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Universal elements are initial in a covariant category of elements and terminal in a presheaf category of elements

Statement

Let $C$ be locally small.

For $F : C \to S e t$ , a pair $(R, u)$ is a universal element if and only if $(R, u)$ is initial in $\int F$ .
For $P : C^{o p} \to S e t$ , a pair $(R, u)$ is a universal element if and only if $(R, u)$ is terminal in $\int P$ .

Facts & Assumptions

Given: A locally small category $C$ and either a functor $F : C \to S e t$ or a presheaf $P$ with the indicated pair $(R, u)$ .

[L1]

Covariant universality says that for every $(c, x)$ there is a unique $f : R \to c$ with $F (f) (u) = x$ ; presheaf universality says there is a unique $f : c \to R$ with $P (f) (u) = x$ (A representation is equivalently a universal element with a unique factorisation property).

[F1]

In $\int F$ , a morphism $(R, u) \to (c, x)$ is an $f : R \to c$ satisfying $F (f) (u) = x$ ; in $\int P$ , a morphism $(c, x) \to (R, u)$ is an $f : c \to R$ satisfying $x = P (f) (u)$ (The category of elements of a covariant functor or a presheaf).

[F2]

An object is initial when it has exactly one morphism to every object and terminal when it has exactly one morphism from every object (Initial object, terminal object, and zero object).

Proof

technique · direct

1.1

By [F1], the morphisms $(R, u) \to (c, x)$ in $\int F$ are exactly the morphisms counted by the covariant condition in [L1].

L1F1

1.2

By [F1], the morphisms $(c, x) \to (R, u)$ in $\int P$ are exactly the morphisms counted by the presheaf condition in [L1].

L1F1

2.1

Thus that condition holds for all $(c, x)$ if and only if $(R, u)$ has exactly one morphism to each object of $\int F$ , which by [F2] is initiality.

step 1.1F2

3.1

Thus that condition holds for all $(c, x)$ if and only if $(R, u)$ has exactly one morphism from each object of $\int P$ , which by [F2] is terminality.

step 1.2F2∎

Depends on

Used by

Freyd's representability theorem for continuous Set-valued functors satisfying a solution set condition Theorem

Cited to discharge well-definedness by The category of elements of a covariant functor or a presheaf.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Proposition 2.4.8 (standard reference, not scraped)