arXiv subjects

Guillaume Laplante-Anfossi

Publications and source records attributed to Guillaume Laplante-Anfossi.

11 recordsLinked to original sources

arXiv2026

A universal characterization of the curved homotopy Lie and associative operads

We study the category of nonsymmetric dg operads valued in strict graded-mixed complexes, equipped with a distinguished arity zero weight one element which generates the weight grading, and whose differential has weight one. We show that the initial object is the curved A-infinity operad, that the forgetful functor to the category of operads under it admits a right adjoint, and that the unit of the adjunction encodes the operation of twisting a curved A-infinity algebra by a Maurer-Cartan element. The corresponding notions for symmetric operads characterize the curved L-infinity operad and the corresponding twisting procedure.

math.AT↗

arXiv2026

Genus Zero Kashiwara-Vergne Solutions from Braids

Using the language of moperads -- monoids in the category of right modules over an operad -- we reinterpret the Alekseev--Enriquez--Torossian construction of Kashiwara--Vergne (KV) solutions from associators. We show that any equivalence between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck--Teichmüller module groups act on the latter, intertwining the actions of the KV symmetry groups. In the other direction, we show that any symmetric KV solution gives rise to a module map from parenthesized braids with a frozen strand to tangential automorphisms of free Lie algebras. This map factors through the moperad of chord diagrams if and only if the associated KV associator is a Drinfeld associator.

math.AT↗

arXiv2025

Operadic twisting as an adjunction

For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with certain splittings (not respecting the differential) of the projection to its quotient by the curvature operation. We derive a similar classification of Tw-coalgebras. For the class of operads whose Koszul dual admits a unital extension, we give explicit formulas for the cTw-coalgebra structures on their curved homotopy resolutions, recovering the convolution Lie algebra's ``gauge group action'' of Dotsenko, Shadrin, and Vallette.

math.AT↗

arXiv2025

Framed Polytopes and Higher Categories

In the early 1990s, Kapranov and Voevodsky proposed a geometric method for constructing higher-categorical pasting diagrams from generically framed convex polytopes. This work revisits their construction and identifies a convex-geometric condition that is both necessary and sufficient for the procedure to yield a well-defined pasting diagram. Our criterion, the absence of cellular loops, relates their construction to the theory of cellular strings, an active area of convex geometry originating in the Baues problem. This paper introduces higher-dimensional cellular strings and uses them to disprove the Kapranov-Voevodsky conjecture in the following strong sense. Not only do we exhibit framed polytopes admitting cellular loops, but we also construct examples for which every admissible frame produces one. As observed by these authors, Street's orientals arise from canonically framed cyclic simplices. We establish that this family is exceptional as any random $n$-simplex, canonically framed, almost surely exhibits cellular loops in the large $n$-limit.

math.CT↗

arXiv2025

Kashiwara-Vergne solutions degree by degree

We show that solutions to the Kashiwara-Vergne problem can be extended degree by degree. This can be used to simplify the computation of a class of Drinfel'd associators, which under the Alekseev-Torossian conjecture, may comprise all associators. We also give a proof that the associated graded Lie algebra of the Kashiwara-Vergne group is isomorphic to the graded Kashiwara-Vergne Lie algebra.

math.QA↗

arXiv2025

Term rewriting on nestohedra

We define term rewriting systems on the vertices and faces of nestohedra, and show that the former are confluent and terminating. While the associated posets on vertices generalize Barnard--McConville's flip order for graph-associahedra, the preorders on faces generalize the facial weak order for permutahedra and the generalized Tamari order for associahedra. Moreover, we define and study contextual families of nestohedra, whose local confluence diagrams satisfy a certain uniformity condition. Among them are associahedra and operahedra, whose associated proofs of confluence for their rewriting systems reproduce proofs of categorical coherence theorems for monoidal categories and categorified operads.

math.CT↗

arXiv2024

Steenrod operations via higher Bruhat orders

The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-$i$ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining this correspondence involves interpreting the coproducts geometrically in terms of zonotopal tilings, which allows us to give conceptual proofs of their properties and show that all reasonable coproducts arise from our construction.

math.AT↗

arXiv2024

Topological proofs of categorical coherence

We give a short topological proof of coherence for categorified non-symmetric operads by using the fact that the diagrams involved form the 1-skeleton of simply connected CW complexes. We also obtain a "one-step" topological proof of Mac Lane's coherence theorem for symmetric monoidal categories, as suggested by Kapranov in 1993. Our analysis is based on a notion of combinatorial homotopy, which we further study in the special case of polyhedral complexes, leading to a second geometrical proof of coherence which is very close to Mac Lane's original argument. We use Morse theory to show that this second method is (strictly) less general than the first. We provide a detailed analysis of how both methods allow us to deduce these two categorical coherence results and discuss possible generalizations to higher categories.

math.AT↗

arXiv2023

Cellular diagonals of permutahedra

We provide a systematic enumerative and combinatorial study of geometric cellular diagonals on the permutahedra. In the first part of the paper, we study the combinatorics of certain hyperplane arrangements obtained as the union of $\ell$ generically translated copies of the classical braid arrangement. Based on Zaslavsky's theory, we derive enumerative results on the faces of these arrangements involving combinatorial objects named partition forests and rainbow forests. This yields in particular nice formulas for the number of regions and bounded regions in terms of exponentials of generating functions of Fuss-Catalan numbers. By duality, the specialization of these results to the case $\ell = 2$ gives the enumeration of any geometric diagonal of the permutahedron. In the second part of the paper, we study diagonals which respect the operadic structure on the family of permutahedra. We show that there are exactly two such diagonals, which are moreover isomorphic. We describe their facets by a simple rule on paths in partition trees, and their vertices as pattern-avoiding pairs of permutations. We show that one of these diagonals is a topological enhancement of the Sanbeblidze-Umble diagonal, and unravel a natural lattice structure on their sets of facets. In the third part of the paper, we use the preceding results to show that there are precisely two isomorphic topological cellular operadic structures on the families of operahedra and multiplihedra, and exactly two infinity-isomorphic geometric universal tensor products of homotopy operads and A-infinity morphisms.

math.CO↗

arXiv2023

The diagonal of the multiplihedra and the tensor product of A-infinity morphisms

We define a cellular approximation for the diagonal of the Forcey--Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A-infinity morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology.

math.AT↗

arXiv2022

The diagonal of the operahedra

The primary goal of this article is to set up a general theory of coherent cellular approximations of the diagonal for families of polytopes by developing the method introduced by N. Masuda, A. Tonks, H. Thomas and B. Vallette. We apply this theory to the study of the operahedra, a family of polytopes ranging from the associahedra to the permutahedra, and which encodes homotopy operads. After defining Loday realizations of the operahedra, we make a coherent choice of cellular approximations of the diagonal, which leads to a compatible topological cellular operad structure on them. This gives a model for topological and algebraic homotopy operads and an explicit functorial formula for their tensor product.

math.AT↗