* Re: when does preservation of monos imply left exactness?
@ 2011-10-26 11:59 Dmitry Roytenberg
2011-10-26 21:52 ` Richard Garner
` (2 more replies)
0 siblings, 3 replies; 10+ messages in thread
From: Dmitry Roytenberg @ 2011-10-26 11:59 UTC (permalink / raw)
To: Categories list
I'll try again...
On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg
<starrgazerr@gmail.com> wrote:
> Dear category theorists,
>
> It is well known that any functor that preserves finite limits
> preserves monomorphisms, and that for an additive right-exact functor
> between abelian categories, the converse is also true. Is it known how
> far this extends to the non-additive setting? In other words, what
> exactness properties of two categories and a functor between them
> would suffice to conclude that the functor preserves finite limits if
> and only if it preserves monos? For instance, is it enough to assume
> that the categories be Barr-exact and that the functor preserve all
> colimits, finite products and monos to conclude that it also preserves
> equalizers?
>
> Any references would be extremely helpful.
>
> Thanks,
> Dmitry
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: when does preservation of monos imply left exactness? 2011-10-26 11:59 when does preservation of monos imply left exactness? Dmitry Roytenberg @ 2011-10-26 21:52 ` Richard Garner 2011-10-27 10:32 ` George Janelidze [not found] ` <C86754D7A15D4A118F901BAD51AA3331@ACERi3> 2 siblings, 0 replies; 10+ messages in thread From: Richard Garner @ 2011-10-26 21:52 UTC (permalink / raw) To: Dmitry Roytenberg; +Cc: Categories list Dear Dmitry, There is a result along these lines in Mike Barr's paper "On categories with effective unions", Springer LNM 1348. His Theorem 4.1 says: Suppose F: C --> D is a functor such that C has finite limits, cokernel pairs and effective unions, and F preserves finite products, regular monomorphisms and cokernel pairs. Then F preserves finite limits. In the statement of this result, a category is said to have effective unions if the union of two regular subobjects always exists and is calculated as the pushout over the intersection. Best, Richard On 26 October 2011 22:59, Dmitry Roytenberg <starrgazerr@gmail.com> wrote: > I'll try again... > > On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg > <starrgazerr@gmail.com> wrote: >> Dear category theorists, >> >> It is well known that any functor that preserves finite limits >> preserves monomorphisms, and that for an additive right-exact functor >> between abelian categories, the converse is also true. Is it known how >> far this extends to the non-additive setting? In other words, what >> exactness properties of two categories and a functor between them >> would suffice to conclude that the functor preserves finite limits if >> and only if it preserves monos? For instance, is it enough to assume >> that the categories be Barr-exact and that the functor preserve all >> colimits, finite products and monos to conclude that it also preserves >> equalizers? >> >> Any references would be extremely helpful. >> >> Thanks, >> Dmitry >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: when does preservation of monos imply left exactness? 2011-10-26 11:59 when does preservation of monos imply left exactness? Dmitry Roytenberg 2011-10-26 21:52 ` Richard Garner @ 2011-10-27 10:32 ` George Janelidze 2011-10-27 22:08 ` Steve Lack 2011-10-28 12:27 ` Dmitry Roytenberg [not found] ` <C86754D7A15D4A118F901BAD51AA3331@ACERi3> 2 siblings, 2 replies; 10+ messages in thread From: George Janelidze @ 2011-10-27 10:32 UTC (permalink / raw) To: Dmitry Roytenberg, Categories list All right, somebody should answer... Dear Dmitry, First, may I suggest that looking at category theory as merely a 'generalized abelian category theory' is not really a good idea? Specifically, in the abelian context 'right exact + preserves monos => left exact' is simple but extremely important, while for general categories it is indeed wrong, and adding conditions to make this work seems to be a strange thing to do! Now, more specifically, concerning Barr exact: Let 2 be the ordered set {0,1} considered as a category, let S be the category of sets, and let F : S --> 2 be the left adjoint of the canonical embedding 2 --> S (hence, for a set X, F(X) = 0 if and only if X is empty). Then: (a) 2 and S are Barr exact. (b) F preserves all colimits since it is a left adjoint. (c) F preserves products (moreover, the fact that it preserves all products is one of the standard forms of the Axiom of Choice). (d) F preserves monomorphisms since every morphism in 2 is a monomorphism. (e) F does not preserve all equalizers since two parallel morphisms between non-empty sets might have the empty equalizer. Best regards George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Wednesday, October 26, 2011 1:59 PM To: "Categories list" <categories@mta.ca> Subject: