Has anybody considered (and are there any references with standard results) categories that do *not* have *all* pullbacks but nevertheless have some nice exactness properties? For example, instead of saying that regular epis are stable under pullback (so that the pullback of a regular epi along any map is also regular-epic), I might say that any pullback of a regular epi is regular-epic *if* it exists. (I might instead use a weaker variant, requiring this only in the case that *all* pullbacks of the regular epi in question exist; or else requiring that all pullbacks of *all* regular epis exist, yielding a stronger variant). For a more specific example, the category of smooth manifolds misses many pullbacks but has the property above (at least the weaker form; as I recall, the surjective submersions are precisely those regular epis that have all pullbacks, but I forget if any other regular epis exist; in any case, the pullback of a surjective submersion along any smooth map exists and is also surjective-submersive). --Toby
My recollection is that in the original definition only pullbacks of regular epis as well as kernel pairs were assumed to exist. Although you could just assume that when the pullback of a regular epi exists it is a regular epic, I think that would vitiate the definition. However, one possibility that I have known of for a long time but not written about is to suppose that when A --> B is regular epic and B' --> B is arbitrary and you look at all pairs A' --> A, A' --> B' that make the evident square commute, then the family of all those A' --> B' is an effective epic family. In that category, a pullback, if it exists, is terminal. On Thu, 18 Jan 2007, Toby Bartels wrote:
Has anybody considered (and are there any references with standard results) categories that do *not* have *all* pullbacks but nevertheless have some nice exactness properties?
For example, instead of saying that regular epis are stable under pullback (so that the pullback of a regular epi along any map is also regular-epic), I might say that any pullback of a regular epi is regular-epic *if* it exists. (I might instead use a weaker variant, requiring this only in the case that *all* pullbacks of the regular epi in question exist; or else requiring that all pullbacks of *all* regular epis exist, yielding a stronger variant).
For a more specific example, the category of smooth manifolds misses many pullbacks but has the property above (at least the weaker form; as I recall, the surjective submersions are precisely those regular epis that have all pullbacks, but I forget if any other regular epis exist; in any case, the pullback of a surjective submersion along any smooth map exists and is also surjective-submersive).
--Toby
Michael Barr wrote:
Toby Bartels wrote:
Has anybody considered (and are there any references with standard results) categories that do *not* have *all* pullbacks but nevertheless have some nice exactness properties?
My recollection is that in the original definition only pullbacks of regular epis as well as kernel pairs were assumed to exist.
By "the original definition", you mean the definitions here?: Michael Barr, Exact categories, in Exact Categories and Categories of Sheaves, Lecture Notes in Mathematics 236, Springer-Verlag, 1971. I've never read this, since you-exact categories are now standard, but I guess that one should always go back to the source! --Toby
Has anybody considered (and are there any references with standard results) categories that do *not* have *all* pullbacks but nevertheless have some nice exactness properties?
For example, instead of saying that regular epis are stable under pullback (so that the pullback of a regular epi along any map is also regular-epic),
grothendieck notion of strict epi (SGA4) is equivalent to the notion of regular epi in the presence of the kernel-pair, but it makes sense in the absence of pull-backs. you can say that a strict epi is "stable under pullbacks" also in the absence of pullbacks: Z_i -------> X | | |f_i |f \/ h \/ Z --------> Y a strict epi f is universal if given any h there exists a strict epi family f_i as indicated in the diagram. this exactness property is as good as stability under pullbacks see the links http://arXiv.org/abs/math/0611701 http://arXiv.org/abs/math/0612727 i am afraid thought that you have different examples in mind. eduardo j. dubuc
Of course I meant the definition in LNM#236. However, I don't have the original source at home anyway, so I would have to wait to check it.
M. Barr wrote (in part, concerning the question of defining the stability of a regular epi under pull-backs without pull-backs)
However, one possibility that I have known of for a long time but not written about is to suppose that when A --> B is regular epic and B' --> B is arbitrary and you look at all pairs A' --> A, A' --> B' that make the evident square commute, then the family of all those A' --> B' is an effective epic family. In that category, a pullback, if it exists, is terminal.
refer to this property as (*) Well, (*) is the same of what I wrote in my posting in the subject: (*):
you can say that a strict epi is "stable under pullbacks" also in the absence of pullbacks:
Z_i -------> X | | |f_i |f \/ h \/ Z --------> Y
a strict epi f is universal if given any h there exists a strict epi family f_i as indicated in the diagram.
this exactness property is as good as stability under pullbacks see the links
Of course, it is the same if we are talking of the same thing. That we are. When I say "strict", I mean it in the sense of SGA4 Expose I, 10.2 10.3, and we should assume that it coincides with what M. Barr calls "effective". Contrary to M. Barr terminology, "effective" is also utilizad in SGA4, presicely, when the kernel pair exists ! Concerning the above notion (*) of "stability under pull-backs without pull-backs" (an instance of "universality"), it is also defined in SGA4 Expose II 2.5, and it is simply the following: an arrow F: X ---> Y (singleton family) is a strict universal epimorphism if it is a cover for the canonical topology. In Proposition 2.6 it is stablished the characterization of strict universal epimorphisms by the property (*) above. e.d.
participants (3)
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Eduardo Dubuc -
Michael Barr -
Toby Bartels