# nLab SemiLat

$Semi Lat$ is the category whose objects are semilattices and whose morphisms are semilattice homomorphisms, that is functions which preserve finitary joins (equivalently, binary joins and the bottom element).

We mention joins above as if the objects are join-semilattices; one can just as well consider them meet-semilattices and say that the homomorphisms preserve finitary meets (including the top element). For $Semi Lat$ in itself, this is purely a difference in notational convention. We can avoid this choice by saying $Semi Lat$ is the category whose objects are idempotent commutative monoids and whose morphisms are monoid homomorphisms. However, the choice corresponds to using either of two inclusion functors representing $Semi Lat$ as a subcategory of Pos.

There is a forgetful functor from $Semi Lat$ to $Set$, and its left adjoint sends any set $X$ to the free semilattice on that set: namely, the finite powerset $\mathcal{P}_{fin}(X)$ of $X$, i.e. the set of finite subsets of $X$, thought of as a join-semilattice. (Note that this exists even in predicative mathematics, as long as we are allowed to define sets by recursion over natural numbers, although you have to construct it by general nonsense instead of as a subset of the full powerset. For purposes of constructive mathematics, by ‘finite’ we mean Kuratowski finite?.)

$Semi Lat$ is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories.

category: category

Last revised on March 9, 2019 at 21:22:20. See the history of this page for a list of all contributions to it.