functors $\text{Vect} \to \text{Vect}$ that preserve filtered and sifted colimits

Question

I'm considering various functors from the category $\text{Vect}$ of real vector spaces to itself, and would like to know that they preserve filtered colimits and possibly even sifted colimits. The functors I'm interested in send $V$ to the power $V^k$, the tensor power $V^{\otimes k}$, the exterior power $\Lambda^k(V)$, and the free vector space $F(V)$ on the underlying set of $V$. I have proved by hand that some of these preserve filtered colimits or sifted colimits, but I'm looking for references and/or conceptual arguments.

Of course, I'd also like to know if some of these don't preserve filtered and/or sifted colimits.

(I doubt it matters that I'm working over the reals.)

Answer 1

Maybe the following tools can help.

1. $G: \mathcal{C}_1 \times \cdots\times \mathcal{C}_k\to \mathcal{D}$ preserves sifted colimits separately in each variable if and only if it preserves sifted colimits. Indeed, given a diagram $p: K \to \mathcal{C}_1\times\cdots \times \mathcal{C}_k$ with $K$ sifted, observe that $p= (q_i)\circ \delta$ where $(q_i): \prod_{i=1}^kK \to \prod_{i=1}^k\mathcal{C}_i$ is the product of $q_i=\mathrm{proj}_i\circ p$ and $\delta$ is the diagonal. Since $K$ is sifted, $\delta$ is final, and the colimit may be computed over $K^{\times k}$ instead of over $K$. But then we can apply the hypothesis $k$ times to see that this colimit is preserves by $G$.
2. The diagonal map $\mathcal{C} \to \mathcal{C}^{\times j}$ preserves all colimits.
3. The forgetful functor $\mathbf{Vect} \to \mathbf{Set}$ preserves all sifted colimits (since it preserves filtered colimits and reflexive coequalizers by inspection).
4. The free vector space functor preserves all colimits, being a left adjoint.

Combining (4) and (3) gives the "F(V)" example you wanted, so let's move on to the others.

Notice that the tensor power functor $V \mapsto V^{\otimes n}$ is a composite of the diagonal and the tensor product functor, which preserves all colimits separately in each variable, so it preserves sifted colimits. Moreover, there is an action of $\Sigma_n$ on $V^{\otimes n}$ so this functor refines to $\mathbf{Vect} \to \mathbf{Fun}(\mathrm{B}\Sigma_n, \mathbf{Vect})$. This also preserves sifted colimits since colimits in functor categories are computed pointwise.

It's also true that the product $V \mapsto V^{\times n}$, as a functor to vector spaces with a $\Sigma_n$ action, preserves sifted colimits, since the cartesian product of sets preserves all colimits of sets in each variable, and sifted colimits of vector spaces may as well be computed on the underlying set, so the same reasoning applies (even though the cartesian product functor does not preserve all colimits of vector spaces separately in each variable).

So now, given a functor $\mathbf{Fun}(\mathrm{B}\Sigma_n, \mathbf{Vect})\to \mathbf{Vect}$ that preserves sifted colimits, we can apply it to the tensor power functor to get new functors. For example, we can take $\Sigma_n$-coinvariants (i.e. symmetric powers). We can tensor with the sign representation and then take $\Sigma_n$-coinvariants (which gives the exterior power). Any Schur functor works too, since we're just tensoring over $\mathbb{C}[\Sigma_n]$ (or $\mathbb{R}[\Sigma_n]$ over the reals) with the corresponding representation, and that preserves colimits.