solution set condition



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


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.

Revised on October 31, 2010 07:35:46 by Kevin V? (