Every 2-Segal space is unital

Feller, M. N.; Garner, Richard; Kock, Joachim; Proulx, May U.; Weber, Mark

doi:10.48550/arxiv.1905.09580

Cited by 2 publications

(5 citation statements)

References 4 publications

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“…In particular, the squares of [9,Figure (8), Lemma 3.10] sent to pullbacks by a decomposition space include all basic pushouts except for the pushout squares of cofaces in [9,Lemma 2.10] between outer face maps and the middle squares in (3.4). As decomposition spaces agree with the 2-Segal spaces of [3] by [4], the squares which factor into these restricted basic pushout squares ought to be those pushouts which are preserved by the standard functor from ∆ to Connes' cycle category Λ, according to [20,Theorem 2].…”

Section: [M]mentioning

confidence: 99%

“…The following definitions and the characterization of Theorem 5.5 are well-known, but they do not seem to be easy to find in the literature in this exact form. Much of the related literature is in the area of relational database theory, where often more general hypergraphs rather than simplicial complexes are considered 4 , resulting in greater generality and complexity than what we need here.…”

Section: Lifting Conditions and Acyclic Configurationsmentioning

confidence: 99%

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Weak cartesian properties of simplicial sets

Constantin¹,

Fritz²,

Perrone³

et al. 2021

Preprint

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Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category ∆ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in ∆ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design, and provide simpler characterizations of these properties using factorization results for pushouts squares in ∆, which we characterize completely, along with several other classes of squares in ∆. Examples of simplicial sets with certain completeness conditions include quasicategories, Kan complexes, and bar constructions for algebras of certain classes of monads. The latter is our motivating example which we discuss in a companion paper.

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Section: [M]mentioning

confidence: 99%

Section: Lifting Conditions and Acyclic Configurationsmentioning

confidence: 99%

Weak cartesian properties of simplicial sets

Constantin¹,

Fritz²,

Perrone³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 1.17 ([9] , [11], ( [15], Theorem 4.10)). For a simplicial groupoid X : ∆ op → Grpd, the following are equivalent 1.…”

Section: Decalagementioning

confidence: 99%

“…In equivalent terms, for 0 < i < n, a simplicial groupoid is upper 2-Segal when squares as to the left are required to be pullbacks, and lower 2-Segal when this is only required for squares as to the right, 11], Proposition 2.1). Let X be an upper 2-Segal groupoid.…”

Section: Slicesmentioning

confidence: 99%

“…They are certain simplicial objects subject to an axiom that expresses decomposition, in the same way as the Segal condition (which characterises categories among simplicial sets) expresses composition. Decomposition spaces are the same thing as the 2-Segal spaces of Dyckerhoff and Kapranov [9] (see [11] for the last piece of this equivalence). It seems likely that all combinatorial coalgebras, bialgebras and Hopf algebras (with nonnegative section coefficients) arise from the incidence coalgebra construction of decomposition spaces.…”

Section: Introductionmentioning

confidence: 98%

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The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces

Forero¹

2021

Preprint

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Kock, and Tonks [17] constructed a decomposition space U of all Möbius intervals, as a recipient of Lawvere's interval construction for Möbius categories, and conjectured that U enjoys a certain universal property: for every Möbius decomposition space X, the space of CULF functors from X to U is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture.The discrete case is general enough to cover all locally finite posets, Cartier-Foata monoids, and Möbius categories. The proof is 2-categorical. First, we construct a local strict model of U , which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other selfmodifications than the identity.

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Every 2-Segal space is unital

Cited by 2 publications

References 4 publications

Weak cartesian properties of simplicial sets

Weak cartesian properties of simplicial sets

The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces

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