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For a category $\mathbf{C}$ with finite products, denote by $\mathbf{C}_{\text{Grp}}$ the category of group objects in $\mathbf{C}$.

Using the fact that $G\in \operatorname{Obj}(\mathbf{C})$ is a group object iff $\operatorname{Mor}_{\mathbf{C}}(C,G)$ is a group (equipped with the induced maps) for all $C\in \operatorname{Obj}(\mathbf{C})$ it should follow that $\mathbf{C}_{\text{Grp}}$ has products because $\operatorname{Mor}_{\mathbf{C}}(\cdot ,G)\times \operatorname{Mor}_{\mathbf{C}}(\cdot ,G)=\operatorname{Mor}_{\mathbf{C}}(\cdot ,G\times H)$.

This argument of course fails if one replaces "product" with "coproduct". Indeed, taking $\mathbf{C}$ to be the category of finite sets, it seems that in general we should not expect the category of group objects to possess coproducts. This then yields the question: what are sufficient conditions on $\mathbf{C}$ to guarantee that $\mathbf{C}_{\text{Grp}}$ has coproducts? What about all (small) colimtis? To what extent does this generalize to the category of models of an algebraic theory in a category?

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The following is due to Linton [Coequalizers in categories of algebras]:

Let $\mathcal{C}$ be a category with colimits of small diagrams and let $\mathbb{T}$ be a monad on $\mathcal{C}$. The following are equivalent:

  • $\mathcal{C}^\mathbb{T}$ has colimits of all small diagrams.
  • $\mathcal{C}^\mathbb{T}$ has coequalisers of all parallel pairs.
  • $\mathcal{C}^\mathbb{T}$ has coequalisers of all reflexive pairs.

Basically, the idea is that the coproduct of $\mathbb{T}$-algebras is presented by the "coproduct" of the presentations – this is why we need coequalisers.

In order to apply Linton's theorem to your problem, we need to answer two questions:

  • Is the category of group objects in $\mathcal{C}$ monadic over $\mathcal{C}$?
  • Does the category of group objects in $\mathcal{C}$ have coequalisers of all reflexive pairs?

For the first question, one notes that all the conditions of Beck's monadicity theorem are automatically satisfied except perhaps for the existence of the left adjoint – so one hopes to apply a suitable adjoint functor theorem. The second question is significantly harder. A sufficient condition is that $\mathcal{C}$ be a locally presentable category and the monad be accessible. Or, if $\mathcal{C}$ is cartesian closed, then one can prove by hand that the forgetful functor creates coequalisers of reflexive pairs.

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  • $\begingroup$ I have a couple of follow-up questions about this, which for convenience and reasons of size, I will break-up into separate comments. Actually one of the things I got stuck on last night was the application of the (general) adjoint functor theorem to the forgetful functor, in particular, the verification of the solution set condition. It seems that for $\mathbf{C}$ a general cocomplete category, this is hopeless (though I didn't bother looking for a counter-example)? What are sufficient conditions to make this work? $\endgroup$ – Jonathan Gleason Mar 29 '16 at 17:36
  • $\begingroup$ If $\mathcal{C}$ is locally presentable then you have a left adjoint. $\endgroup$ – Zhen Lin Mar 29 '16 at 17:37
  • $\begingroup$ Perhaps I could verify the other conditions of Beck's Theorem by hand, but I don't know enough category theory to make their deduction "automatic". What fact tells us immediately these conditions are satisfied without having to check? $\endgroup$ – Jonathan Gleason Mar 29 '16 at 17:39
  • $\begingroup$ It's not automatic – you do have to check it. $\endgroup$ – Zhen Lin Mar 29 '16 at 17:43
  • $\begingroup$ Finally, if I understand you correctly, then $\mathbf{C}$ cocomplete, cartesian closed, and locally presentable is sufficient to guarantee that the category of group objects is cocomplete? $\endgroup$ – Jonathan Gleason Mar 29 '16 at 17:44

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