Related papers: A prop structure on partitions

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal…

We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple…

We define a naturality construction for the operations of weak omega-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product with a directed interval, and behaves…

We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and definable subcategories, and as such appear in a wide range…

We describe four natural operad structures on the vector space generated by isomorphism classes of finite posets. The three last ones are set-theoretical and can be seen as a simplified version of the first, the same way the NAP operad…

We develop methods for proving that certain extensions of polynomial functors do not split naturally. As an application we give a functorial description of the third and the fourth stable homotopy groups of the classifying spaces of free…

We will introduce the notion of higher derived bracket construction in the category of operads and prove that the higher derived bracket construction of Lie operad is equivalent to the cobar construction of Leibniz operad. The theorem is…

It is well known that strict $\omega$-categories, strict $\omega$-functors, strict natural $\omega$-transformations, and so on, form a strict $\omega$-category. A similar property for weak $\omega$-categories is one of the main hypotheses…

From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space" is the complement of an arrangement of…

A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a…

Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in…

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of…

We define the notion of a (P,P-tilde)-structure on a universe p in a locally cartesian closed category category C with a binary product structure and construct a (Pi,lambda)-structure on the C-systems CC(C,p) from a (P,P-tilde)-structure on…

Let p be a prime. Let FpSp be the group algebra of the symmetric group over the finite field Fp with |Fp|=p$. Let Fp be the trivial FpSp-module. We present a projective resolution PRes Fp of the module Fp and equip the Yoneda algebra…

This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to…

We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct…

The combinatorial properties of partitions with various restrictions on their hooksets are explored. A connection with numerical semigroups extends current results on simultaneous s/t-cores. Conditions that suffice for a partition to…

Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting…

We develop the homotopy theory of semisimplicial sets constructively and without reference to point-set topology to obtain a constructive model for $\omega$-groupoids. Most of the development is folklore, but for a few results the author is…

A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category…