categories: Re: when does preservation of monos imply left exactness? > I'll try again... > > On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg > <starrgazerr@gmail.com> wrote: >> Dear category theorists, >> >> It is well known that any functor that preserves finite limits >> preserves monomorphisms, and that for an additive right-exact functor >> between abelian categories, the converse is also true. Is it known how >> far this extends to the non-additive setting? In other words, what >> exactness properties of two categories and a functor between them >> would suffice to conclude that the functor preserves finite limits if >> and only if it preserves monos? For instance, is it enough to assume >> that the categories be Barr-exact and that the functor preserve all >> colimits, finite products and monos to conclude that it also preserves >> equalizers? >> >> Any references would be extremely helpful. >> >> Thanks, >> Dmitry >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: when does preservation of monos imply left exactness? 2011-10-27 10:32 ` George Janelidze @ 2011-10-27 22:08 ` Steve Lack 2011-10-28 12:27 ` Dmitry Roytenberg 1 sibling, 0 replies; 10+ messages in thread From: Steve Lack @ 2011-10-27 22:08 UTC (permalink / raw) To: George Janelidze; +Cc: Dmitry Roytenberg, Categories list Dear All, While I agree that category theory is not just generalized abelian category theory, there are nonetheless some theorems along the lines that Dmitry suggests. Richard mentioned one; here is another. Of course neither is nearly as simple as the abelian case. If the opposite of the categories A and B are exact Mal'cev, then any functor f:A->B which preserves finite colimits and regular monos preserves equalizers of coreflexive pairs. Thus any functor f:A->B which preserves finite colimits, finite products and regular monos preserves all finite limits. In particular, A and B could be toposes, then if f:A->B preserves finite colimits, finite products, and monomorphisms, it preserves all finite limits. From the point of view of George's example, the problem is that in the category 2, the map 0->1 is mono but not regular mono. Best wishes, Steve Lack. On 27/10/2011, at 9:32 PM, George Janelidze wrote: > All right, somebody should answer... > > Dear Dmitry, > > First, may I suggest that looking at category theory as merely a > 'generalized abelian category theory' is not really a good idea? > > Specifically, in the abelian context > > 'right exact + preserves monos => left exact' > > is simple but extremely important, while for general categories it is indeed > wrong, and adding conditions to make this work seems to be a strange thing > to do! > > Now, more specifically, concerning Barr exact: > > Let 2 be the ordered set {0,1} considered as a category, let S be the > category of sets, and let F : S --> 2 be the left adjoint of the canonical > embedding 2 --> S (hence, for a set X, F(X) = 0 if and only if X is empty). > Then: > > (a) 2 and S are Barr exact. > > (b) F preserves all colimits since it is a left adjoint. > > (c) F preserves products (moreover, the fact that it preserves all products > is one of the standard forms of the Axiom of Choice). > > (d) F preserves monomorphisms since every morphism in 2 is a monomorphism. > > (e) F does not preserve all equalizers since two parallel morphisms between > non-empty sets might have the empty equalizer. > > Best regards > > George Janelidze > > -------------------------------------------------- > From: "Dmitry Roytenberg" <starrgazerr@gmail.com> > Sent: Wednesday, October 26, 2011 1:59 PM > To: "Categories list" <categories@mta.ca> > Subject: categories: Re: when does preservation of monos imply left > exactness? > >> I'll try again... >> >> On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg >> <starrgazerr@gmail.com> wrote: >>> Dear category theorists, >>> >>> It is well known that any functor that preserves finite limits >>> preserves monomorphisms, and that for an additive right-exact functor >>> between abelian categories, the converse is also true. Is it known how >>> far this extends to the non-additive setting? In other words, what >>> exactness properties of two categories and a functor between them >>> would suffice to conclude that the functor preserves finite limits if >>> and only if it preserves monos? For instance, is it enough to assume >>> that the categories be Barr-exact and that the functor preserve all >>> colimits, finite products and monos to conclude that it also preserves >>> equalizers? >>> >>> Any references would be extremely helpful. >>> >>> Thanks, >>> Dmitry >>> > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: when does preservation of monos imply left exactness? 