nLab SupLat

SupLatSup Lat


SupLatSup 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, InfLatInf Lat is the category whose objects are inflattices and whose morphisms are inflattice homomorphisms, which preserve all meets.

Actually, SupLatSup Lat and InfLatInf 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 SupLatSup Lat and InfLatInf Lat as replete subcategories of Pos; similarly, CompLat can be viewed as a replete wide subcategory of SupLatSup Lat and InfLatInf Lat in two different ways.

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


SupLatSup 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 XX is the power set 𝒫X\mathcal{P}X of XX with the operation of union; an element aa of XX appears as the singleton subset {a}\{a\} in 𝒫X\mathcal{P}X.

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

SupLatSup 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 SupLatSup Lat is a quantale, including frames as a special case.

In fact, SupLatSup 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, SupLatSup Lat is not available in predicative mathematics. The definition goes through, but we cannot prove that SupLatSup 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 CompLatComp Lat is not wide as a subcategory of SupLatSup Lat, and SupLatSup Lat is not monadic (since 𝒫X\mathcal{P}X is generally large).

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

category: category

Last revised on August 9, 2020 at 20:21:30. See the history of this page for a list of all contributions to it.