\( \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\uHom}{\,\underline{\!Hom\!}\,} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\map}{map} \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\comp}{comp} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\true}{true} \DeclareMathOperator{\Sub}{Sub} \DeclareMathOperator{\Lan}{Lan} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\PSh}{PSh} \DeclareMathOperator{\Sh}{Sh} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\Glue}{Glue} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\oneim}{1im} \DeclareMathOperator{\twoim}{2im} \DeclareMathOperator{\charr}{char} \DeclareMathOperator{\Spec}{Spec} \newcommand{\ProFinSet}{\mathrm{ProFinSet}} \newcommand{\sSet}{\mathrm{sSet}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\deltacat}{\boldsymbol{\Delta}} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Set}{\mathrm{Set}} \newcommand{\Ring}{\mathrm{Ring}} \newcommand{\CatMon}{\mathrm{CatMon}} \newcommand{\Cof}{\mathrm{Cof}} \newcommand{\Fib}{\mathrm{Fib}} \newcommand{\Frm}{\mathrm{Frm}} \newcommand{\Loc}{\mathrm{Loc}}\)

Frames

Table of Contents

1. Frames

A frame \( A \) is

  • a lattice \( A \) (partially ordered set with finite meets and joins),

such that

  • \( A \) is complete (arbitrary joins exist)
  • infinite distributive law:
\begin{equation*} a \wedge \bigvee _{i \in I} b_i = \bigvee _{i \in I}(a \land b_i) \end{equation*}

Let \(A, B \) be frames. A morphism of frames (or frame homomorphism) \(h \colon A \rightarrow B \) is a morphism of sets that preserves finite meets and arbitrary joins.

We write \( \Frm \) for the categories of frames and morphisms of frames.

2. Locales

A locale is the same thing as a frame. Let \( X \) be a locale. Then we write \(\mathcal O (X) \) for the frame corresponding to \(X \).

Let \(X, Y \) be locales. A morphisms of locales \(f \colon X \rightarrow Y \) is a morphism of frames \(f^* \colon \mathcal O (Y) \rightarrow \mathcal O (X) \).

We write \(\Loc \) for the category of locales, ie for the category \(\Frm ^{op} \).

Author: Frederik Gebert

Created: 2025-02-10 Mon 21:45