Is the category of group actions LCCC?

Is the category of group actions LCCC?

[Timothy Revell]

Dear All,

I'm wondering whether the category of ALL group actions is locally
Cartesian closed. This is NOT the functor category [G,Set] for some
category G with one object, since we allow G to vary. To be more
specific the category is as follows.

  - The objects are pairs (G,X), where G is a group and X is a G-Set.

  - A morphism (G,X) -> (G', X') is given by a pair (h,f), where h:G->G'
is a group homomorphism and f: X -> X' is a function (a morphism in Set)
such that for all  g in G, x in X

    h(g) * f(x) = f(g * x)

where * on the left denotes the group action of G' on X' and * on the
right denotes the group action of G on X.


All the best,
Tim


-- 
Timothy Revell,
Department of Computer and Information Sciences,
University of Strathclyde.
The University of Strathclyde is a charitable body, registered in
Scotland, with registration number SC015263.


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Re: Is the category of group actions LCCC?

[Steve Lack]

Dear Tim,

Write Act for the category you mention. Then Act/(G,0) is clearly equivalent to Grp/G.
Since Grp/G is not cartesian closed, neither is Act/(G,0).

Regards,

Steve Lack.

On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:

> Dear All,
> 
> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. This is NOT the functor category [G,Set] for some
> category G with one object, since we allow G to vary. To be more
> specific the category is as follows.
> 
>  - The objects are pairs (G,X), where G is a group and X is a G-Set.
> 
>  - A morphism (G,X) -> (G', X') is given by a pair (h,f), where h:G->G'
> is a group homomorphism and f: X -> X' is a function (a morphism in Set)
> such that for all  g in G, x in X
> 
>    h(g) * f(x) = f(g * x)
> 
> where * on the left denotes the group action of G' on X' and * on the
> right denotes the group action of G on X.
> 
> 
> All the best,
> Tim
> 
> 
> -- 
> Timothy Revell,
> Department of Computer and Information Sciences,
> University of Strathclyde.
> The University of Strathclyde is a charitable body, registered in
> Scotland, with registration number SC015263.
> 


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Re: Is the category of group actions LCCC?

[Ross Street]

On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:

> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. 

This is what I answered Timothy:
======
No, it’s not.
Since the category has a terminal object (1,1), being a LCCC would imply it
was cartesian closed. However, that would imply (G,X) \times — preserved
the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0).
======

But it seems there is more to the story. 
The thing stopping the category of actions from
being cartesian closed is that the category Gp of groups is not. However,
the category Gpd of groupoids and the category Cat of categories are.
The (2-)category Cat//’Set’ of all category actions is defined as follows:
objects (A,F) are functors F : A —> Set and morphisms (f,t) : (A,F) —>  (B,G)
are functors f : A —> B with natural transformation t : F ==> G f.
This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is
([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H g).

However Cat//’Set’ is not locally cartesian closed basically because Cat
is not. It is not even locally cartesian closed as a bicategory.
The 2-category Gpd is cartesian closed; it is not locally cartesian closed;
it is locally cartesian closed as a bicategory. 

Similarly, Gpd//’Set’ is locally cartesian closed as a bicategory.
Often, in dealing with groups, we find groupoids help.
This case is a good example and I hope helps in the applications
you have in mind, Timothy.

Ross



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Re: Is the category of group actions LCCC

[pjf]

On page	-15 (yes, a negative page number) of the TAC reprinting of
Abelian Categories (in the 2003 Forward) I wrote:

    The very large category [described] in Exercise 6-A -- with a few
    variations -- has been a great source of counterexamples over the
    years....In its category of abelian-group objects  Ext(A,B)  is a
    proper class iff there???s a non-zero group homomorphism from  A to  B
    (it needn???t respect the actions) hence the only injective object is
    the zero object (which settled a once-open problem about whether
    there are enough injectives in the category of abelian groups in
    every elementary topos with natural-numbers object.
      http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf


On 2014-09-01 05:12, Timothy Revell wrote:
> Dear All,
>
> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. This is NOT the functor category [G,Set] for some
> category G with one object, since we allow G to vary. To be more
> specific the category is as follows.
>
>   - The objects are pairs (G,X), where G is a group and X is a G-Set.
>
>   - A morphism (G,X) -> (G', X') is given by a pair (h,f), where
> h:G->G'
> is a group homomorphism and f: X -> X' is a function (a morphism in
> Set)
> such that for all  g in G, x in X
>
>     h(g) * f(x) = f(g * x)
>
> where * on the left denotes the group action of G' on X' and * on the
> right denotes the group action of G on X.
>
>
> All the best,
> Tim


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Re: Is the category of group actions LCCC?

[Clemens.BERGER]

Hi Timothy, hi Ross,

    this message to highlight the importance of the construction of the
category of group actions (Grp,C) over an arbirary base category C
(while Timothy is just considering the case C=Sets). Indeed, the
category of group actions (Grp,C) has as objects triples (G,X,\phi)
consisting of a group G, an object X of C, and a monoid morphism
G->C(X,X), and as maps (g,f):(G,X)->(H,Y) pairs consisting of a group
morphism g:G->H and an arrow f:X->Y in C such that an obvious pentagonal
diagram in Sets commutes. Composition in this category is defined by
gluing together two pentagons along a commutative square.

    This construction seems to have quite interesting preservation
properties (alas not inclucing cartesian nor local cartesian
closedness). These properties may have their use in the theory of
semi-abelian categories.

    Just a few samples:

The category (Grp,Grp) of group actions in groups is equivalent to
Bourn's category of split epimorphisms in Grp, also known as the
category of points in Grp. The latter is known to inherit several
important properties of the category Grp, such as protomodularity,
regularity and exactness. In general, if C is regular (exact) then
(Grp,C) as well.

In the theory of cocommutative Hopf algebras over a field k, a structure
thm of Gabriel-Cartier may be interpreted as saying that over an
algebraically closed field of characteristic 0, the category
CocommHopf_k is equivalent to (Grp,Lie_k), where Lie_k is the category
of Lie k-algebras. Since the latter is exact, this shows that
CocommHopf_k is exact as well. An adjunction argument implies then that
CocommHopf_k is exact for any field of characteristic 0. This is the
most difficult step in showing that CocommHopf_k is actually
semi-abelian for any field of characterstic zero.

    All the best,
                 Clemens.


Le 2014-09-04 02:19, Ross Street a ??crit??:
> On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk>
> wrote:
>
>> I'm wondering whether the category of ALL group actions is locally
>> Cartesian closed.
>
> This is what I answered Timothy:
> ======
> No, it???s not.
> Since the category has a terminal object (1,1), being a LCCC would
> imply it
> was cartesian closed. However, that would imply (G,X) \times ???
> preserved
> the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0).
> ======
>
> But it seems there is more to the story.
> The thing stopping the category of actions from
> being cartesian closed is that the category Gp of groups is not.
> However,
> the category Gpd of groupoids and the category Cat of categories are.
> The (2-)category Cat//???Set??? of all category actions is defined as
> follows:
> objects (A,F) are functors F : A ???> Set and morphisms (f,t) : (A,F) ???>
> (B,G)
> are functors f : A ???> B with natural transformation t : F ==> G f.
> This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)]
> is
> ([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H
> g).
>
> However Cat//???Set??? is not locally cartesian closed basically because
> Cat
> is not. It is not even locally cartesian closed as a bicategory.
> The 2-category Gpd is cartesian closed; it is not locally cartesian
> closed;
> it is locally cartesian closed as a bicategory.
>
> Similarly, Gpd//???Set??? is locally cartesian closed as a bicategory.
> Often, in dealing with groups, we find groupoids help.
> This case is a good example and I hope helps in the applications
> you have in mind, Timothy.
>
> Ross
>

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Re: Is the category of group actions LCCC?

