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Register a new userSegal spaces, spans, and semicategories
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Authors
Rune Haugseng
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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.
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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 $infty$-category of $n$-globular spaces. The proof works for a broad class of $infty$-categories that includes all $infty$-topoi.
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 augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.
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, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_bullet$-constructions.
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 of structures that occur as such Segal $mathcal{O}$-spaces for an algebraic pattern $mathcal{O}$ include $infty$-categories, $(infty,n)$-categories, $infty$-operads, $infty$-properads, and algebras for an $infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal objects, including conditions under which the latter are preserved by left and right Kan extensions. In particular, we obtain necessary and sufficent conditions on a pattern $mathcal{O}$ for free Segal $mathcal{O}$-spaces to be described by an explicit colimit formula, in which case we say that $mathcal{O}$ is extendable. In the second part of the paper we explore the relationship between extendable algebraic patterns and polynomial monads, by which we mean cartesian monads on presheaf $infty$-categories that are accessible and preserve weakly contractible limits. We first show that the free Segal $mathcal{O}$-space monad for an extendable pattern $mathcal{O}$ is always polynomial. Next, we prove an $infty$-categorical version of Webers Nerve Theorem for polynomial monads, and use this to define a canonical extendable pattern from any polynomial monad, whose Segal spaces are equivalent to the algebras of the monad. These constructions yield functors between polynomial monads and extendable algebraic patterns, and we show that these exhibit full subcategories of saturated algebraic patterns and complete polynomial monads as localizations, and moreover restrict to an equivalence between the $infty$-categories of saturated patterns and complete polynomial monads.
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