Following the general concept of -category, a -category is a category whose hom-objects are (-1)-groupoids, hence which for every pair of objects have either no morphism or an essentially unique one.
More in detail, recall that:
an (n,1)-category is an (∞,1)-category such that every hom ∞-groupoid is (n-1)-truncated.
a 0-truncated ∞-groupoid is equivalently a set;
a (-1)-truncated ∞-groupoid is a subsingleton.
Therefore:
(relation between preorders and (0,1)-categories)
An -category is equivalently preordered.
Since every hom--groupoid is (-1)-truncated, it is a subsingleton, so there is at most one morphism from any object to any other; hence the -category is preordered by the relation on objects that there exists an element of the hom--groupoid between two objects.
In the higher category theory literature, there is a distinction between -categories and -categories which satisfy a Segal completeness or Rezk completeness condition and those which do not, which leads to a distinction between whether -categories and -categories by default are Segal / Rezk complete or not:
Those authors who start with -categories and define -categories from (-1)-truncations of the hom--groupoids tend to assume Segal / Rezk completeness by default, where a -category which are not Segal complete / Rezk complete is then called a -precategory, or when defined internally, a Segal -category.
Those authors who build -categories from components tend not to assume Segal / Rezk completeness by default, where a -category which does satisfy Segal completeness or Rezk completeness is called a univalent -category, or when defined internally, a complete Segal -category or Rezk -category. Furthermore, since the hom--groupoids are subsingletons, one can equivalently call such a -category gaunt or skeletal, since those conditions are equivalent to Segal completeness / Rezk completeness for -categories.
This distinction is important even for -categories because it denotes the difference between a preorder and a partial order as a -category, and also any mathematical structures derived from preorders vs partial orders, such as the order-theoretic definitions of presemilattices vs semilattices, prelattices vs lattices, or Heyting prealgebras vs Heyting algebras. In the case of -groupoids, this is the difference between a setoid and a set, and thus between algebraic structures defined using equivalence relations vs using equality, such as the algebraic definitions of presemilattices vs semilattices, prelattices vs lattices, or Heyting prealgebras vs Heyting algebras. See also the relation between preorders and (0,1)-categories for additional details.
Here we assume that -categories are Segal / Rezk complete:
A -category with the structure of a site is a (0,1)-site: a posite.
A -category with the structure of a cartesian monoidal category is a meet-semilattice.
A -category with the structure of a cocartesian monoidal category is a join-semilattice.
A -category with the structure of both a cartesian monoidal category and a cocartesian monoidal category is a lattice.
A -category with the structure of a topos is a (0,1)-topos: a Heyting algebra.
A -category with the structure of a Grothendieck topos is a Grothendieck (0,1)-topos: a frame or locale.
A -category which is also a groupoid (that is, every morphism is an isomorphism) is a -category (which may think of as either a -category or as a -groupoid), which is the same as a set (up to equivalence) or a symmetric proset (up to isomorphism).
(0,1)-category
Last revised on July 7, 2026 at 17:37:29. See the history of this page for a list of all contributions to it.