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We define a model structure on the category GCat of small categories with an action by a finite group G by lifting the Thomason model structure on Cat. We show there is a Quillen equivalence between GCat with this model structure and GTop with the standard model structure.
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Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact etale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.
Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from these resolutions. We also give an alternative description of the morphism sets in terms of Yoneda Ext groups.
We use Segal-Mitchisons cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided.
In a triangulated category, cofibre fill-ins always exist. Neeman showed that there is always at least one good fill-in, i.e., one whose mapping cone is exact. Verdier constructed a fill-in of a particular form in his proof of the $4 times 4$ lemma, which we call Verdier good. We show that for several classes of morphisms of exact triangles, the notions of good and Verdier good agree. We prove a lifting criterion for commutative squares in terms of (Verdier) good fill-ins. Using our results on good fill-ins, we also prove a pasting lemma for homotopy cartesian squares.
Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A Boardman-Vogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.
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