arXiv subjects

Brandon Shapiro

Publications and source records attributed to Brandon Shapiro.

5 recordsLinked to original sources

arXiv2024

A combinatorial construction of homology via ACGW categories

2-Segal spaces arise not only from $S_\dotp$-constructions associated to Waldhausen and (proto) exact categories, but also from $S_\dotp$-constructions associated to certain double-categorical structures. A major step in this direction is due to the work of Bergner--Osorno--Ozornova--Rovelli--Scheimbauer, who propose augmented stable double Segal objects as a natural input for an $S_\dotp$-construction. More recently, another such input has been put forth: ACGW categories. ACGW categories have the advantage that they are combinatorial in nature (as opposed to homotopical or algebraic), and thus have fewer difficult coherence issues to work with. The goal of this paper is to introduce the reader to the key ideas and techniques for working with ACGW categories. To do so, we focus on how homology theory generalizes to ACGW categories, particularly in the central example of finite sets. We show how the ACGW formalism can be used to produce various classical homological algebra results such as the Snake lemma and long exact sequences of relative pairs.

math.KT↗

arXiv2022

Enrichment of Algebraic Higher Categories

We provide a definition of enrichment that applies to a wide variety of categorical structures, generalizing Leinster's theory of enriched $T$-multicategories. As a sample of newly enrichable structures, we describe in detail the examples of enriched monoidal categories and enriched double categories, with a focus on monoidal double categories as broadly convenient bases of enrichment.

math.CT↗

arXiv2021

Familial Monads as Higher Category Theories

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any higher category structure which arises as algebras for a familially representable monad on a presheaf category, then use this to describe several examples relating to higher category theory and cubical sets. The proof of this characterization avoids tedious naturality arguments by passing through the theory of categorical polynomials; along the way, we give descriptions of pullbacks, composites, and exponentiations of split opfibrations in terms of their classifying functors which may be of independent interest.

math.CT↗

arXiv2021

Weak cartesian properties of simplicial sets

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 G\'{a}lvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category $\Delta$ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in $\Delta$ 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. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in $\Delta$, which we characterize completely, along with several other classes of squares in $\Delta$. Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.

math.CT↗

arXiv2017

Densities of Hyperbolic Cusp Invariants

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing density for links is bounded above by 3.1263.... Moreover, there is a sequence of links with cusp crossing density approaching 3. The least upper bound for cusp crossing density remains an open question. For two-component hyperbolic links, cusp crossing density is shown to be dense in the interval [0,1.6923...] and for all hyperbolic links, cusp crossing density is shown to be dense in [0, 2.120...].

math.GT↗