Is the $\mathrm{Ind}$-category of groups equivalent to the category of group objects of $\mathrm{Ind}(\mathbf{Set})$

category-theorylimits-colimitsuniversal-algebra

Let $\mathcal{T}$ be a Lawvere theory and let $\mathcal{C}$ be a category admitting finite products. Denote by $\mathrm{Mod}(\mathcal{T}, \mathcal{C})$ the category of models of $\mathcal{T}$ in $\mathcal{C}$.

Then, one can show that $\mathrm{Mod}(\mathcal{T}, \mathrm{Ind}(\mathcal{C}))$ admits small filtered colimits and the canonical embedding $\iota \colon \mathcal{C} \to \mathrm{Ind}(\mathcal{C})$ preserves finite products.

Since $\mathrm{Ind}$ is the free cocompletion with respect to filtered colimits, there is a unique (up to isomorphism) functor $\tilde{\iota_\ast} \colon \mathrm{Ind}(\mathrm{Mod}(\mathcal{T}, \mathcal{C})) \to \mathrm{Mod}(\mathcal{T}, \mathrm{Ind}(\mathcal{C}))$ such that $\tilde{\iota_\ast}$ preserves small filtered colimits and the diagram

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commutes. In particular, for any $\alpha \colon I \to \mathrm{Mod}(\mathcal{T}, \mathcal{C})$ where $I$ is small and filtered, we have
$\tilde{\iota_\ast}(\mathrm{''colim'' \alpha)} \cong \mathrm{colim} (\iota_\ast \circ \alpha)$.

Now, I was able to show that $\tilde{\iota_\ast}$ is fully faithful.

But e.g. for $\mathcal{C} = \mathbf{FinSet}$ and $\mathcal{T}$ the theory of groups, $\tilde{\iota_\ast}$ fails to be essentially surjective.
In that case, the functor $\tilde{\iota_\ast}$ is given by

$\begin{matrix} \tilde{\iota_\ast} & \colon & \mathrm{Ind}(\mathbf{FinGrp}) & \to & \mathbf{Grp}\\ & & \mathrm{''colim''}(\alpha) & \mapsto & \mathrm{colim}(\alpha) \end{matrix}$

This functor is not essentially surjective since $\mathbb{Z}$ cannot be the colimit of finite groups.

Does anyone know a counterexample in the case $\mathcal{C} = \mathbf{Set}$?

Best Answer

We can use more or less the same idea to construct a counterexample in the case $\mathcal{C} = \mathbf{Set}$.

We define a functor $$ \alpha \colon \mathbb{N}_0 \to \mathbf{Set}, n \mapsto, \{-n, \ldots, 0 , \ldots, n\}$$ where $\mathbb{N}_0$ is regarded as a filtered category by its order.

Then, $''\mathrm{colim}'' \alpha$ together with the obvious morphisms defines a group object in $\mathrm{Ind}(\mathbf{Set})$.

Now, by a lot of technical arguments one can show that this group object is not in the essential image of $\tilde{\iota_\ast}$.

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