2–Segal spaces as invertible infinity-operads

Walde, Tashi

doi:10.2140/agt.2021.21.211

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…The operads uAs and sOp are canonically decomposable, and the left functors in the sequences (Cat), (Op), (Alg) and (SPsh) and the middle functors in the sequences (Cat) and (Op) are (∞, 1)-localizations. As a corollary, we get Theorem 3.0.1 of [Wal21], which implies Theorem 1.1 of [BdBM20] and the asphericity of Ω, proved in [ACM19].…”

Section: Introductionmentioning

confidence: 51%

Twisted arrow categories, operads and Segal conditions

Burkin¹

2021

Preprint

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We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category ∆, Segal's category Γ, Connes cyclic category Λ, Moerdijk-Weiss dendroidal category Ω, and categories similar to graphical categories of Hackney-Robertson-Yau are twisted arrow categories of symmetric or cyclic operads. Twisted arrow categories of operads admit Segal presheaves and 2-Segal presheaves, or decomposition spaces. Twisted arrow category of an operad P is the (∞, 1)-localization of the corresponding category Ω/P by the boundary preserving morphisms. Under mild assumptions, twisted arrow categories of operads, and closely related universal enveloping categories, are generalized Reedy.

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Section: Introductionmentioning

confidence: 51%

Twisted arrow categories, operads and Segal conditions

Burkin¹

2021

Preprint

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“…In this section we specialize Definition 4.2 to the case when the codomain G is simply-connected, which implies that the domain is also a tree by [HRY20a, Proposition 5.2]. The resulting categories of graphs are related to cyclic operads [GK95] and higher cyclic operads [HRY19,Wal21]. Our goal is to give a description of new graph maps between trees that is mirrors the 'complete morphisms' from Definition 1.12 of [HRY19].…”

Section: Appendix a Categories Of Trees For Cyclic Operadsmentioning

confidence: 99%

Categories of graphs for operadic structures

Hackney

2021

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We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalized operads can be realized at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and betweeen wheeled properads and modular operads.

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“…Subsequent work by Date: April 19, 2022. The author was partially supported by NSF RTG grant DMS-1839968 and NSF grant DMS-1906281. other authors has further expanded the scope of the theory; in [2] it is shown that 2-Segal objects can be viewed equivalently through the lens of double categories, in [24] they are seen to be equivalent to algebras of spans, while [25] shows that they can also be viewed as invertible operads. There has also been further development in the direction of Hall algebras in [26], as well as continued work exploring Möbius inversion from Gálvez-Kock-Tonks and others, such as [4].…”

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confidence: 99%

Quasi-2-Segal sets

Feller¹

2022

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We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and Gálvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar relationship as the one Segal spaces have with quasi-categories. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called décalage) are quasi-categories, as well as an edgewise subdivision criterion.

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2–Segal spaces as invertible infinity-operads

Cited by 7 publications

References 24 publications

Twisted arrow categories, operads and Segal conditions

Twisted arrow categories, operads and Segal conditions

Categories of graphs for operadic structures

Quasi-2-Segal sets

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