Example of a functor preserving only finite coproducts

Question

What is an example of a functor $$F : \mathsf{Set} \to \mathsf{Set}$$ which preserves finite coproducts, but not infinite coproducts?

The functors preserving infinite coproducts are given by $T \times - : \mathsf{Set} \to \mathsf{Set}$ for some set $T$. If a functor preserves finite coproducts, then the natural map $F(1) \times X \to F(X)$ is an isomorphism if $X$ is finite. Thus, a counterexample will involve infinite sets.

For the category $\mathsf{Ab}$ of abelian groups, a counterexample is simply $X \mapsto X^{\mathbb{N}}$.

Edit.

Jakob Werner has suggested the functor $F(X) = \mathrm{Spec}(A^X)$ for any commutative ring $A$. For fields $A$ this gives the functor of ultrafilters in my answer. I am still curious if there are other, more basic classes of examples.

Answer 1

The following example seems to work. Consider the composition $$F : \mathsf{Set} \xrightarrow{D} \mathsf{Top} \xrightarrow{\beta} \mathsf{CompHaus} \xrightarrow{U} \mathsf{Set},$$ where $D$ is the discrete topology, $\beta$ is the Stone-Cech-compactification, and $U$ is the forgetful functor. In other words, $F(X)$ is the set of ultrafilters on $X$. Since $D$ and $\beta$ are left adjoint and $U$ preserves finite coproducts, it follows that $F$ preserves finite coproducts. But it does not preserve arbitrary coproducts, since $F(\mathbb{N})$ is not isomorphic to $\coprod_{n \in \mathbb{N}} F(\{n\}) \cong \mathbb{N}$.

Answer 2

Here is a class of examples, generalizing yours: Look at the composition $$\mathbf{Set}\xrightarrow{\mathfrak{P}}\mathbf{CRing}^{\operatorname{op}}\xrightarrow{\operatorname{Spec}}\mathbf{Set}\,.$$ The first functor takes a set $X$ to the power set ring $\mathfrak{P}(X)\cong\mathbb{F}_2^X$. The interesting thing happens at the second arrow. One can check that the compositions is isomorphic to your functor $F$, as Asaf Karigala points out. But of course you can replace $\mathbb{F}_2$ by any other fixed commutative ring.

Answer 3

As noted, the ultrafilter functor is an example; in fact, it is the terminal example, as shown by Reinhard Börger in this paper:

http://www.sciencedirect.com/science/article/pii/0022404987900417

All finite-coproduct-preserving endofunctors of $\mathbf{Set}$ are actually quite closely related to ultrafilters. Another basic example is as follows. If $\mathscr U$ is an ultrafilter on a set $A$, we can consider the ultrapower functor $(-)^{\mathscr U} \colon \mathbf{Set} \to \mathbf{Set}$. This sends a set $X$ to the quotient of $X^A$ by the equivalence relation wherein $\vec{x} \equiv \vec{y}$ just when $\{a \in A : x_a = y_a\} \in \mathscr U$ (i.e., when $\vec x = \vec y$ "$\mathscr U$-almost everywhere").

The ultrapower functors preserve finite coproducts (as well as finite limits). In fact, the ultrapower functors are basic in the sense that any finite-coproduct-preserving functor $\mathbf{Set} \to \mathbf{Set}$ is a colimit in $[\mathbf{Set}, \mathbf{Set}]$ of ultrapower functors.

Here is another example, again involving ultrafilters. Let $\mathbb{T}$ be any first-order theory and let $M$ and $N$ be models for $\mathbb{T}$. We can define a finite-coproduct-preserving functor $F_{M,N} \colon \mathbf{Set} \to \mathbf{Set}$ by taking

$$F_{M,N}(X) = \sum_{\mathscr U \in \beta X} \mathbf{Emb}(M^\mathscr{U}, N) $$

where here $\beta X$ is the set of ultrafilters on $X$; $M^{\mathscr U}$ is the ultrapower model of $\mathbb T$ (which is a model by Łoś's theorem); and $\mathbf{Emb}(M^{\mathscr U}, N)$ denotes the set of elementary embeddings of $M^{\mathscr U}$ in $N$.

Answer 4

I think the example that HeinrichD gave himself is probably the canonical one.

But I'll just point out that it's not minimal, in that it has proper subfunctors that also work.

Heinrich's example can be restated as "$F(X)$ is the set of maximal ideals of $\mathbb{F}_2^X$". But you could also take the subset of maximal ideals for which the quotient field is bounded in size by some cardinal.

Or a similar idea: fix a commutative ring $R$ and an integral domain $S$, and let $F(X)$ be the set of ring homomorphisms $R^X\to S$.