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Idea

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

Definition

Fix a meaning of $(\mathrm{\infty },\mathrm{\infty })$ \infty (\infty,\infty)-category, however weak or strict you wish. Then an $n$-poset is an a$\mathrm{\infty }(n-1)$ \infty (n-1) -category - such that alltruncation of an $(\infty,\infty)$-category. As all $j$-morphisms for $j \geq n$ are equivalent in an $(n-1)$-truncated $(\infty,r)$-category, all parallel pairs of $j$-morphisms are equivalent for $j \geq n$ . Thus, by up definition, to which equivalence, means there that is no point in mentioning anything beyond$(n-1,\mathrm{\infty })$ n (n-1,\infty) -morphisms, -categories not are even the whether same two as given parallel$(n-1,n)$ n (n-1,n) -morphisms -categories. are Thus, equivalent. up This to definition equivalence, makes there sense is as no low point as in mentioning anything beyond$n=-1$ n = -1 ; -morphisms, the not statement even that whether two given parallel$(n-1)$ (-1) n -morphisms are equivalent equivalent. simply This means definition that makes there sense exists as an low object as (a$0n=-1$ 0 n = -1; the statement that parallel $(-1)$-morphisms are equivalent simply means that there exists an object (a $0$-morphism).

Special cases

• The concept of $(-1)$-poset is trivial.

• A $0$-poset is a truth value.

• A $1$-poset or (0,1)-category is simply a poset.

  Because, by the definition of $(0,1)$-category, we have that any two $1$-morphisms with the same source and target are equivalent. Hence there is, up to equivalence, at most one morphism for every ordered pair of objects. The rest of the axioms say that this is all the information there is in a $(0,1)$-category. Therefore, by the discussion at poset – As a category with extra properties, a $(0,1)$-category is a poset. (See also thin category.)

• A $2$-poset or (1,2)-category is a locally posetal 2-category.

• In general, an $n$-poset is an $n$-category in which all parallel pairs of $n$-morphisms are equal.

• An $\infty$-poset is the same thing as an $(\mathrm{\infty },\mathrm{\infty })$ \infty (\infty,\infty)-category.

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 $(\mathrm{\infty },\mathrm{\infty })$ \infty (\infty,\infty)-category or $n$-category is; it is actually simpler to define $n$-posets from scratch as given above than to define them in terms of $n$-categories.

Basic theorems

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