39 research outputs found
The Combinatorics of Iterated Loop Spaces
It is well known since Stasheff's work that 1-fold loop spaces can be
described in terms of the existence of higher homotopies for associativity
(coherence conditions) or equivalently as algebras of contractible
non-symmetric operads. The combinatorics of these higher homotopies is well
understood and is extremely useful.
For the theory of symmetric operads encapsulated the corresponding
higher homotopies, yet hid the combinatorics and it has remain a mystery for
almost 40 years. However, the recent developments in many fields ranging from
algebraic topology and algebraic geometry to mathematical physics and category
theory show that this combinatorics in higher dimensions will be even more
important than the one dimensional case.
In this paper we are going to show that there exists a conceptual way to make
these combinatorics explicit using the so called higher nonsymmetric
-operads.Comment: 23 page
Centers and homotopy centers in enriched monoidal categories
We consider a theory of centers and homotopy centers of monoids in monoidal
categories which themselves are enriched in duoidal categories. Duoidal
categories (introduced by Aguillar and Mahajan under the name 2-monoidal
categories) are categories with two monoidal structures which are related by
some, not necessary invertible, coherence morphisms. Centers of monoids in this
sense include many examples which are not `classical.' In particular, the
2-category of categories is an example of a center in our sense. Examples of
homotopy center (analogue of the classical Hochschild complex) include the
Gray-category Gray of 2-categories, 2-functors and pseudonatural
transformations and Tamarkin's homotopy 2-category of dg-categories,
dg-functors and coherent dg-transformations.Comment: 52 page
Regular patterns, substitudes, Feynman categories and operads
We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal. Second, we subsume the Getzler and Kaufmann-Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan extensions and a given monad, in this case the free-symmetric-monoidal-category monad. Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction
Symmetrisation of -operads and compactification of real configuration spaces
It is well known that the forgetful functor from symmetric operads to
nonsymmetric operads has a left adjoint given by product with the
symmetric group operad. It is also well known that this functor does not affect
the category of algebras of the operad. From the point of view of the author's
theory of higher operads, the nonsymmmetric operads are 1-operads and
is the first term of the infinite series of left adjoint functors
called symmetrisation functors, from -operads to symmetric operads with the
property that the category of one object, one arrow, . . ., one -arrow
algebras of an -operad is isomorphic to the category of algebras of
. In this paper we consider some geometrical and homotopical aspects
of the symmetrisation of -operads. We construct an -operadic analogue of
Fulton-Macpherson operad and show that its symmetrisation is exactly the operad
of Fulton and Macpherson. This implies that a space with an action of a
ontractible -operad has a natural structure of an algebra over an operad
weakly equivalent to the little -disks operad. A similar result holds for
chain operads. These results generalise the classical Eckman-Hilton argument to
arbitrary dimension. Finally, we apply the techniques to the Swiss Cheese type
operads introduced by Voronov and get analogous results in this case.Comment: 48 page
Locally constant n-operads as higher braided operads
We introduce a category of locally constant -operads which can be
considered as the category of higher braided operads. For the
homotopy category of locally constant -operads is equivalent to the homotopy
category of classical nonsymmetric, braided and symmetric operads
correspondingly.Comment: to appear in "Noncommutative Geometry
Crossed interval groups and operations on the Hochschild cohomology
We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades