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We show that Segal spaces, and more generally category objects in an $infty$-category $mathcal{C}$, can be identified with associative algebras in the double $infty$-category of spans in $mathcal{C}$. We use this observation to prove that having identities is a property of a non-unital $(infty,n)$-category.
We give a new proof of the equivalence between two of the main models for $(infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $Theta_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the $infty$-cat
We construct higher categories of iterated spans, possibly equipped with extra structure in the form of local systems, and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the fram
It is known by results of Dyckerhoff-Kapranov and of Galvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of
In a previous paper, we showed that a discrete version of the $S_bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, b
Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an algebraic pattern, bywhich we mean an $infty$-category equipped with a factorization system and a collection of elementary objects. Examples