nLab
discrete object classifier

Contents

Context

2-Category Theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Universes

Contents

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

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

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

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

Examples

In GrpdGrpd

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

See also

References

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

Last revised on March 2, 2021 at 11:24:34. See the history of this page for a list of all contributions to it.