discrete object classifier

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

(in category theory/type theory/computer science)

**of all homotopy types**

**of (-1)-truncated types/h-propositions**

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*.

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

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

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 $B:U \rightarrow G$.

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.

- 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.