nLab discrete object classifier

Idea

Just as a subobject classifier in a (1,1)-category classifies the monomorphisms or (-1)-truncated morphisms (and thus the subobjects) of the category, a discrete object classifier in a (2,1)-category should classify the faithful or 0-truncated morphisms (and thus the discrete objects) in the (2,1)-category, see at n-truncated morphisms – between groupoids.

Definition

In a (2,1)-category $C$ with terminal object $*$, interval object $I$, and finite (2,1)-pullbacks, a discrete object classifier is a morphism $inhabited: [I, Set] \rightarrow Set$ whose target is a nonterminal object $Set$ and whose source is the internal hom-object $[I, Set]$ such that for every faithful morphism $B \colon U \rightarrow G$ in $C$, there is a unique morphism $F:G \rightarrow Set$ such that there is a (2,1)-pullback diagram of the form

$$\array{U & \to & [I,Set] \\ ^{B}\downarrow & \cong & \downarrow^{inhabited}\\ G & \underset{\chi_U}{\to} & Set}$$

Remark

If $Set$ exists, it is typically called the universe of sets or groupoid of sets. A global element $F:G \rightarrow Set$ is typically called an indexed family, an element $A:* \rightarrow Set$ is typically called a set, and a set $A:* \rightarrow Set$ is inhabited if there exists a morphism $B:* \rightarrow [I, Set]$ such that the $A$ factors into $inhabited \circ B$. All these terms refer to the internal set theory of the (2,1)-category $C$.

The morphism $\chi_U$ is also called the classifying morphism of the discrete object $U$ and morphism $B:U \rightarrow G$.

Examples

In $Grpd$

In the (2,1)-category Grpd of groupoids and functors between groupoids, the discrete object classifier $Set$ is the groupoids of sets, or, as $Set$ is a groupoid with a category structure, more commonly known as the category of sets.

See also

• subobject classifier

• (sub)object classifier in an (infinity,1)-topos

• classifying morphism

• type of types, universe in a topos

• class object

References

• David Corfield: 101 things to do with a 2-classifier (blog)