46 citations · 197 across the 31 of their papers we have counts for
31 papers
On partial groups of small order
Philip Hackney
We explain a computer enumeration of all partial groups (in the sense of Chermak) of order at most 10. An accompanying dataset contains a full list, consisting of 123,650 partial g…
Binary partial groups
Philip Hackney, Justin Lynd, Edoardo Salati
There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse.…
Free decomposition spaces
Philip Hackney, Joachim Kock
We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon…
Culf maps and edgewise subdivision
Philip Hackney, Joachim Kock
We show that, for any simplicial space , the -category of culf maps over is equivalent to the -category of right fibrations over , the…
Segal conditions for generalized operads
Philip Hackney
This note is an introduction to several generalizations of the dendroidal sets of Moerdijk--Weiss. Dendroidal sets are presheaves on a category of rooted trees, and here we conside…
Labelled cospan categories and properads
Jonathan Beardsley, Philip Hackney
We prove Steinebrunner's conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, stri…
Categories of graphs for operadic structures
Philip Hackney
We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on…
An -categorical pasting theorem
Philip Hackney, Viktoriya Ozornova, Emily Riehl +1
We show that any pasting diagram in any -category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a pasting sche…
Induced model structures for higher categories
Philip Hackney, Martina Rovelli
We give a new criterion guaranteeing existence of model structures left-induced along a functor admitting both adjoints. This works under the hypothesis that the functor induces id…
On rectification and enrichment of infinity properads
Hongyi Chu, Philip Hackney
We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk compl…
Coextension of scalars in operad theory
Gabriel C. Drummond-Cole, Philip Hackney
The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functo…
Modular operads and the nerve theorem
Philip Hackney, Marcy Robertson, Donald Yau
We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modula…
A graphical category for higher modular operads
Philip Hackney, Marcy Robertson, Donald Yau
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 Qui…
Dwyer--Kan homotopy theory for cyclic operads
Gabriel C. Drummond-Cole, Philip Hackney
We introduce a general definition for colored cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from colored cyc…
A criterion for existence of right-induced model structures
Gabriel C. Drummond-Cole, Philip Hackney
Suppose that is a functor whose target is a Quillen model category. We give a succinct sufficient condition for the existence of the right-induced…
Configuration spaces form a Segal semi-dendroidal space
Philip Hackney
The purpose of this short note is to illustrate the utility of (semi-) dendroidal objects in describing certain 'up-to-homotopy' operads. Specifically, we exhibit a semi-dendroidal…
On factorizations of graphical maps
Philip Hackney, Marcy Robertson, Donald Yau
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…
Lecture notes on infinity-properads
Philip Hackney, Marcy Robertson
These are notes for three lectures on higher properads given at a program at the mathematical institute MATRIX in Australia in June 2016. The first lecture covers the case of opera…
Higher cyclic operads
Philip Hackney, Marcy Robertson, Donald Yau
We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category of trees, which…
Shrinkability, relative left properness, and derived base change
Philip Hackney, Marcy Robertson, Donald Yau
For a connected pasting scheme , under reasonable assumptions on the underlying category, the category of -colored -props admits a cofibrantly…
A simplicial model for infinity properads
Philip Hackney, Marcy Robertson, Donald Yau
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 s…
Relative left properness of colored operads
Philip Hackney, Marcy Robertson, Donald Yau
The category of -colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relat…
Infinity Properads and Infinity Wheeled Properads
Philip Hackney, Marcy Robertson, Donald Yau
A theory of -properads is developed, extending both the Joyal-Lurie -categories and the Cisinski-Moerdijk-Weiss -operads. Every connected wheel-free graph g…
Diagrams encoding group actions on -spaces
Julia E. Bergner, Philip Hackney
We introduce, for any group , a category such that diagrams satisfying a Segal condition correspond to infinite loop spaces with a -actio…