category object in an (∞,1)-category, groupoid object
A semi-Segal space is like a Segal space but without specified identities/degeneracies. It is to semicategories as Segal spaces are to categories.
Let $\mathcal{C} =$ sSet, equipped with the standard model structure on simplicial sets.
A semi-Segal space is a semi-simplicial object in $\mathcal{C}$ such that
it is a fibrant object in the Reedy model structure on $\mathcal{C}^{\Delta^{op}_+}$;
it satisfies the Segal conditions be weak equivalences.
A semi-simplicial object $X_\bullet$ being Reedy fibrant means that for each $n \in \mathbb{N}$ the morphisms
are fibrations.
Equivalently this says that it is a semi-simplicial object which satisfies the Segal conditions by homotopy pullbacks. This is just as for Segal spaces, see there for details.
A complete semi-Segal space is a semi-Segal space $X_\bullet$ such that the two morphisms
are weak equivalences.
This is the completeness/univalence condition just as for complete Segal spaces.
A semi-Segal space is quasiunital if (…)
A morphism of complete semi-Segal spaces $f_\bullet \colon X_\bullet \to Y_\bullet$ is quasi-unital if it preserves the weak equivalences, hence if
The notion is mentioned in
More details are spelled out in
See also
Hiro Lee Tanaka, Functors (between $\infty$-categories) that aren’t strictly unital (arXiv:1606.05669)
Wolfgang Steimle, Degeneracies in quasi-categories, arxiv:1702.08696
Last revised on March 1, 2017 at 12:23:07. See the history of this page for a list of all contributions to it.