Related papers: Cooperads as Symmetric Sequences
In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse…
Assuming complex functions defined on complex curves satisfy recursion relations with respect to number of parameters, we express the corresponding cohomology theory via generalizations of holomorphic connections. In examples provided, the…
We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…
We introduce new algebraic structures associated with heptagon relations -- higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of…
The theory of the functional sequences and series is presented; uniformly convergent, convergent in the sense of a mean square and weakly convergent sequences and series are considered. Sequential approach to constructing generalized…
The (co)homology theory of n-ary (co)compositions is a functor associating to $n$-ary (co)composition a complex. We present unified approach to the cohomology theory of coassociative and Lie coalgebras and for $2n$-ary cocompositions. This…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads,…
We introduce the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence, and we prove a variant of Cobham's theorem for the newly introduced class of sequences.
Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories…
The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…
Synchronization, that occurs both for non-chaotic and chaotic systems, is a striking phenomenon with many practical implications in natural phenomena. However, even before synchronization, strong correlations occur in the collective…
We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of lambda theories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation…
Recently, the theory of symmetric spaces has come to play an increased role in the physics of integrable systems and in quantum transport problems. In addition, it provides a classification of random matrix theories. In this paper we give a…
We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.
We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The…
This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for…
We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
We introduce two constructions of a coassociative comultiplication in the algebra of phrases in a given alphabet. As a preliminary step we give two constructions of a pre-Lie comultiplication in the module generated by words.
Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…