Abstract. In recent work, Hess and Shipley [18] defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBökstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
In this paper, we establish a multiplicative equivalence between two multiplicative algebraic K-theory constructions, Elmendorf and Mandell's version of Segal's K-theory and Blumberg and Mandell's version of Waldhausen's S• construction. This equivalence implies that the ring spectra, algebra spectra, and module spectra constructed via these two classical algebraic K-theory functors are equivalent as ring, algebra or module spectra, respectively. It also allows for comparisions of spectrally enriched categories constructed via these definitions of Ktheory. As both the Elmendorf-Mandell and Blumberg-Mandell multiplicative versions of K-theory encode their multiplicativity in the language of multicategories, our main theorem is that there is multinatural transformation relating these two symmetric multifunctors that lifts the classical functor from Segal's to Waldhausen's construction. Along the way, we provide a slight generalization of the Elmendorf-Mandell construction to symmetric monoidal categories.Second, we focus on K-theory as a construction on Waldhausen categories, rather than any of the classes of infinity categories. Since the classical Waldhausen and Segal constructions apply only to more specific types of input, their uniqueness is not addressed by any of the uniqueness or universality results above. The present paper provides this kind of multiplicative comparison between the classical versions of algebraic K-theory.
Abstract. For any finite group G, there are several well-established definitions of a G-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal G-spectrum for each finite group G. We use the framework of enriched indexed categories to make this precise. We also consider equivariant K-theory and Spin c -cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.
Abstract. We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg-MacLane spectra for Mackey functors and for suspension spectra for finite G-sets.
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