nLab terminal object in a quasi-category



(,1)(\infty,1)-Category theory

Limits and colimits



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.


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

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


A quick survey is on page 159 of

For more details see definition, p. 46 in

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