Construction of Generators

Lemma.

In a category let $S$ be a generating set which is strongly connected (between any two objects in $S$ there is a morphism). If the coproduct $U \coloneqq \coprod_{G \in S} G$ exists, then it is a generator.

Proof. This is a straight forward generalization of this result. We remark that the assumption about $S$ implies that each inclusion $G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f h = g h$ for all $h : U \to A$ . If $G \in S$ , any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $fh = gh$ holds for all $h : G \to A$ and $G \in S$ . Since $S$ is a generating set, this implies $f = g$ . $\square$

Author: Martin Brandenburg

Context

This page is referenced by the following categories.

category of simplicial sets