来自数论、物理与拓扑的三个霍普夫代数及其共同背景 II：一般范畴论表述

拓扑与辛几何中范畴维数的关联

Relating categorical dimensions in topology and symplectic geometry

本文研究拓扑与辛几何中三角范畴的维数概念，建立Rouquier维数与几何量（如Morse临界值数、LS范畴等）的新界，并引入dg-范畴的LS范畴及构造精确Lagrangian余边。

Categorical aspects of toric topology

We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as …

Hopf Algebras and Invariants of the Johnson Cokernel

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case,…

Hopf algebras and homotopy invariants

In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(ΩΣX; \coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(ΩΣX; \coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(ΩΣX; \coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these invariants are obstructions to t…

On Spineless Cacti, Deligne's Conjecture and Connes--Kreimer's Hopf Algebra

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.

Renormalization groupoids in algebraic topology

Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $Mξ$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a grou…

Centrality and the KRH Invariant

The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in previous work of the authors. The present paper gives an abstract description of these structures and shows how the Hopf algebraic image of a knot lies in the center of the corresponding Hopf algebra. The paper also shows how all the axiomatic proper…

Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations

Given a $C_\infty$ coalgebra $C_*$, a strict dg Hopf algebra $H_*$, and a twisting cochain $τ:C_* \rightarrow H_*$ such that $Im(τ) \subset Prim(H_*)$, we describe a procedure for obtaining an $A_\infty$ coalgebra on $C_* \otimes H_*$. This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an $A_\infty$ coalgebra model of the chains on the free loop space $LM$ based on the $C_\infty$ coalgebra structure of $H_*(M)$ induced by the diagonal map $M \rightarrow M \times M$ and the Hopf algebra model of the based loop space given by $T(H_*(M)[-1])$. When …

The Fundamental Infinity-Groupoid of a Parametrized Family

Given an infinity-category C, one can naturally construct an infinity-category Fam(C) of families of objects in C indexed by infinity-groupoids. An ordinary categorical version of this construction was used by Borceux and Janelidze in the study of generalized covering maps in categorical Galois theory. In this paper, we develop the homotopy theory of such "parametrized families" as generalization of the classical homotopy theory of spaces. In particular, we study homotopy-theoretical constructions that arise from the fundamental infinity-groupoids of families in an infinity-category. In the sa…

Power maps in algebra and topology

Given any twisting cochain t:C -->A, where C is a connected, coaugmented chain coalgebra and A is an augmented chain algebra over an arbitrary PID R, we construct a twisted extension of chain complexes A --> H(t) --> C. We show that both the well-known Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases of H(t), which we therefore call the Hochschild complex of t. We explore the extent of the naturality of the Hochschild complex construction and apply the results of this exploration to determining conditions under which H(t) …