2025/10/07 by Boris Shoikhet, Shoikhet, Boris
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Quantum Algebra (math.QA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2510.05813
openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose a method for constructing a colored (d+1)-operad seqd in Sets, in the sense of Batanin [Ba1,2], whose category of colors (=the category of unary operations) is the category Θd, dual to the Joyal category of d-disks [J], [Be2,3]. For d=1 it is the Tamarkin Δ-colored 2-operad seq, playing an important role in his paper [T3] and in the solution loc.cit. to the Deligne conjecture for Hochschild cochains. We expect that for higher d these operads provide a key to solution to the the higher Deligne conjecture, in the (weak) d-categorical context. For general d the construction is based on two combinatorial conjectures, which we prove to be true for d=2,3. We introduce a concept of a generalised Joyal disk, so that the category of generalised Joyal d-disks admits an analogue of the funny product of ordinary categories. (For d=1, a generalised Joyal disk is a category with a ``minimal'' and a ``maximal'' object). It makes us possible to define a higher analog Ld of the lattice path operad [BB] with Θd as the category of unary operations. The Θd-colored (d+1)-operad seqd is found ``inside'' the desymmetrisation of the symmetric operad Ld. We construct ``blocks'' (subfunctors of Ld) labelled by objects of the cartesian d-power of the Berger complete graph operad [Be1], and prove the contractibility of a single block in the topological and the dg condensations. In this way, we essentially upgrade the known proof given by McClure-Smith [MS3] for the case d=1, so that the refined argument is generalised to the case of Θd. Then we prove that seqd is contractible in topological and dg condensations (for d=2,3, and for general d modulo the two combinatorial conjectures).