Re: when does preservation of monos imply left exactness?

Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg]

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
>


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Re: when does preservation of monos imply left exactness?

[Richard Garner]

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/ ]

Re: when does preservation of monos imply left exactness?

[George Janelidze]

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/ ]

Re: when does preservation of monos imply left exactness?

[Steve Lack]

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
>>> 
> 

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Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg]

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.

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Re: when does preservation of monos imply left exactness?

[George Janelidze]

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/ ]

Correcting a misprint in my previous message

[George Janelidze]

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/ ]

Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg]

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/ ]

Re: when does preservation of monos imply left exactness?

[George Janelidze]

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/ ]

Re: Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg]

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/ ]