizers are implied by just pullbacks and the rest, you need to assume them too. $\endgroup$ Feb 12, 2018 at 23:32
  • $\begingroup$ It's interesting. I've been building up my own notes on a related idea---proving everything along the way. I keep expecting to need more limits but I haven't yet. Fortunately or not, it means I need to keep proving everything for myself. Wish I had a copy of the Elephant :) $\endgroup$ Feb 12, 2018 at 23:41
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    $\begingroup$ @Tyler the 'Baby Elephant' is available as an inexpensive Dover reprint, and despite what Johnstone says ("don't buy it, buy the Elephant instead") it's still a handy book. $\endgroup$
    – David Roberts
    Feb 13, 2018 at 0:44
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    $\begingroup$ @Peter "I resisted this for quite a while, because I don't really want people to buy `Topos Theory' (I'd much rather they bought copies of the Elephant), but in the end I had to concede that Dover knew their business better than I did, and if they said there was a market for it they were probably right." Peter Johnstone, categories mailing list, 6 March 2014 $\endgroup$
    – David Roberts
    Feb 13, 2018 at 9:50

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