nLab
n-poset

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

An nn-poset is any of several concepts that generalize posets in higher category theory. In fact, nn-posets are the same as (n1,)(n-1,\infty)-categories.

Definition

Fix a meaning of (,)(\infty,\infty)-category, however weak or strict you wish. Then an nn-poset is a (n1)(n-1)-truncation of an (,)(\infty,\infty)-category. As all jj-morphisms for jnj \geq n are equivalent in an (n1)(n-1)-truncated (,r)(\infty,r)-category, all parallel pairs of jj-morphisms are equivalent for jnj \geq n by definition, which means that (n1,)(n-1,\infty)-categories are the same as (n1,n)(n-1,n)-categories. Thus, up to equivalence, there is no point in mentioning anything beyond nn-morphisms, not even whether two given parallel nn-morphisms are equivalent. This definition makes sense as low as n=1n = -1; the statement that parallel (1)(-1)-morphisms are equivalent simply means that there exists an object (a 00-morphism).

Special cases

In the light of the general definition, one must interpret ‘is’ up to equivalence of categories. The last statement also depends on how strict your definition of (,)(\infty,\infty)-category or nn-category is; it is actually simpler to define nn-posets from scratch as given above than to define them in terms of nn-categories.

Basic theorems

The (,)(\infty,\infty)-category of (small) nn-posets, as a full sub-(∞,∞)-category of the (,)(\infty,\infty)-category of (,)(\infty,\infty)-categories, is an (n+1)(n+1)-poset. That is, nn-posets form an (n+1)(n+1)-poset. This is well known for small values of nn.

Last revised on March 1, 2021 at 01:08:50. See the history of this page for a list of all contributions to it.