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  • 1.
    Bergner, Julia
    et al.
    University of California, Riverside, USA.
    Hackney, Philip
    Stockholm University, Faculty of Science, Department of Mathematics. University of California, Riverside, USA.
    Group actions on Segal operads2014In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 202, no 1, 423-460 p.Article in journal (Refereed)
    Abstract [en]

    We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give a Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.

  • 2.
    Bergner, Julia
    et al.
    University of California, Riverside, USA.
    Hackney, Philip
    Stockholm University, Faculty of Science, Department of Mathematics. University of California, Riverside, USA.
    Reedy categories which encode the notion of category actions2015In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, 193-222 p.Article in journal (Refereed)
    Abstract [en]

    We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.

  • 3.
    Hackney, Philip
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Robertson, Marcy
    Yau, Donald
    Relative left properness of colored operads2016In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 16, no 5, 2691-2714 p.Article in journal (Refereed)
    Abstract [en]

    The category of C-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness condition, ie that the class of weak equivalences between Sigma-cofibrant operads is closed under cobase change along cofibrations. We also provide an example of Dwyer which shows that the model structure on C-colored symmetric operads is not left proper.

  • 4.
    Hackney, Philip
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Robertson, Marcy
    University of California, Los Angeles, USA.
    Yau, Donald
    Ohio State University at Newark.
    Relative left properness of colored operadsManuscript (preprint) (Other academic)
    Abstract [en]

    The categories of C-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a weaker version of left properness. We also provide an example of Dwyer which shows that this category is not left proper.

  • 5.
    Hackney, Philip
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Robertson, Marcy
    Yau, Donald
    Shrinkability, relative left properness, and derived base change2017In: New York journal of mathematics, ISSN 1076-9803, E-ISSN 1076-9803, Vol. 23, 83-117 p.Article in journal (Refereed)
    Abstract [en]

    For a connected pasting scheme G, under reasonable assumptions on the underlying category, the category of C-colored G-props admits a cofibrantly generated model category structure. In this paper, we show that, if G is closed under shrinking internal edges, then this model structure on 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 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 c-props. The final section gives illuminating examples that justify the conditions on base model categories.

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