Hall monoidal categories and categorical modules

Walde, Tashi

doi:10.48550/arxiv.1611.08241

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…In the present paper, the main use of this concept is that monoidal structures on decomposition spaces are required to be culf: this is what ensures that the multiplication resulting from taking cardinality is comultiplicative so as to yield a bialgebra. The second use of culfness is that the objective analogue of comodules is given by certain culf maps [101], [107], [12], cf. 3.1.2 below.…”

Section: Simplicial Groupoidsmentioning

confidence: 99%

“…Comodules arising in combinatorics are often the cardinality of certain slice-level comodules given by so-called comodule configurations of simplicial groupoids, first studied by Walde [101] and Young [107]. We follow the terminology of Carlier [12], [13].…”

Section: Comodule Configurationsmentioning

confidence: 99%

“…The interaction between P * and P ⊙ is a bit more complicated now, since the two bialgebras are no longer supported at the same groupoid, and we need to consider comodules more general than just those underlying a bialgebra itself. The objective account of comodules is given by certain culf simplicial maps [101], [107], [12]. We need a third simplicial groupoid which is a comodule over Bar S (P ⊙ ) and having the same base groupoid as Bar S (P * ).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Simplicial Groupoidsmentioning

confidence: 99%

Section: Comodule Configurationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The incidence comodule bialgebra of the Baez-Dolan construction

Kock

2019

Preprint

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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].…”

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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“…Remark. After the first version of this paper was completed, a preprint by Tashi Walde [44] was posted to the arXiv which also aims at developing a theory of modules over higher Segal spaces. We comment where appropriate on the overlap.…”

Section: Introductionmentioning

confidence: 99%

Relative 2–Segal spaces

Young

2018

Algebr. Geom. Topol.

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We introduce a relative version of the 2-Segal simplicial spaces defined by Dyckerhoff and Kapranov and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2-Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the R•-construction from Grothendieck-Witt theory. We show that a relative 2-Segal space defines a categorical representation of the Hall algebra associated to the base 2-Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2-Segal spaces.

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Hall monoidal categories and categorical modules

Cited by 5 publications

References 11 publications

The incidence comodule bialgebra of the Baez-Dolan construction

The incidence comodule bialgebra of the Baez-Dolan construction

Decomposition Spaces and Restriction Species

Relative 2–Segal spaces

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