CatDat

Implication Details

Claim: If a category has an extremal generating set and is left cancellative and is locally finite and is semi-strongly connected, then it is essentially small.

Proof: Suppose a category $\C$ is locally finite, left cancellative, semi-strongly connected, and has an extremal generating set $S$. We then claim that $$\Ob(\C) \to \IN^S, \, X \mapsto \bigl(G \mapsto \card(\Hom(G, X))\bigr)$$ is injective on isomorphism classes of $\Ob(\C)$. To see this, suppose two objects $X$ and $Y$ map to the same cardinality tuple. Since $\C$ is semi-strongly connected, we may assume without loss of generality that there is a morphism $f : X \to Y$. Then since $f$ is a monomorphism, for each $G \in S$ we have $$f \circ {-} : \Hom(G, X) \to \Hom(G, Y)$$ is an injective function between finite sets of equal cardinality, and therefore is also a bijection. By the assumption that $S$ is an extremal generating set, we thus have $f$ is an isomorphism.
This shows that the collection of isomorphism classes of objects of $X$ is in bijection with a set. Together with the assumption that the category is locally finite, this implies the category is essentially small.