arXiv subjects
Publications and source records attributed to Martin Markl.
We show that abelian categories can be characterized as structures consisting of a colax algebra and a lax algebra connected by a lax mixed rewriting rule. To this end we develop a theory of lax rewriting rules for pairs of lax-lax and colax-lax algebras over 2-monads.
We prove that a pointed category has kernels if and only if it is a lax algebra for the arrow 2-monad, and that this holds if and only if it is the d\'ecalage of a supercoherent structure. We will then interpret categories with kernels as the sought-after weak version of unary operadic categories.
We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and Markl. Our approach is particularly natural in applications to the operadic category of graphs and the related category of modular operads and their clones.
Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of minimal models that are unique up to isotopy, a stronger result obviously known to T. Kadeishvili and certainly to others, yet seemingly overlooked by mankind. We will explore how this stronger result can help in the study of Massey products. First, we show that the attempts to extract a local information from the ternary operation $\mu_3$ of our minimal model leads directly to the rediscovery of the triple Massey product. The motto is: "The triple Massey product is an invariant manifestation of $\mu_3$." We then prove that, under reasonable assumptions, the higher Massey product $\langle x_1,\ldots,x_n\rangle$ equals the set of all values $\mu_n(x_1,\ldots,x_n)$, where $\mu_n$ runs over the $n$-ary products of our minimal models. We believe that this note will help to elucidate the still somewhat enigmatic relationship between minimal models and Massey products.
We show that the functor which assigns to an A-infinity morphism between isotopy classes of A-infinity algebras whose linear part is a chain homotopy equivalence its underlying chain map is a discrete Grothendieck bifibration. We then generalize our results to P-infinity structures over a field of characteristic zero, for any quadratic Koszul operad P. An immediate application is a categorical framework in which the transfers of e.g. A-infinity, L-infinity and C-infinity structures are strictly functorial. A by product of our reasoning is a general transfer theorem for P-infinity algebras, which we prove in the last section.
We develop a self-dual, bivariant extension of the concept of an operadic category, its associated operads and their algebras. Our new theory covers, besides all classical subjects, also generalized traces and bivariant versions of Kapranov's charades. It is, moreover, combinatorially rich and aesthetically pleasing.
We will show that the Morrison-Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.
We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie-admissible and L-infinity structures. As a main application, and the original motivation for the present work, we show how a certain generalization of the well-known big bracket construction of Lecomte\textendash Roger and Kosmann-Schwarzbach encompassing the case of homotopy involutive Lie bialgebras can be obtained.
The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.
We characterize $A_\infty$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_\infty$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal P}_\infty$-structures over a field of characteristic zero, for any quadratic Koszul operad ${\mathcal P}$.
The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the second part we study Koszulity of Per-operads. Among other things we prove that the terminal Per-operad is Koszul self-dual. We then describe strongly homotopy permutads as algebras of its minimal model. Our paper shall advertise analogous future results valid in general operadic categories, and the prominent role of operadic (op)fibrations in the related theory.
We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.
We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincaré series.
By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.
This is the first paper of a series which aims to set up the cornerstones of Koszul duality for operads over operadic categories. To this end we single out additional properties of operadic categories under which the theory of quadratic operads and their Koszulity can be developped, parallel to the traditional one by Ginzburg and Kapranov. We then investigate how these extra properties interact with discrete operadic (op)fibrations, which we use as a powerful tool to construct new operadic categories from old ones. We pay particular attention to the operadic category of graphs, giving a full description of this category (and its variants) as an operadic category, and proving that it satisfies all the additional properties. Our present work provides an answer to a question formulated in Loday's last talk in 2012:``What encodes types of operads?''. In the second and third papers of our series we continue Loday's program by answering his second question: ``How to construct Koszul duals to these objects?'', and proving Koszulity of some of the most relevant operads.
We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive.
We develop a new formalism for the Quantum Master Equation $Δe^{S/\hbar} = 0$ and the category of ${\rm IBL}_\infty$-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of ${\rm IBL}_\infty$-algebras and ${\rm L}_\infty$-algebras, and on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.
We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two-colored operad and characterize its algebras via morphisms of Frobenius algebras, recovering some previously known results of Kaufmann, Penner and others. As an important auxiliary tool we characterize inclusions of cyclic operads that induce inclusions of their modular completions.