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There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.
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In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.
Let $mathcal{A}$ be a small abelian category. The purpose of this paper is to introduce and study a category $overline{mathcal{A}}$ which implicitly appears in construction of some TQFTs where $overline{mathcal{A}}$ is determined by $mathcal{A}$. If $mathcal{A}$ is the category of abelian groups, then the TQFTs obtained by Dijkgraaf-Witten theory of abelian groups or Turaev-Viro theory of bicommutative Hopf algebras factor through $overline{mathcal{A}}$ up to a scaling. In this paper, we go further by giving a sufficient condition for an $mathcal{A}$-valued Brown functor to extend to a homotopy-theoretic analogue of $overline{mathcal{A}}$-valued TQFT for arbitrary $mathcal{A}$. The results of this paper and our subsequent paper reproduces TQFTs obtained by DW theory and TV theory.
We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.
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