nLab terminal object in a quasi-category

Idea

In a quasi-category the notion of terminal object known from ordinary category theory is relaxed in the homotopy theoretic sense to the suitable notion in (∞,1)-category theory:

instead of demanding that from any other object there is a unique morphism into the terminal object, in a quasi-category there is a contractible space of such morphisms, i.e. the morphism to the terminal object is unique up to homotopy.

Definition

Let $C$ be a quasi-category and $c \in C$ one of its objects (a vertex in the corresponding simplicial set). The object $c$ is a terminal object in $C$ if the following equivalent conditions hold:

• The projection from the over quasi-category $C_{/c} \to C$ is a trivial fibration of simplicial sets.

• For every object $d$ of $C$ the right hom Kan-complex into $d$ is contractible:

$$Hom_C^R(d,c) \simeq {*} \,.$$

Related concepts

• initial object, terminal object

• h-initial object

• terminal object in a 2-category

• initial object in a 2-category

References

A quick survey is on page 159 of

• André Joyal, The theory of quasicategories and its applications, lectures at Simplicial Methods in Higher Categories, (pdf)

For more details see definition 1.2.12.3, p. 46 in

• Jacob Lurie, Higher Topos Theory