Coassembly is a homotopy limit map

We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.Comment: Accepted version. Several improvements from the referee, including a more elegant proof of Lemma 3.

A convenient category of parametrized spectra

We describe a point-set category of parametrized orthogonal spectra, a model structure on this category, and a separate, more geometric class of cofibrant-and-fibrant objects. The structures we describe are "convenient" in that they are preserved by the most common operations. They allow us to reduce sophisticated statements about the homotopy category to straightforward claims at the point-set level. We use this framework to give a construction of the bicategory of parametrized spectra, one that is far more direct than earlier approaches. This gives a clean bridge between the concrete index theory pioneered by Dold, and the formal bicategorical theory developed by May and Sigurdsson, Ponto, and Shulman.Comment: 97 pages with references. This article is a condensation of arXiv:1906.0477

Coherence for indexed symmetric monoidal categories

Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise generalizations. In this paper, we extend existing coherence theorems to the setting of indexed symmetric monoidal categories. The most central theorem states that a large family of operations on a bicategory defined from an indexed symmetric monoidal category are all canonically isomorphic. As a part of this theorem, we introduce a rigorous graphical calculus that specifies when two such operations admit a canonical isomorphism.Comment: 100 pages, 64 figures, 13 table

Parametrized spectra, a low-tech approach

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. We cover most of the central results from the book of May and Sigurdsson, sometimes with weaker hypotheses, and give a new construction of the bicategory $\mathcal Ex$ of parametrized spectra. We also give a careful account of coherence results at the level of homotopy categories. The potential audience for this work may extend outside the boundaries of modern homotopy theory, so our treatment is structured to use as little technology as possible. In particular, many of the results are stated without using model categories. We also illustrate some applications to fixed-point theory.Comment: 182 pages including references and index, 5 figures. v2: comments incorporated, expository sections streamlined, a brief discussion of Thom spectra added. A short user's guide now accompanies the text, see https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pd

The Morita equivalence between parametrized spectra and module spectra

We give a Quillen equivalence between May and Sigurdsson's model category of parametrized spectra over BG, and Mandell, May, Schwede, and Shipley's model category of modules over the orthogonal ring spectrum \Sigma^\infty_+ G, for each topological group G. More generally, for a topological category C we introduce an "aggregate" model structure on the category of diagrams of spectra indexed by C, and prove that it is Quillen equivalent to spectra over BC. This lifts several earlier results, and leads to a complete characterization of the dualizable parametrized spectra, answering a question of May and Sigurdsson.Comment: 24 pages, to appear in Contemporary Mathematic

The Transfer is Functorial

We prove that the Becker-Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a "multiplicative'" description of the transfer, different from the standard presentation as the trace of a diagonal map.Comment: This is the final preprint version. The article is to appear in the Advances in Mathematic