rigid (monoidal) categories that is cocomplete

Question

I am looking for examples of rigid (monoidal) categories that are cocomplete. I found some good examples of rigid categories in Etingof's book, but they seem that they are not cocomplete. Any help would be appreciated.

Thanks,

Answer 1

Why there aren't many examples.

One reason there are very few examples is because a rigid category with coproducts of size $\kappa$ must have biproducts of size $\kappa$ -- cf. here. Plenty of familiar categories have finite biproducts, such as the category of abelian groups, but very few have infinitary biproducts.

Another related reason there are few examples is that a rigid category is self-dual. Most good cocomplete categories are locally presentable, but a locally presentable category is self-dual iff it is a preorder.

An example. There should be examples of preorders which are rigid, but I'm struggling to come up with one beyond the one-element poset. So here's just one example, which is not a poset:

The category $\mathsf{SupLat}$ of suplattices is an example of a rigid cocomplete category which is not a preorder.

Here, $\mathsf{SupLat}$ is a non-full subcategory of the category of posets. An object $S$ of $\mathsf{SupLat}$ is a sup-lattice, i.e. a (small) poset $S$ such that every subset of $S$ has a supremum in $S$. A morphism $S \to T$ is a map which preserves all suprema. The internal hom is the obvious one: $[S,T]$ is the set of morphisms $S \to T$ under the usual pointwise ordering. The unit is the 2-element suplattice $\top$. The dual of a suplattice $S$ is $[S,\top] = S^{op}$, i.e. $S$ with the reverse ordering. The tensor product may be defined as $S \otimes T = [S,T^{op}]^{op}$.

Some comments:

• Any morphism $f: S \to T$ is order-preserving, since the order can be defined as $x \leq y \leftrightarrow x \vee y = y$, so if $x \leq y$, then $f(x) \vee f(y) = f(x \vee y) = f(y)$ so that $f(x) \leq f(y)$.

• Any (small) sup-lattice $S$ is also an inf-lattice (i.e. any subset of $S$ has an inf). But the morphisms in $\mathsf{SupLat}$ are not required to preserve infs.

• The tensor product $S\otimes T$ has the following universal property. Let $f: S \times T \to U$ be an order-preserving map between suplattices which is bilinear in the sense that $\vee_{a \in A} f(a,t) = f(\vee A,t)$ and $\vee_{b \in B} f(s,b) = f(s,\vee B)$ for all $s \in S, t \in T, A \subseteq S, B \subseteq T$.

• The nlab only states that $\mathsf{SupLat}$ is star-autonomous which is weaker than being rigid, in that the unit need not be self-dual. I think the reason for this is that in weak foundations, the unit of the monoidal structure is the object of truth values; in weak foundations this need not be self-dual, but in ordinary mathematics it is just the two-element sup-lattice, which is definitely self-dual.

For more discussion of $\mathsf{SupLat}$ see here.