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, an II-indexed family (X i) iI(X_i)_{i \in I} of objects of CC and an II-indexed family (f i:YF(X i)) iI(f_i\colon Y \to F(X_i))_{i \in I} of morphisms in DD such that each morphism h:YF(X)h\colon Y \to F(X) in DD can be factored, for some index ii and some morphism t:X iXt\colon X_i \to X in CC, as

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

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

  • To define the projective tensor product of Banach spaces using the adjoint functor theorem, let CC and DD each be Ban (the category of Banach spaces and short linear maps), fix a Banach space BB, and let F(X)F(X) be BdLin(B,X)BdLin(B,X), the Banach space of bounded linear maps from BB to XX.

    If f:X 1X 2f\colon X_1 \to X_2 is a short linear map and ϕ:BX 1\phi\colon B \to X_1 is a bounded linear map, then BdLin(B,f)(ϕ)fϕ:BX 2BdLin(B,f)(\phi) \coloneqq f \circ \phi\colon B \to X_2 is a bounded linear map with norm BdLin(B,f)(ϕ)fϕ{\|BdLin(B,f)(\phi)\|} \leq {\|f\|} {\|\phi\|}, which proves that BdLin(B,f)f1{\|BdLin(B,f)\|} \leq {\|f\|} \leq 1, so that BdLin(B,f)BdLin(B,f) is also a short map, as it must be for FF to be a functor. (That FF is linear and preserves identities and composition is trivial.)

    The solution set condition now states: For every Banach space YY there exists a small set II, an II-indexed family (X i) iI(X_i)_{i \in I} of Banach spaces and an II-indexed family (f i:YBdLin(B,X i)) iI(f_i\colon Y \to BdLin(B,X_i))_{i \in I} of short linear maps such that each short linear map h:YBdLin(B,X)h\colon Y \to BdLin(B,X) can be factored, for some index ii and some short linear map t:X iXt\colon X_i \to X, as h=BdLin(B,t)f ih = BdLin(B,t) \circ f_i.

    (to be completed)

Revised on August 23, 2016 16:20:09 by Toby Bartels (64.89.52.42)