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have a quick suggestion for a definition at embedding,
This was the first idea that came to mind when reading Toby’s initial remark there, haven’t really thought much about it.
Ah, what I am describing is of course just effective monomorphism (in the sense remarked at effective epimorphism)
wrote effective monomorphism (which was in fact requested already by a handful of entries) and linked back and forth to embedding.
We should really have
$effective mono \leftrightarrow embedding$
$effective epi \leftrightarrow covering$
In a category with finite limits and finite colimits, effective monomorphisms are the same thing as regular monomorphisms. I recorded this fact (with proof) at regular monomorphism.
But regular monomorphisms are slightly more general than effective monomorphisms, since there are notable examples where we don’t have all limits and colimits. A relevant example is the category of smooth manifolds. Here I believe it is a fact that smooth embeddings $i: M \to N$ are indeed regular monomorphisms, but not effective since there is no pushout $N +_M N$.
Thus I propose that it should be
regular mono $\leftrightarrow$ embedding
regular epi $\leftrightarrow$ cover
and in fact I think this is a fairly widely adopted convention among category theorists, to define embedding and cover in this way.
Thanks, Todd. I rephrased embedding a bit to reflect this better.
okay, so there is evidently some polishing indicated on some $n$Lab entries here.
i did brief edits to make the pattern
regular epimorphism $\leftrightarrow$ covering
regular monomorphism $\leftrightarrow$ embedding
be reflected better in the $n$Lab entries. But have to rush off now.
At some point I’ll see whether I can prove that claim about smooth embeddings of manifolds being regular monomorphisms, and stick it somewhere in the Lab, but I’m not sure where yet.
In the absence of kernel pairs, there is also the notion which the nLab currently calls a strict epimorphism. I have seen some sources define “regular epi” or “effective epi” to mean this notion in the absence of kernel pairs. In particular I’m thinking of the remarks after C2.1.11 in the Elephant which define a sieve to be “effective-epimorphic” just when the morphisms composing it are jointly “strictly epimorphic” in the sense used here. It would be interesting to see examples separating strict epis from regular ones.
Regarding regular epimorphisms of smooth manifolds: these are more general that surjective submersions, and include all split epis, which are not very useful (in general) when working with manifolds. It seems regular epis really are only good, as Mike says, in the presence of kernel pairs.
I do not like that $n$Lab defines “covering” in a specific way topos theorists and alike strains of category theorists do. Covering/cover is a much more wide notion which can take in various functorial setups different versions.
If somebody wants to make an equality between regular epi and covering, than say regular epi, without assuming local conventions from a narrow subfield of mathematics, and without incorporating them in addition to a number of other entries. It is for example, inappropriate for categorical situations related to “sheaves” on noncommutative spaces.
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