arXiv subjects
Publications and source records attributed to Gabriel C. Drummond-Cole.
We correct an error in the paper referred to in the title. Part of the argument is organized as a general method for establishing when (derived) functors factor through a fixed Serre subcategory, which may be of some more general interest.
Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of curved algebras cannot follow the same path as differential graded algebras. In this article, we propose to study curved algebras by means of curved operads. We develop the theory of bar and cobar constructions adapted to this new notion as well as Koszul duality theory. To be able to provide meaningful definitions, we work in the context of objects which are filtered and complete and become differential graded after applying the associated graded functor. This setting brings its own difficulties but it nevertheless permits us to define a combinatorial model category structure that we can transfer to the category of curved operads and to the category of algebras over a curved operad using free-forgetful adjunctions. We address the case of curved associative algebras. We recover the notion of curved Aoo-algebras, and we show that the homotopy categories of curved associative algebras and of curved Aoo-algebras are Quillen equivalent.
We introduce a general definition for colored cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from colored cyclic operads to colored operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski--Moerdijk model structure on the category of colored operads enriched in simplicial sets to the category of colored cyclic operads enriched in simplicial sets.
In a recent paper, the second author and Joana Cirici proved a theorem that says that given appropriate hypotheses, $n$-formality of a differential graded algebraic structure is equivalent to the existence of a chain-level lift of a homology-level degree twisting automorphism using a unit of multiplicative order at least $n$. Here we give another proof of this result of independent interest and under slightly different hypotheses. We use the homotopy transfer theorem and an explicit inductive procedure in order to kill the higher operations. As an application of our result, we prove formality with coefficients in the $p$-adic integers of certain dg-algebras coming from hyperplane and toric arrangements and configuration spaces.
The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which roughly says that operations in the codomain operad can be written essentially uniquely as operations in arity one followed by operations in the domain operad.
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism prop. In this note, we show that more generally, given an category C enriched in a monoidal category V, the functor that associates to a monoid in V its category of representations in C is adjoint to the functor that computes the endomorphism monoid of any functor with domain C. After describing the first results of the theory we give several examples of applications.
The purpose of this document is to provide data about known Betti numbers of unordered configuration spaces of small graphs in order to guide research and avoid duplicated effort. It contains information for connected multigraphs having at most nine edges which contain no loops, no bivalent vertices, and no internal (i.e., non-leaf) bridges.
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to Świątkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.
We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.
We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been implicitly studied. We ask whether the endomorphism operad of the forgetful functor from algebras over an operad to the ground category recovers that operad. The answer is positive for operads in vector spaces over an infinite field, but negative both in vector spaces over finite fields and in sets. Several examples are computed.
For a family of Jacobians of smooth pointed curves there is a notion of tautological algebra. There is an action of $\mathfrak{sl}_2$ on this algebra. We define and study a lifting of the Polishchuk operator, corresponding to $f\in \mathfrak{sl}_2$, on an algebra consisting of punctured Riemann surfaces. As an application we prove that a collection of tautological relations on moduli of curves, discovered by Faber and Zagier, come from a class of relations on the universal Jacobian.
Suppose that $F: \mathcal{N} \to \mathcal{M}$ is a functor whose target is a Quillen model category. We give a succinct sufficient condition for the existence of the right-induced model category structure on $\mathcal{N}$ in the case when $F$ admits both adjoints. We give several examples, including change-of-rings, operad-like structures, and anti-involutive structures on infinity categories. For the last of these, we explore anti-involutive structures for several different models of $(\infty, 1)$-categories, and show that known Quillen equivalences between base model categories lift to equivalences.
We give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type.
Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. A natural example extends ordinary probability theory on a finite volume Riemannian manifold M. In this example, initial conditions for fluid flow on M are identified with collections of homotopy random variables and solutions to the Euler equation are identified with homotopies between collections of homotopy random variables. Several ideas about using homotopy probability theory to study fluid flow are introduced.
An operadic framework is developed to explain the inversion formula relating moments and cumulants in operator-valued free probability theory.
We construct a cooperad which extends the framework of homotopy probability theory to free probability theory. The cooperad constructed, which seems related to the sequence and cactus operads, may be of independent interest.
We construct a space of string diagrams, which are a type of fatgraph with some additional data, and show that there are string topology operations on the chains of the free loop space of a closed Riemannian manifold which are parameterized by the chains on the space of string diagrams. These operations are shown to recover known structure on homology of the free loop space.