2011-10-27 10:32 ` George Janelidze 2011-10-27 22:08 ` Steve Lack @ 2011-10-28 12:27 ` Dmitry Roytenberg 2011-10-28 21:36 ` George Janelidze 1 sibling, 1 reply; 10+ messages in thread From: Dmitry Roytenberg @ 2011-10-28 12:27 UTC (permalink / raw) To: Steve Lack; +Cc: George Janelidze, richard.garner, Categories list Dear colleagues, First, let me thank everyone who has replied so far: I am learning a lot from this discussion (not being a category theorist myself). Far be it from me to make sweeping statements about general category theory, I am merely interested in particular functors between particular categories. It just seems that preservation of monos would be easier to check than preservation of equalizers, which is why I am looking for general criteria. Consider the following example. Let k-Alg denote the category of commutative k-algebras over some ground ring k, and let F:k-Alg-->k-Alg denote taking coproduct with an object A. Then F preserves all colimits and finite products; furthermore, if F preserves monos, it preserves all finite limits (which is the case iff A is flat as a k-module). The only proof I know takes advantage of the convenient fact that the coproduct in k-Alg extends to the tensor product in the abelian category k-Mod, as well as the existence of a fully faithful functor from k-Mod to k-Alg given by the square-zero extension. If anyone knows a proof which does not involve going to k-Mod, just using some general properties of k-Alg, I'd be very happy to learn about it. Notice that k-Alg is not co-Mal'cev, nor is every monomorphism regular. Thanks again, Dmitry On Fri, Oct 28, 2011 at 12:08 AM, Steve Lack <steve.lack@mq.edu.au> wrote: > Dear All, > > While I agree that category theory is not just generalized abelian > category theory, there are nonetheless some theorems along the > lines that Dmitry suggests. Richard mentioned one; here is another. > Of course neither is nearly as simple as the abelian case. > > If the opposite of the categories A and B are exact Mal'cev, then any > functor f:A->B which preserves finite colimits and regular monos preserves equalizers > of coreflexive pairs. Thus any functor f:A->B which preserves finite colimits, finite products > and regular monos preserves all finite limits. > > In particular, A and B could be toposes, then if f:A->B preserves finite colimits, finite > products, and monomorphisms, it preserves all finite limits. > > From the point of view of George's example, the problem is that in the category 2, the > map 0->1 is mono but not regular mono. > > Best wishes, > > Steve Lack. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: when does preservation of monos imply left exactness? 2011-10-28 12:27 ` Dmitry Roytenberg @ 2011-10-28 21:36 ` George Janelidze 2011-10-29 6:01 ` Correcting a misprint in my previous message George Janelidze 0 siblings, 1 reply; 10+ messages in thread From: George Janelidze @ 2011-10-28 21:36 UTC (permalink / raw) To: categories Dear Dmitry, Let me write then F(X) = A@X, denoting the tensor product over k by @. Hence, briefly, @ is + (coproduct) in k-Alg. The fact that, when A is flat as a k-module, A+(-) : k-Alg --> k-Alg preserves finite limits is absolutely classical, as well as its proof via modules. However, you do not need the fully faithful functor from k-Mod to k-Alg given by the square-zero extension for that: simply use the fact the forgetful functor k-Alg --> k-Mod not only preserves limits, but also reflects isomorphisms (Some people prefer to say/use "creates isomorphisms"). The preservation of finite limits by A+(-) : k-Alg --> k-Alg (already for modules) implies, for example, Grothendieck's Descent Theorem saying that if A is flat, then k --> A is an effective descent morphism. However, much stronger result is known now: k --> A is an effective descent morphism if and only if it is pure as a monomorphism of k-modules. This stronger result, commonly known as an unpublished theorem of A. Joyal and M. Tierney, was claimed several times by various authors, and I can tell you more if you are interested. Anyway, since the category k-Alg has very bad exactness properties in a sense, modules were used by category-theorists themselves. And, in spite of nice results mentioned by Richard Garner and Steve Lack, I still suggest not to go through a generalization of the abelian case. Well, such a suggestion surely cannot be "universal", and if you need a better suggestion, you should tell us why are you doing this - if I may say so. Best regards, George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Friday, October 28, 2011 2:27 PM To: "Steve Lack" <steve.lack@mq.edu.au> Cc: "George Janelidze" <janelg@telkomsa.net>; <richard.garner@mq.edu.au>; "Categories list" <categories@mta.ca> Subject: categories: Re: when does preservation of monos imply left exactness? > Dear colleagues, > > First, let me thank everyone who has replied so far: I am learning a > lot from this discussion (not being a category theorist myself). Far > be it from me to make sweeping statements about general category > theory, I am merely interested