2-Segal sets and the Waldhausen construction

Bergner, Julia E.; Osorno, Angélica M.; Ozornova, Viktoriya; Rovelli, Martina; Scheimbauer, Claudia

doi:10.1016/j.topol.2017.12.009

Cited by 17 publications

(24 citation statements)

References 30 publications

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“…We furthermore require this bisimplicial ∞groupoid to be stable, see Section 2.3. This stability condition is a pullback condition on certain squares, and is a ∞-categorical reformulation of the notion of Bergner-Osorno-Ozornova-Rovelli-Scheimbauer [6], suitable for ∞-groupoids.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Remark 2.3.2. Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer introduced the notion of stable double category (bisimplicial set) in [6]: they define a double category to be stable if every square is uniquely determined by its span of source morphisms and, independently by its cospan of target morphisms. The present definition is a categorical reformulation of their notion suitable for ∞-groupoids.…”

Section: Stabilitymentioning

confidence: 99%

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Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

Carlier

2020

Algebr. Geom. Topol.

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In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain augmented double Segal spaces subject to some exactness conditions. We establish a Möbius inversion principle for (co)modules, and a Rota formula for certain more involved structures called Möbius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of adjunctions between Möbius decomposition spaces, in the spirit of Rota's original formula.

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Section: Outline Of the Papermentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

Carlier

2020

Algebr. Geom. Topol.

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“…It is the restriction species of R-substructures of X. See Bergner et al [5] for examples of slices of the decomposition space of graphs. The fact that slicing a restriction species produces again restriction species reflects the local nature of coalgebras: every element in a coalgebra generates a coalgebra.…”

Section: Slices Of Examplesmentioning

confidence: 99%

“…A map operates as a function on the set of dots when considered a map in ∆ while it operates as a function on the walls when considered a map in ∆ gen . Here is a picture of a certain map 5 → 4 in ∆ and of the corresponding map [5] ← [4] in ∆ gen .…”

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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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“…A key source of examples of such structures is the output of Waldhausen's S • -construction when applied to an exact category, as shown in [DK12] and [GCKT18]. In [BOORS18] and [BOORS] we show that any 2-Segal space which satisfies a unitality condition arises from such a construction for a suitably general input.…”

Section: Introductionmentioning

confidence: 99%