TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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In a locally small category, separating and coseparating sets are equivalently jointly faithful families of representables

Statement

Let $C$ be locally small and let $G$ be a set of objects. Then $G$ is separating if and only if the family of covariant representables $C (G, -) : C \to S e t (G \in G)$ is jointly faithful. Dually, a set $H$ is coseparating if and only if the family $C (-, H) : C^{o p} \to S e t$ is jointly faithful.

Here a family of functors $(F_{i})_{i \in I}$ with common domain is jointly faithful when, for every parallel pair $f, g : X \to Y$ in that domain, $F_{i} (f) = F_{i} (g)$ for all $i \in I$ implies $f = g$ . For a one-member family this is faithfulness in the sense of Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors.

Facts & Assumptions

Given: A locally small category $C$ and supplied sets of objects $G$ and $H$ .

[L1]

Local smallness makes every hom-collection $C (X, Y)$ a set (Small, locally small, and large categories).

[L2]

In a locally small category the hom-assignments $C (G, -)$ and $C (-, H)$ are functors to $S e t$ (The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$ ).

[L3]

A functor $F$ is faithful when every induced $F_{A, B} : C (A, B) \to D (F A, F B)$ is injective (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors). Joint faithfulness of a family is the condition stated in this item's own Statement, and is not taken from the cited definition.

[L4]

A separating set detects distinct maps by precomposition, while a coseparating set detects them by postcomposition (Separating and coseparating sets of objects).

Proof

technique · direct

1.1L1L2L3L4

If $G$ is separating and $f, g : X \to Y$ have equal images under every $C (G, -)$ , then $f \circ h = g \circ h$ for every $G \in G$ and $h : G \to X$ ; [L4] forces $f = g$ . Conversely, joint faithfulness says that distinct $f, g$ have unequal images under some $C (G, -)$ , which means some $h : G \to X$ satisfies $f h \neq g h$ . Thus the separating and joint-faithfulness conditions are equivalent.

2.1step 1.1L2L3L4∎

Applying the same argument in the opposite category exchanges precomposition with postcomposition and proves that $H$ is coseparating exactly when the contravariant representables $C (-, H)$ are jointly faithful.

Depends on

Used by

The cancellation and epimorphism descriptions of a generator agree Theorem

Dependency tree · two levels

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Sources

E. Riehl, Category Theory in Context, section 4.7 (standard reference, not scraped)