in particular functors between > particular categories. It just seems that preservation of monos would > be easier to check than preservation of equalizers, which is why I am > looking for general criteria. > > Consider the following example. Let k-Alg denote the category of > commutative k-algebras over some ground ring k, and let > > F:k-Alg-->k-Alg > > denote taking coproduct with an object A. Then F preserves all > colimits and finite products; furthermore, if F preserves monos, it > preserves all finite limits (which is the case iff A is flat as a > k-module). The only proof I know takes advantage of the convenient > fact that the coproduct in k-Alg extends to the tensor product in the > abelian category k-Mod, as well as the existence of a fully faithful > functor from k-Mod to k-Alg given by the square-zero extension. If > anyone knows a proof which does not involve going to k-Mod, just using > some general properties of k-Alg, I'd be very happy to learn about it. > Notice that k-Alg is not co-Mal'cev, nor is every monomorphism > regular. > > Thanks again, > > Dmitry > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Correcting a misprint in my previous message 2011-10-28 21:36 ` George Janelidze @ 2011-10-29 6:01 ` George Janelidze 0 siblings, 0 replies; 10+ messages in thread From: George Janelidze @ 2011-10-29 6:01 UTC (permalink / raw) To: categories; +Cc: Dmitry Roytenberg, Steve Lack Correcting a misprint in my previous message: Not "creates isomorphisms" but "creates limits" (obviously). George [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
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* Re: when does preservation of monos imply left exactness? [not found] ` <C86754D7A15D4A118F901BAD51AA3331@ACERi3> @ 2011-10-29 21:55 ` Dmitry Roytenberg [not found] ` <CAAHD2LKCe2kg=t7=FzkOmzHesZoQWX_itcAgjTRVh6SrL31tSA@mail.gmail.com> 1 sibling, 0 replies; 10+ messages in thread From: Dmitry Roytenberg @ 2011-10-29 21:55 UTC (permalink / raw) To: George Janelidze; +Cc: Steve Lack, richard.garner, categories Dear George, First, a small correction: A@- should be considered as a functor to A-Alg, not k-Alg, in order for what I said to be correct (I thank Steve Lack for pointing that out). The square-zero extension is used to show that preservation of monomorphisms in k-Alg by A@- -- a priori a weaker condition than flatness -- actually implies preservation of monomorphisms in k-Mod, i.e flatness. After that it's the classical story you recalled. As for why - fair enough: I'm interested to know whether this property of commutative algebras is shared by other types of algebras (e.g algebras over k-linear operads, or more general algebraic theories like analytic or C-infinity rings). The fact that the coproduct coincides with the tensor product of underlying modules is a very special property of commutative algebras, so the classical proof fails already for associative algebras. So I wonder what general exactness results are available. For instance, I find Michael Barr's theorem mentioned by Richard very useful. Best, Dmitry On Fri, Oct 28, 2011 at 11:36 PM, George Janelidze <janelg@telkomsa.net> wrote: > Dear Dmitry, > > Let me write then F(X) = A@X, denoting the tensor product over k by @. > Hence, briefly, @ is + (coproduct) in k-Alg. The fact that, when A is flat > as a k-module, A+(-) : k-Alg --> k-Alg preserves finite limits is absolutely > classical, as well as its proof via modules. However, you do not need the > fully faithful functor from k-Mod to k-Alg given by the square-zero > extension for that: simply use the fact the forgetful functor k-Alg --> > k-Mod not only preserves limits, but also reflects isomorphisms (Some people > prefer to say/use "creates isomorphisms"). > > The preservation of finite limits by A+(-) : k-Alg --> k-Alg (already for > modules) implies, for example, Grothendieck's Descent Theorem saying that if > A is flat, then k --> A is an effective descent morphism. However, much > stronger result is known now: k --> A is an effective descent morphism if > and only if it is pure as a monomorphism of k-modules. This stronger result, > commonly known as an unpublished theorem of A. Joyal and M. Tierney, was > claimed several times by various authors, and I can tell you more if you are > interested. > > Anyway, since the category k-Alg has very bad exactness properties in a > sense, modules were used by category-theorists themselves. And, in spite of > nice results mentioned by Richard Gardner and Steve Lack, I still suggest > not to go through a generalization of the abelian case. Well, such a > suggestion surely cannot be "universal", and if you need a better > suggestion, you should tell us why are you doing this - if I may say so. > > Best regards, > > George Janelidze > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