[Richard Garner]

It seems that the following is in fact true:

Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
cartesian closed with exponents stable under pullback, and Pi's exist
along product projections (and satisfy BCC), then E is cartesian closed.

The product of (a, phi) with (b, psi) in E is of course (a x b,
pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the
product projections in B.

The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi),
ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and 
ev: [b,c] x b ----> c in B.

This in particular applies to Cat//’Set’ as in Ross' message, seen as a
fibration over Cat with reindexing along f:A--->B given by
[f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to
pullbacks, but they don't satisfy BCC; however, right adjoints to
pullback along product projections are given just by (conical) limit
functors, and these do satisfy BCC. So the preceding construction
applies (and a bit of fiddling about shows that this does indeed agree
with Ross' prescription).

As for local cartesian closure: if B is lccc, each fibre is lccc with
fibrewise Pi's stable under pullback, and E--->B has all products, then
it seems that each slice fibration p/A: E/A--->B/pA will satisfy the
conditions in the second paragraph, whence E is also lccc.

Richard

On Thu, Sep 4, 2014, at 10:19 AM, Ross Street wrote:
> On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk>
> wrote:
> 
>> I'm wondering whether the category of ALL group actions is locally
>> Cartesian closed. 
> 
> This is what I answered Timothy:
> ======
> No, it’s not.
> Since the category has a terminal object (1,1), being a LCCC would imply
> it
> was cartesian closed. However, that would imply (G,X) \times — preserved
> the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0).
> ======
> 
> But it seems there is more to the story. 
> The thing stopping the category of actions from
> being cartesian closed is that the category Gp of groups is not. However,
> the category Gpd of groupoids and the category Cat of categories are.
> The (2-)category Cat//’Set’ of all category actions is defined as
> follows:
> objects (A,F) are functors F : A —> Set and morphisms (f,t) : (A,F) —> 
> (B,G)
> are functors f : A —> B with natural transformation t : F ==>  G f.
> This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is
> ([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H
> g).
> 
> However Cat//’Set’ is not locally cartesian closed basically because Cat
> is not. It is not even locally cartesian closed as a bicategory.
> The 2-category Gpd is cartesian closed; it is not locally cartesian
> closed;
> it is locally cartesian closed as a bicategory. 
> 
> Similarly, Gpd//’Set’ is locally cartesian closed as a bicategory.
> Often, in dealing with groups, we find groupoids help.
> This case is a good example and I hope helps in the applications
> you have in mind, Timothy.
> 
> Ross
> 

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Re: Is the category of group actions LCCC?

[Richard Garner]

A small correction. As well as:

Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
cartesian closed with exponents stable under pullback, and Pi's exist
along product projections (and satisfy BCC), then E is cartesian closed.

the following stronger statement is true (weakening the stability
required of the exponents):

Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
cartesian closed with exponents stable under pullback along product
projections, and Pi's exist along product projections (and satisfy BCC),
then E is cartesian closed.

In order to capture Ross' example, this stronger form is needed, since
[f,1]:[B,Set] ---> [A,Set] does not in general preserve exponentials,
while [pi_2,1]:[B,Set] ---> [A*B,Set] does so.

Richard


> 
> The product of (a, phi) with (b, psi) in E is of course (a x b,
> pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the
> product projections in B.
> 
> The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi),
> ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and 
> ev: [b,c] x b ----> c in B.
> 
> This in particular applies to Cat//’Set’ as in Ross' message, seen as a
> fibration over Cat with reindexing along f:A--->B given by
> [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to
> pullbacks, but they don't satisfy BCC; however, right adjoints to
> pullback along product projections are given just by (conical) limit
> functors, and these do satisfy BCC. So the preceding construction
> applies (and a bit of fiddling about shows that this does indeed agree
> with Ross' prescription).
> 
> As for local cartesian closure: if B is lccc, each fibre is lccc with
> fibrewise Pi's stable under pullback, and E--->B has all products, then
> it seems that each slice fibration p/A: E/A--->B/pA will satisfy the
> conditions in the second paragraph, whence E is also lccc.
> 
> Richard
> 

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Re: Is the category of group actions LCCC?

