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 $\mathcal G$, under reasonable assumptions on the underlying category, the category of $\mathfrak C$-colored $\mathcal G$-props admits a cofibrantly generated model category structure. In this paper, we show that, if $\mathcal G$ is closed under shrinking internal edges, then this model structure on $\mathcal G$-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 $\mathcal G$ 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 $\mathcal G$-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 $\overline{\mathcal{M}}_{0,\bullet+1}$, 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 $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $\Omega$ plays for operads. We carefully study properties of $\mathbf{U}$, 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 $\mathbf{U}$.Comment: Minor expository changes throughout, including a new section on "Further directions", Remark 4.3, Example 5.4, Figures 2 and