nLab Frm

FrmFrm is the category whose objects are frames and whose morphisms are frame homomorphisms, that is lattice homomorphisms that preserve directed joins (and thus all joins). FrmFrm is a subcategory of Pos, in fact a replete subcategory of both DistLat and SupLat.

The opposite category of Frm\mathbf{Frm} is the category Loc of locales; this is an example of the duality between space and quantity.


The category Frm\mathbf{Frm} is algebraic (see Stone Spaces).

Free frames

FrmFrm 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 frame on a set XX is the lattice of upper subsets of the free join-semilattice on XX, that is 𝒰𝒫 finX\mathcal{U}\mathcal{P}_{fin}X.

(This construction is constructively valid, but it does not go through in predicative mathematics, where infinite frames are generally large objects.)

Note that free frames are quite different from free locales (which are discrete spaces from a localic perspective).

category: category

Last revised on December 6, 2021 at 00:39:44. See the history of this page for a list of all contributions to it.