94 research outputs found
Cluster categories for topologists
We consider triangulated orbit categories, with the motivating example of
cluster categories, in their usual context of algebraic triangulated
categories, then present them from another perspective in the framework of
topological triangulated categories.Comment: 11 pages, short example added, to appear in proceedings of CATS
On factorizations of graphical maps
We study the categories governing infinity (wheeled) properads. The graphical
category, which was already known to be generalized Reedy, is in fact an
Eilenberg-Zilber category. A minor alteration to the definition of the wheeled
graphical category allows us to show that it is a generalized Reedy category.
Finally, we present model structures for Segal properads and Segal wheeled
properads.Comment: Final author version, to appear in HH
Shrinkability, relative left properness, and derived base change
For a connected pasting scheme , under reasonable assumptions on
the underlying category, the category of -colored -props admits a cofibrantly generated model category structure. In this
paper, we show that, if is closed under shrinking internal edges,
then this model structure on -props satisfies a (weaker version) of
left properness. Connected pasting schemes satisfying this property include
those for all connected wheeled graphs (for wheeled properads), wheeled trees
(for wheeled operads), simply connected graphs (for dioperads), unital trees
(for symmetric operads), and unitial linear graphs (for small categories). The
pasting scheme for connected wheel-free graphs (for properads) does _not_
satisfy this condition.
We furthermore prove, assuming is shrinkable and our base
categories are nice enough, that a weak symmetric monoidal Quillen equivalence
between two base categories induces a Quillen equivalence between their
categories of -props. The final section gives illuminating examples
that justify the conditions on base model categories.Comment: Final version, to appear in NYJ
A simplicial model for infinity properads
We show how the model structure on the category of simplicially-enriched
(colored) props induces a model structure on the category of
simplicially-enriched (colored) properads. A similar result holds for
dioperads.Comment: Final versio
Spaces of Operad Structures
The purpose of this paper is to study the derived category of simplicial
multicategories with arbitrary sets of objects (also known as, colored operads
in simplicial sets). Our main result is a derived Morita theory for
operads-where we describe the derived mapping spaces between two
multicategories P and Q in terms of the nerve of a certain category of
P-Q-bimodules. As an application, we show that the derived category possesses
internal Hom-objects.Comment: Comments Welcome
The Homotopy Theory of Simplicially Enriched Multicategories
In this paper we construct a cofibrantly generated model category structure
on the category of all small symmetric multicategories enriched in simplicial
sets
Operads of genus zero curves and the Grothendieck-Teichm\"{u}ller group
We show that the group of homotopy automorphisms of the profinite completion
of the genus zero surface operad is isomorphic to the (profinite)
Grothendieck-Teichm\"{u}ller group. Using a result of Drummond-Cole, we deduce
that the Grothendieck-Teichm\"{u}ller group acts nontrivially on
, the operad of stable curves of genus
zero. As a second application, we give an alternative proof that the framed
little 2-disks operad is formal.Comment: 36 page
On the category of props
The category of (colored) props is an enhancement of the category of colored
operads, and thus of the category of small categories. The titular category has
nice formal properties: it is bicomplete and is a symmetric monoidal category,
with monoidal product closely related to the Boardman-Vogt tensor product of
operads. Tools developed in this article, which is the first part of a larger
work, include a generalized version of multilinearity of functors, a free prop
construction defined on certain "generalized graphs", and the relationship
between the category of props and the categories of permutative categories and
of operads
Representations of derived A-infinity algebras
The notion of a derived A-infinity algebra arose in the work of Sagave as a
natural generalisation of the classical A-infinity algebra, relevant to the
case where one works over a commutative ring rather than a field. We develop
some of the basic operadic theory of derived A-infinity algebras, building on
work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras
over the Koszul dual cooperad of the operad dAs, and provide a simple
description of these. We study representations of derived A-infinity algebras
and explain how these are a two-sided version of Sagave's modules over derived
A-infinity algebras. We also give a new explicit example of a derived
A-infinity algebra.Comment: 27 pages. To appear in the Proceedings of the August 2013 "Women in
Topology" workshop at BIR
A graphical category for higher modular operads
We present a homotopy theory for a weak version of modular operads whose
compositions and contractions are only defined up to homotopy. This homotopy
theory takes the form of a Quillen model structure on the collection of
simplicial presheaves for a certain category of undirected graphs. This new
category of undirected graphs, denoted , plays a similar role for
modular operads that the dendroidal category plays for operads. We
carefully study properties of , including the existence of certain
factorization systems. Related structures, such as cyclic operads and stable
modular operads, can be similarly treated using categories derived from
.Comment: Minor expository changes throughout, including a new section on
"Further directions", Remark 4.3, Example 5.4, Figures 2 and
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