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We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Gamma$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
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Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $mathcal{D}$ is an infinite loop space, the zeroth space of the $K$-theory spectrum $Kmathcal{D}$. This spectrum has stable homotopy groups concentrated in levels 0, 1, and 2. In this paper, we describe part of the Postnikov data of $Kmathcal{D}$ in terms of categorical structure. We use this to show that there is no strict skeletal Picard 2-category whose $K$-theory realizes the 2-truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard 2-category $Sigma C$ from a Picard 1-category $C$, and show that it commutes with $K$-theory in that $KSigma C$ is stably equivalent to $Sigma K C$.
We show that Mandells inverse $K$-theory functor is a categorically-enriched multifunctor. In particular, it preserves algebraic structures parametrized by operads. As applications, we describe how ring categories, bipermutative categories, braided ring categories, and $E_n$-monoidal categories arise as the images of inverse $K$-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.
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthys map from $K$-theory of endomorphisms to topological restriction homology, defining it for any Waldhausen category with a compatible enrichment in orthogonal spectra. In particular, this extends their construction from rings to ring spectra. We also give a revisionist treatment of the original Dennis trace map from $K$-theory to topological Hochschild homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories).
We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras. We include examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes.
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