nLab SupLat

Definitions

$Sup Lat$ 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, $Inf Lat$ is the category whose objects are inflattices and whose morphisms are inflattice homomorphisms, which preserve all meets.

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

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

Properties

$Sup Lat$ 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 $\mathcal{P}X$ of $X$ with the operation of union; an element $a$ of $X$ appears as the singleton subset $\{a\}$ in $\mathcal{P}X$.

The free inflattice on $X$ is slightly less natural; of course, we can take it to be $\mathcal{P}X$ 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 $\mathcal{P}X$ with the operation of intersection; this uses the fact that complementation is an automorphism of $\mathcal{P}X$. Then the generator $a$ appears as $X \setminus \{a\}$ in the lattice.

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

In fact, $Sup Lat$ is even a star-autonomous category, and a fortiori a linearly distributive category. The dual of a suplattice is its opposite poset, which is also a suplattice since every suplattice is also an inflattice.

In weak foundations

For all practical purposes, $Sup Lat$ is not available in predicative mathematics. The definition goes through, but we cannot prove that $Sup Lat$ 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 $Comp Lat$ is not wide as a subcategory of $Sup Lat$, and $Sup Lat$ is not monadic (since $\mathcal{P}X$ is generally large).

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

category: category