[Claudio Hermida]

On 2014-09-04, 10:05 PM, Richard Garner wrote:
>  It seems that the following is in fact true:
>
>  Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
>  cartesian closed with exponents stable under pullback, and Pi's exist
>  along product projections (and satisfy BCC), then E is cartesian closed.
>
>  The product of (a, phi) with (b, psi) in E is of course (a x b,
>  pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the
>  product projections in B.
>
>  The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi),
>  ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and
>  ev: [b,c] x b ----> c in B.

This is indeed the case and it appears as Corollary 4.12 in

Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure 
and Applied Algebra, Volume 134, Issue 1, 5 January 1999, Pages 83-109, ISSN 
0022-4049, http://dx.doi.org/10.1016/S0022-4049(97)00129-1.
(http://www.sciencedirect.com/science/article/pii/S0022404997001291)


>
>  This in particular applies to Cat//'Set' as in Ross' message, seen as a
>  fibration over Cat with reindexing along f:A--->B given by
>  [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to
>  pullbacks, but they don't satisfy BCC; however, right adjoints to
>  pullback along product projections are given just by (conical) limit
>  functors, and these do satisfy BCC. So the preceding construction
>  applies (and a bit of fiddling about shows that this does indeed agree
>  with Ross' prescription).
>
>  As for local cartesian closure: if B is lccc, each fibre is lccc with
>  fibrewise Pi's stable under pullback, and E--->B has all products, then
>  it seems that each slice fibration p/A: E/A--->B/pA will satisfy the
>  conditions in the second paragraph, whence E is also lccc.

That is also correct, but Cat is not lccc. To get this to work, one must
restrict Cat to the broad subcategory whose morphisms satisfy the
Conduche condition (which is the same as exponentiability in Cat), as
exposed in the nLab page

http://nlab.mathforge.org/nlab/show/Conduche+functor

Claudio



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Re: Is the category of group actions LCCC?

[Claudio Hermida]

On 2014-09-04, 10:05 PM, Richard Garner wrote:
> It seems that the following is in fact true:
>
> Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
> cartesian closed with exponents stable under pullback, and Pi's exist
> along product projections (and satisfy BCC), then E is cartesian closed.
>
> The product of (a, phi) with (b, psi) in E is of course (a x b,
> pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the
> product projections in B.
>
> The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi),
> ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and
> ev: [b,c] x b ----> c in B.

This is indeed the case and it appears as Corollary 4.12 in

Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure and Applied Algebra, Volume 134, Issue 1, 5 January 1999, Pages 83-109, ISSN 0022-4049, http://dx.doi.org/10.1016/S0022-4049(97)00129-1.
(http://www.sciencedirect.com/science/article/pii/S0022404997001291)


>
> This in particular applies to Cat//'Set' as in Ross' message, seen as a
> fibration over Cat with reindexing along f:A--->B given by
> [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to
> pullbacks, but they don't satisfy BCC; however, right adjoints to
> pullback along product projections are given just by (conical) limit
> functors, and these do satisfy BCC. So the preceding construction
> applies (and a bit of fiddling about shows that this does indeed agree
> with Ross' prescription).
>
> As for local cartesian closure: if B is lccc, each fibre is lccc with
> fibrewise Pi's stable under pullback, and E--->B has all products, then
> it seems that each slice fibration p/A: E/A--->B/pA will satisfy the
> conditions in the second paragraph, whence E is also lccc.

That is also correct, but Cat is not lccc. To get this to work, one must
restrict Cat to the broad subcategory whose morphisms satisfy the
Conduche condition (which is the same as exponentiability in Cat), as
exposed in the nLab page

http://nlab.mathforge.org/nlab/show/Conduche+functor

Claudio


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