Global orthogonal spectra

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.Comment: 17 page

Graded Tambara Functors

We define the notion of an $\mathcal{RO}(G)$-graded Tambara functor and prove that any $G$-spectrum with norm multiplication gives rise to such an $\mathcal{RO}(G)$-graded Tambara functor.Comment: 25 pages; elaborated on the description of graded Tambara functors and clarified some of the category theory. Added more detailed reference

A Presheaf Interpretation of the Generalized Freyd Conjecture

We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out quite trivially.Comment: 8 pages; formerly titled "Thinking about the Freyd conjecture

A comparison of norm maps

We present a spectrum-level version of the norm map in equivariant homotopy theory based on the algebraic construction in work of Greenleess-May. We show that this new norm map is same as the construction in work Hill-Hopkins-Ravenel on the Kervaire invariant problem. Our comparison of the two norm maps gives a conceptual understanding of the choices inherent in the definition of the multiplicative norm map.Comment: 11 pages; with appendix by Anna Marie Bohmann and Emily Rieh

Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories

The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.Comment: 20 pages. This updated paper discusses the work on Morita equivalence of Lawvere theories that appeared in version one. In order to better highlight the two separate directions of the results in that first version, the material on assembly maps has been incorporated into a second paper, arXiv:2112.0700

A model structure on GCat

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.Comment: 12 pages. Final version. Will appear in Proceedings for WIT (Women in Topology Workshop

Naive-commutative structure on rational equivariant KK-theory for abelian groups

In this paper, we calculate the image of the connective and periodic rational equivariant complex $K$-theory spectrum in the algebraic model for naive-commutative ring $G$-spectra given by Barnes, Greenlees and K\k{e}dziorek for finite abelian $G$. Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. We further deduce a structure theorem for module spectra over rational equivariant complex $K$-theory.Comment: 19 page

A trace map on higher scissors congruence groups

Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $K$-theory. In this paper we address the particular case of the $K$-theory of polyhedra, also called scissors congruence $K$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason's general framework of $K$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $K$-theory.Comment: 32 pages, 3 figures. Revision of the paper previously entitled "A Farrell--Jones isomorphism for $K$-theory of polyhedra.

Segal and Waldhausen K-theory: a multiplicative comparison

Fundamental work of Segal and Waldhausen gives us two versions of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf--Mandell and Blumberg--Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.Non UBCUnreviewedAuthor affiliation: Vanderbilt UniversityResearche