Category theory; homological algebra Definition

Solution set condition

Definition

Let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor between categories. $G$ satisfies the solution set condition if the following holds for every object $c$ of $\mathcal{C}$ . There are a set $I$ , objects $d_i$ of $\mathcal{D}$ and morphisms $f_i \colon c \to G d_i$ for $i \in I$ with this property. Let $d$ be any object of $\mathcal{D}$ and $h \colon c \to G d$ any morphism. Then $h = G(t) \circ f_i$ for some $i \in I$ and some morphism $t \colon d_i \to d$ .

Remarks

The family $(f_i)_{i \in I}$ is called a solution set for $c$ . The factorization need not be unique, and $I$ must be a set, not a proper class.

Example: let $G$ be the forgetful functor from groups to sets, and let $c = X$ be a set. Every map $h \colon X \to G H$ lands in the subgroup $H'$ generated by $h(X)$ , which has at most $\max(\abs{X}, \aleph_0)$ elements. Each such $H'$ is isomorphic to a group whose underlying set is a subset of $X \sqcup \N$ . All such groups, with all maps from $X$ into them, form a solution set for $X$ .

Non-example: the forgetful functor from complete Boolean algebras to sets fails it, since it preserves limits but has no left adjoint. Indeed, there is no free complete Boolean algebra on a countably infinite set.

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Solution set condition definition

Tag	`H5QDJACKMS`
URI	`m2://H5QDJACKMS`
Kind	definition
Writing	W0 stub
MSC	18A40
Defines	Solution set condition
Uses	Category, Functor, Morphism

Solution set condition

Definition

Remarks

Reading path

Further reading

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