nLab
solution set condition

Contents

Idea

The solution set condition appears as part of the hypothesis in Freyd’s General Adjoint Functor Theorem.

Statement

A functor F:CDF : C \to D satisfies the solution set condition if for every object YY of DD there exists a small set II and an II-indexed family of morphisms {f i:YF(X i)} iI\{f_i : Y \to F(X_i)\}_{i \in I} such that any morphism h:YF(X)h : Y \to F(X) can be factored as

F(t)f i:Yf iF(X i)F(t)F(X) F(t) \circ f_i : Y \stackrel{f_i}{\to} F(X_i) \stackrel{F(t)}{\to} F(X)

for some t:X iXt : X_i \to X and some ii.

This is a smallness condition in that the family is required to be indexed by a small set.

A restatement of this condition is that the comma categories (dF)(d \downarrow F) all admit weakly initial families of objects.

Here is the connection with the adjoint functor theorem: when small products exist in those comma categories, this is equivalent to saying that they all admit weakly initial objects. When small equalizers exist in those comma categories also, this is equivalent to saying that they all admit initial objects, and this is equivalent to FF being a right adjoint.

Examples

Revised on May 12, 2016 04:51:00 by Anonymous Coward (24.205.10.89)