Automorphism group objects

Question

Consider a monoidal category C with operation $\otimes$, unit object $1$, and diagonal map $\delta:A \to A \otimes A$ for all $A \in C$ (with naturality conditions on the diagonal map).

We can define a notion of "group object" in the category, where a group object is an object G of C along with a map from $1$ to G (playing the role of the identity element), a map from G to G (playing the role of the inverse map) and a map $G \otimes G \to G$ (playing the role of the multiplication). Then, we put compatibility conditions on these operations that are commutative diagrams corresponding to associativity, identity elements and inverses. Note that we need to use the diagonal morphism to formulate the condition on inverses.

Wikipedia defines group objects only in the case where the monoidal operation is the categorical product, so if necessary, we can restrict attention to just those cases.

We can define group object morphisms, etc. Some examples of group objects are: topological groups (category of topological spaces with continuous maps and Cartesian product), Lie groups (category of differential manifolds with smooth maps and manifold product), abelian groups (category of groups with group homomorphisms and direct product, by the Eckmann-Hilton principle), and groups (category of sets).

We can then proceed to define a "group object action" of a group object G on an object A by the analogues of the two conditions for a group action. My questions:

1. Under what situations does there exist a group object that plays the role played by the symmetric group on a set? i.e., A group object $\operatorname{Sym}(X)$ for each X in C with an action on X such that any group object action of G on X corresponds to a group object homomorphism $G \to \operatorname{Sym}(X)$ with the desired compatibility conditions?
2. How does this group object $\operatorname{Sym}(X)$, if it exists, relate to the automorphism group $\operatorname{Aut}_C(X)$ (which is an actual group, not a group object)?
3. We can also define a "group object action by automorphisms" as the action of one group object on another satisfying the additional condition of being group automorphisms. (We need to use the diagonal morphism to formulate this compatibility condition). Is there some object called the "automorphism group object of a group object" such that any group object action corresponds to a group object morphism to the automorphism group object?
4. How does this automorphism group object relate to the automorphism group in the group object category?

Answer 1

I can't give you a complete answer (perhaps an expert will come along...), but a Google search reveals that there is an exercise on page 213 of Mac Lane and Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, that basically asks the reader to prove that a sufficient condition is that the category be cartesian closed and admit pullbacks. In this case, the answer to question 2 is that the group of maps from 1 to the automorphism group object is naturally isomorphic to the group of automorphisms of the object. I think the answers to questions 3 and 4 are "yes" and "naturally isomorphic".

You can get endomorphism monoid objects with somewhat weaker hypotheses, namely if your category has representable homs. If for all $Y$ and $Z$, the functor $X \mapsto \operatorname{Hom}(X \otimes Y,Z)$ is representable, we call the representing object $\underline{\operatorname{Hom}}(X,Y)$. There is a rich theory that arises from existence of internal homs. For example, internal homs have a canonical composition structure $\underline{\operatorname{Hom}}(X,Y) \otimes \underline{\operatorname{Hom}}(Y,Z) \to \underline{\operatorname{Hom}}(X,Z)$, and they let you define dual objects $\underline{\operatorname{Hom}}(X,1)$ that satisfy nice compatiblities. Also note that $\operatorname{Hom}(1,\underline{\operatorname{Hom}}(X,Y)) = \operatorname{Hom}(X,Y)$. Unfortunately, the category of finite dimensional vector spaces over a field has internal homs, but only zero has an automorphism group object. This suggests that something like the cartesian closed condition is necessary.

Answer 2

I think that it might be worth refining your conditions to allow for some other interesting examples. Rather than have a diagonal map defined for each object, instead have the diagonal map as part of the group object. This would include vector spaces in your list of examples. The group objects would then by Hopf algebras.

The following example may be of interest: Let H be a Hopf algebra, this may act on an algebra A, we just ask that the algebra multiplication map of A is a morphism of H-modules. Equivalently A is an algebra object in the monoidal category of H-modules; A is called a H-module algebra.

Example: Let V be a representation of a Lie algebra g. Let A=SV be the free commutative algebra on a vs V. Then the universal enveloping algebra Ug acts on SV with g acting as derivations.

One may define a category of Hopf algebras acting on a fixed algebra A. This category has a terminal object, PA. This is very much reminiscent of what you might call Sym(A). PA contains the automorphism group algebra of A.

I'm afraid that I can't relate this example directly to your questions, but it's a good generalisation of the automorphism group, perhaps it might help.