arXiv subjects
Publications and source records attributed to Brandon T. Shapiro.
In this paper, we establish a precise relationship between CGW categories and pointed stable double Segal spaces, both of which were developed as general input for algebraic K-theory. In particular, we show that any CGW category can be regarded as a pointed stable double Segal space, and they can be identified using a classifying diagram construction for double categories with shared isomorphisms.
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce the problem of modeling tilings of the Euclidean or hyperbolic plane as presheaves over such a category. Combinatorially, this amounts to choosing an ``alignment'' for a tiling: a direction for every edge and consistent labels for the edges of each polygonal tile. We show that for a fixed tiling, given a single alignment we can characterize every other alignment of the same tiling by comparison of the edge directions. We then construct a ``reflective'' alignment for any tiling with an even number of polygons at each vertex, and from this generate a large family of alignments with elegant symmetry properties.
The (A)CGW categories of Campbell and Zakharevich show how finite sets and varieties behave like the objects of an exact category for the purpose of algebraic $K$-theory. These structures admit a well-behaved Q-construction akin to Quillen's, and satisfy analogues of the Dévissage and Localization theorems. In this work, we modify Campbell and Zakharevich's axioms to obtain a framework called ECGW categories that allows for an $S_\bullet$-construction akin to Waldhausen's, and show how it produces a K-theory spectrum which satisfies an analogue of the Additivity Theorem. We also define a notion of ``relative ECGW categories'' which have weak equivalences determined by a subcategory of acyclic objects satisfying minimal conditions; these satisfy analogues of the Fibration and Localization Theorems that generalize previous versions in the literature. We illustrate these results with examples including exact categories, extensive categories, algebraic varieties, and polytopes up to scissors congruence.
We provide a categorical framework for mathematical objects for which there is both a sort of "independent" and "dependent" composition. Namely we model them as duoidal categories in which both monoidal structures share a unit and the first is symmetric. We construct the free such category and observe that it is a full subcategory of the category of finite posets. Indeed each algebraic expression in the two monoidal operators corresponds to the poset built by taking disjoint unions and joins of the singleton poset. We characterize these "expressible" posets as precisely those which contain no "zig-zags." We then move on to describe categories equipped with $n$-ary operations for each $n$-element finite poset; we refer to them as "dependence categories" since they allow for combinations of objects based on any network of dependencies between them. These structures model various sorts of dependence including the space-like and time-like juxtaposition of weighted probability distributions in relativistic spacetime, which we model using polynomial endofunctors on the category of sets, as well as the runtimes for multiple computer programs run in parallel and series, which we model using the tropical semiring structure on nonnegative real numbers. With these examples in mind, we conclude by describing ways in which morphisms in a partial monoidal category can be "decorated" in a coherent manner by objects in a dependence category, such as labeling a network of parallel programs with their runtimes.
We show that the double category $\mathbb{C}\mathbf{at}^\#$ of comonoids in the category of polynomial functors (previously shown by Ahman-Uustalu and Garner to be equivalent to the double category of categories, cofunctors, and prafunctors) contains several formal settings for basic category theory and has subcategories equivalent to both the double category $\mathbb{O}\mathbf{rg}$ of dynamic rewiring systems and the double category $\mathbb{P}\mathbf{oly}_{\mathcal{E}}$ of generalized polynomials in a finite limit category $\mathcal{E}$. Also serving as a natural setting for categorical database theory and generalized higher category theory, $\mathbb{C}\mathbf{at}^\#$ at once hosts models of a wide range of concepts from the theory and applications of polynomial functors and category theory.
We show that the construction due to Leinster and Weber of a generalized Lawvere theory for a familially representable monad on a (co)presheaf category, and the associated ``nerve'' functor from monad algebras to (co)presheaves, have an elegant categorical description in the double category $\mathbb{C}\mathbf{at}^{\#}$ of categories, cofunctors, familial functors, and transformations. In $\mathbb{C}\mathbf{at}^{\#}$, which also arises from comonoids in the category of polynomial functors, both a familial monad and a (co)presheaf it acts on can be modeled as horizontal morphisms; from this perspective, the theory category associated to the monad is built using left Kan extension in the category of endomorphisms, and the nerve functor is modeled by a single composition of horizontal morphisms in $\mathbb{C}\mathbf{at}^{\#}$. For the free category monad $path$ on graphs, this provides a new construction of the simplex category as $Δ:= \lens{path}{path \circ path}$. We also explore the free Eilenberg-Moore completion of $\mathbb{C}\mathbf{at}^{\#}$, in which constructions such as the free symmetric monoidal category monad on $\mathbf{Cat}$ can modeled using the rich language of polynomial functors.
Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.
Natural organized systems adapt to internal and external pressures and this happens at all levels of the abstraction hierarchy. Wanting to think clearly about this idea motivates our paper, and so the idea is elaborated extensively in the introduction, which should be broadly accessible to a philosophically-interested audience. In the remaining sections, we turn to more compressed category theory. We define the monoidal double category Org of dynamic organizations, we provide definitions of Org-enriched, or dynamic, categorical structures -- e.g. dynamic categories, operads, and monoidal categories -- and we show how they instantiate the motivating philosophical ideas. We give two examples of dynamic categorical structures: prediction markets as a dynamic operad and deep learning as a dynamic monoidal category.
We define the monoidal category $(Poly_E,y,\triangleleft)$ of polynomials under composition in any category $E$ with finite limits, including both cartesian and vertical morphisms of polynomials, and generalize to this setting the Dirichlet tensor product of polynomials $\otimes$, duoidality of $\otimes$ and $\triangleleft$, closure of $\otimes$, and coclosures of $\triangleleft$. We also prove that $\triangleleft$-comonoids in $Poly_E$ are precisely the internal categories in $E$ whose source morphism is exponentiable, generalizing a result of Ahman-Uustalu equating categories with polynomial comonads, and show that coalgebras in this setting correspond to internal copresheaves. Finally, the double category of ``typed'' polynomials in $E$ is recovered using $\triangleleft$-bicomodules in $Poly_E$.
The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $\mathsf{Set}$ (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on $\mathsf{Set}$ for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in $Δ^n$, but generally not all inner horns.
Nonstandard interactions (NSI) of neutrinos with matter can significantly alter neutrino flavor evolution in supernovae with the potential to impact explosion dynamics, nucleosynthesis, and the neutrinos signal. In this paper, we explore, both numerically and analytically, the landscape of neutrino flavor transformation effects in supernovae due to NSI and find a new, heretofore unseen transformation processes can occur. These new transformations can take place with NSI strengths well below current experimental limits. Within a broad swath of NSI parameter space, we observe symmetric and standard matter-neutrino resonances for supernovae neutrinos, a transformation effect previously only seen in compact object merger scenarios; in another region of the parameter space we find the NSI can induce neutrino collective effects in scenarios where none would appear with only the standard case of neutrino oscillation physics; and in a third region the NSI can lead to the disappearance of the high density Mikheyev-Smirnov-Wolfenstein resonance. Using a variety of analytical tools, we are able to describe quantitatively the numerical results allowing us to partition the NSI parameter according to the transformation processes observed. Our results indicate nonstandard interactions of supernova neutrinos provide a sensitive probe of beyond the Standard Model physics complementary to present and future terrestrial experiments.