# $\mathrm{Sup}\mathrm{Lat}$

## Definitions

$\mathrm{Sup}\mathrm{Lat}$ is the category whose objects are suplattices and whose morphisms are suplattice homomorphisms, that is functions which preserve all joins (including the bottom element). Analogously, $\mathrm{Inf}\mathrm{Lat}$ is the category whose objects are inflattices and whose morphisms are inflattice homomorphisms, which preserve all meets.

Actually, $\mathrm{Sup}\mathrm{Lat}$ and $\mathrm{Inf}\mathrm{Lat}$ are equivalent; the difference between the two is merely the notational choice between $\le$ and $\ge$. However, this choice corresponds to using either of two inclusion functors representing $\mathrm{Sup}\mathrm{Lat}$ and $\mathrm{Inf}\mathrm{Lat}$ as replete subcategories of Pos; similarly, CompLat can be viewed as a replete wide subcategory of $\mathrm{Sup}\mathrm{Lat}$ and $\mathrm{Inf}\mathrm{Lat}$ in two different ways.

One can write $\mathrm{Comp}\mathrm{Semi}\mathrm{Lat}$ (meaning the category of complete semilattices) if one wishes to remain ambiguous about the notation.

## Properties

$\mathrm{Sup}\mathrm{Lat}$ is given by a variety of algebras, or equivalently by an algebraic theory, so it is an equationally presented category; however, it requires operations of arbitrarily large arity. Nevertheless, it is a monadic category (over Set), because it has free objects. Specifically, the free suplattice on a set $X$ is the power set $𝒫X$ of $X$ with the operation of union; an element $a$ of $X$ appears as the singleton subset $\left\{a\right\}$ in $𝒫X$.

The free inflattice on $X$ is slightly less natural; of course, we can take it to be $𝒫X$ with the operation of union again, but then the order on the elements is the opposite of the usual order. However, we can also take it to be $𝒫X$ with the operation of intersection; this uses the fact that complementation is an automorphism of $𝒫X$. Then the generator $a$ appears as $X\setminus \left\{a\right\}$ in the lattice.

$\mathrm{Sup}\mathrm{Lat}$ is a monoidal category; it admits a tensor product which represents binary morphisms: functions which preserve joins separately in each variable. A monoid in $\mathrm{Sup}\mathrm{Lat}$ is a quantale, including frames as a special case.

## In weak foundations

For all practical purposes, $\mathrm{Sup}\mathrm{Lat}$ is not available in predicative mathematics. The definition goes through, but we cannot prove that $\mathrm{Sup}\mathrm{Lat}$ has any infinite objects. (More precisely, the power set of any nontrivial small suplattice must be small.) Generally speaking, predicative mathematics treats infinite suplattices only as large objects. Although they are of little interest, we can ask which of the facts above hold predicatively; the answer is that $\mathrm{Comp}\mathrm{Lat}$ is not wide as a subcategory of $\mathrm{Sup}\mathrm{Lat}$, and $\mathrm{Sup}\mathrm{Lat}$ is not monadic (since $𝒫X$ is generally large).

In impredicative constructive mathematics, we cannot intepret $𝒫X$ with intersection as the free inflattice on $X$, since complementation is not an automorphism. Everything else goes through, however, including the interpretation of $𝒫X$ with reverse inclusion as the free inflattice. In particular, $\mathrm{Sup}\mathrm{Lat}$ (and hence $\mathrm{Sup}\mathrm{Inf}$) is still a monadic category.

category: category

Revised on September 17, 2012 18:10:37 by Toby Bartels (98.23.131.250)