There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and outlined for weak n-categories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves non-abelian cohomology, stacks, torsors, homotopy, and higher-dimensional categories. Many of the ideas are scattered through the literature or are folklore; a few are new.
Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they redevelop the category theory of $\infty$-categories using 2-categorical arguments. In this paper, we begin with a self-contained review of the parts of their theory needed to discuss adjunctions and monadicity. This is applied in order to extend to the $\infty$-categorical context the classical criterion for fully faithfulness of the comparison functor induced by an adjunction. We discuss the relation with previous work in the literature---which primarily uses model-categorical techniques---and indicate applications…
We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois descent makes no reference to Grothendieck descent theory at all. In order to make Galois descent constructions perfectly clear, we also describe their connections with some other related constructions of cat…
It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomor…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…
In this note we present an $\infty$-categorical framework for descent along adjunctions and a general formula for counting conjugates up to equivalence which unifies several known formulae from different fields.
It has become obvious in the recent development that the structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this paper we explicitely put the Ramsey property and the dual Ramsey property in the context of categories of finite structures and investigate the invariance of these properties under some standard categorical constructions. We use elementary category theory to generalize some combinatorial results and using the machinery of very basic category theory provide new combinatorial statements (…
We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach to compositional game theory closely related to open games, in which the controller is a composite of game-theoretic agents. We believe that 'cybernetic' is an appropriate name for the processes that can be described in this framework.
We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathscr{X}$ are equivalent to chain complexes in $\mathscr{X}$; (b) the observation of Church, Ellenberg and Farb that $\mathscr{X}$-valued species are equivalent to $\mathscr{X}$-valued functors from the category of finite sets and injective partial functions; (c) a resu…
Mount Allison University（蒙特爱立森大学）；University of Strathclyde（思克莱德大学）；University College London（伦敦大学学院）
We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of …
Absolute Universes of combinatorial games, as defined in a recent paper by the same authors, include many standard short normal- misère- and scoring-play monoids. In this note we show that the class is categorical, by extending Joyal's construction of arrows in normal-play games. Given $G$ and $H$ in an Absolute Universe $U$, we study instead the Left Provisonal Game $[G, H]$, which is a normal-play game, independently of the particular Absolute Universe, and find that $G\longrightarrow H$ (implying $G\succcurlyeq H$) corresponds to the set of winning strategies for Left playing second in $[G,…
Diagrammatic notation has become a ubiquitous computational tool; early examples include Penrose's graphical notation for tensor calculus, Feynman's diagrams for perturbative quantum field theory, and Cvitanovic's birdtracks for Lie algebras. Category theory provides a robust framework in which to understand the nature of such diagrams, and Joyal and Street formalized this framework by introducing string diagrams, governed by the syntax of monoidal 1-categories. The notion of "manifold diagrams" generalizes string diagrams to higher dimensions, and can be interpreted in higher-categorical term…
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it is a quasicategory which has unique fillers for inner horns of dimension $3$ and greater. Using Duskin's technique, we show how his nerve applies to $(2,1)$-category functors, making it a fully faithful inclusion of $(2,1)$-categories into simplicial sets. Then we consider analogues of this extension of Duskin's result for several different two-dimensional …
We define highest weight categorical actions of sl_2 on highest weight categories and show that basically all known examples of categorical sl_2-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories O of type $A$, categories O for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Ra…
It has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4-manifolds. This idea has been refined recently, by proposing to use 2-groups and their representations as specific examples of 2-categories. The challenge has been to make these proposals fully explicit. Here we give a concrete realization of this program. Building upon our earlier work with Baez and Wise on the representation theory of 2-groups, we construct a four-dimensional state sum model based on a categorified version of the Euclidean group. We define and explicitly…
Varagnolo and Vasserot conjectured an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a truncation of an affine parabolic category O of type A. In this paper we reduce their conjecture to some purely combinatorial conjecture. Our principal tool for the reduction is the theory of categorical sl_2-actions on highest weight categories. Then we prove our combinatorial conjecture in some cases.
Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in $(\infty,1)$-categories. Completeness is a property of Segal spaces introduced by Rezk that characterizes those Segal spaces which are $(\infty,1)$-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study its ramific…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-be…
