Relative 2–Segal spaces

Young, Matthew B.

doi:10.2140/agt.2018.18.975

Cited by 15 publications

(12 citation statements)

References 53 publications

(143 reference statements)

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“…This is a monoidal CULF functor, and altogether there is a monoidal CULF functor from the decomposition space of P -trees to the decomposition space of combinatorial trees. This is an interesting example of a relative 2-Segal space in the sense of Young [54] and Walde [51]. 7.13.…”

Section: Examples: Various Flavours Of Trees (Actually Forests)mentioning

confidence: 99%

Decomposition Spaces and Restriction Species

Gálvez-Carrillo

Kock

Tonks

2018

International Mathematics Research Notices

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We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces. Proposition 7.9. Monoidal directed restriction species naturally induce monoidal decomposition spaces and hence bialgebras.

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Section: Examples: Various Flavours Of Trees (Actually Forests)mentioning

confidence: 99%

Decomposition Spaces and Restriction Species

Gálvez-Carrillo

Kock

Tonks

2018

International Mathematics Research Notices

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“…For example, some work on higher dimensional S • -constructions in the context of higher Segal spaces has been done by Poguntke [Pog17], and it would be interesting to extend our approach to this context as well. We also note that the relative S • -construction we consider here does not use the notion of relative 2-Segal spaces of Walde [Wal16], and Young [You16]. Finally, a slight generalization of N pe should provide a nerve functor for Penney's augmented proto-exact (∞, 1)-categories [Pen17].…”

Section: Introductionmentioning

confidence: 99%

Comparison of Waldhausen constructions

Bergner,

Osorno,

Ozornova

et al. 2019

Preprint

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“…This construction is a common generalisation of classical constructions of incidence (co)algebras of posets [63], [92], monoids [16], and Möbius categories [76], [27], [74], but reveals also most other coalgebras in combinatorics to be of incidence type (see for example [53] and [54]). Decomposition spaces are the same thing as the 2-Segal spaces of Dyckerhoff and Kapranov [32] (see [41] for the last piece of this equivalence), of importance in homological algebra and representation theory, notably in connection with Hall algebras and the Waldhausen Sconstruction [32], [31], [107], [89], [90]. The theory is now being developed in many directions.…”

Section: Introductionmentioning

confidence: 99%

The incidence comodule bialgebra of the Baez-Dolan construction

Kock

2019

Preprint

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Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez-Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid Ω (considered as a onecoloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of Ω-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque-Ebrahimi-Fard-Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier-Hairer-Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez-Dolan construction itself, providing it with a new algebraic perspective.

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Relative 2–Segal spaces

Cited by 15 publications

References 53 publications

Decomposition Spaces and Restriction Species

Decomposition Spaces and Restriction Species

Comparison of Waldhausen constructions

The incidence comodule bialgebra of the Baez-Dolan construction

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