Complete join-semilattice

A complete join-semilattice is a poset with joins of all families, including the empty family. A morphism preserves arbitrary joins.

The category $\mathbf{CSLat}$ has complete join-semilattices as objects and arbitrary-join-preserving maps as morphisms. Taking opposite orders and right adjoints gives an involutive self-duality.

In the category of complete join-semilattices, hom-sets have pointwise join as addition and the constant-bottom map as zero, making it a semi-additive category. For any family $A_i$, the cartesian product with coordinatewise joins is also its coproduct. The injections put an element in one coordinate and bottom in all others. A family of maps $f_i:A_i\to B$ extends uniquely by $(a_i)\mapsto {\bigvee }_i{f}_i(a_i)$, since each tuple is the join of its coordinate injections. Thus all set-indexed canonical product-coproduct comparisons are invertible. The two-element chain has distinct identity and zero morphisms, so this category is not trivial and not additive.