Re: Is the category of group actions LCCC?

[Clemens.BERGER@unice.fr]

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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