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* Re: when does preservation of monos imply left exactness? [not found] ` <CAAHD2LKCe2kg=t7=FzkOmzHesZoQWX_itcAgjTRVh6SrL31tSA@mail.gmail.com> @ 2011-10-29 23:10 ` George Janelidze 2011-10-31 10:45 ` Dmitry Roytenberg 0 siblings, 1 reply; 10+ messages in thread From: George Janelidze @ 2011-10-29 23:10 UTC (permalink / raw) To: Dmitry Roytenberg; +Cc: Steve Lack, richard.garner, categories Dear Dmitry, Absolutely correct (although it does not change anything I said). Thank you for explaining "why". So your real question is about preservation of finite limits by functors of the form A+(-), in the case non-commutative algebras (of various kinds). Well, from this point of view the categories of algebras are 'difficult', and I don't recall any reasonable result at the moment. Moreover, I am surprised that Barr's theorem helps here (which does not mean that the theorem itself is not good of course!), and I would be very interested to learn, what exactly could you deduce from it? Best regards, George -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Saturday, October 29, 2011 11:55 PM To: "George Janelidze" <janelg@telkomsa.net> Cc: "Steve Lack" <steve.lack@mq.edu.au>; <richard.garner@mq.edu.au>; <categories@mta.ca> Subject: Re: categories: Re: when does preservation of monos imply left exactness? > Dear George, > > First, a small correction: A@- should be considered as a functor to > A-Alg, not k-Alg, in order for what I said to be correct (I thank > Steve Lack for pointing that out). > > The square-zero extension is used to show that preservation of > monomorphisms in k-Alg by A@- -- a priori a weaker condition than > flatness -- actually implies preservation of monomorphisms in k-Mod, > i.e flatness. After that it's the classical story you recalled. > > As for why - fair enough: I'm interested to know whether this property > of commutative algebras is shared by other types of algebras (e.g > algebras over k-linear operads, or more general algebraic theories > like analytic or C-infinity rings). The fact that the coproduct > coincides with the tensor product of underlying modules is a very > special property of commutative algebras, so the classical proof fails > already for associative algebras. So I wonder what general exactness > results are available. For instance, I find Michael Barr's theorem > mentioned by Richard very useful. > > Best, > > Dmitry [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Re: when does preservation of monos imply left exactness? 2011-10-29 23:10 ` George Janelidze @ 2011-10-31 10:45 ` Dmitry Roytenberg 0 siblings, 0 replies; 10+ messages in thread From: Dmitry Roytenberg @ 2011-10-31 10:45 UTC (permalink / raw) To: George Janelidze; +Cc: Steve Lack, richard.garner, categories Dear George, Well, I find Barr's theorem useful insomuch as it highlights regular monos as the relevant ones and thereby brings the situation into sharper focus: proving the preservation of the equalizers of cokernel pairs should be easier than arbitrary ones. Of course, characterizing the regular monos and proving that they are preserved by cobase change (I've finally remembered what A@- is called!) could be a difficult matter, depending on the circumstances. So, I thank everyone for the feedback. I will post here if I manage to prove anything of interest. Best, Dmitry On Sun, Oct 30, 2011 at 1:10 AM, George Janelidze <janelg@telkomsa.net> wrote: > Dear Dmitry, > > Absolutely correct (although it does not change anything I said). > > Thank you for explaining "why". So your real question is about preservation > of finite limits by functors of the form A+(-), in the case non-commutative > algebras (of various kinds). Well, from this point of view the categories of > algebras are 'difficult', and I don't recall any reasonable result at the > moment. Moreover, I am surprised that Barr's theorem helps here (which does > not mean that the theorem itself is not good of course!), and I would be > very interested to learn, what exactly could you deduce from it? > > Best regards, > > George > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
end of thread, other threads:[~2011-10-31 10:45 UTC | newest] Thread overview: 10+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2011-10-26 11:59 when does preservation of monos imply left exactness? Dmitry Roytenberg 2011-10-26 21:52 ` Richard Garner 2011-10-27 10:32 ` George Janelidze 2011-10-27 22:08 ` Steve Lack 2011-10-28 12:27 ` Dmitry Roytenberg 2011-10-28 21:36 ` George Janelidze 2011-10-29 6:01 ` Correcting a misprint in my previous message George Janelidze [not found] ` <C86754D7A15D4A118F901BAD51AA3331@ACERi3> 2011-10-29 21:55 ` when does preservation of monos imply left exactness? Dmitry Roytenberg [not found] ` <CAAHD2LKCe2kg=t7=FzkOmzHesZoQWX_itcAgjTRVh6SrL31tSA@mail.gmail.com> 2011-10-29 23:10 ` George Janelidze 2011-10-31 10:45 ` Dmitry Roytenberg
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