Abstract
We begin this paper by noting that, in a 1969 paper in the Transactions, M.C. McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ-spaces - indeed this is roughly what Segal did — and then to a more modern situation:K ⊗ RwhereKis a based space andRis a unital, augmented, commutative, associative S-algebra.
The model comes with an easy-to-describe filtration. If one letsK = S nand then stabilizes with respect to n, one gets a filtered model for the Topological André-Quillen Homology of R. When R = E∑∞(SlΩ∞X) + one arrives at a filtered model for the connective cover of a spectrumX constructed from its Oth space.
Another example comes by lettingKbe a finite complex, andRthe S-dual of a finite complexZ.Dualizing again, one arrives at G. Arone’s model for the Goodwillie tower of the functor sendingZto ∑∞ MapT(K,Z).
Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E1-terms. A few nontrivial examples are given.
In an appendix, we describe the construction for unital, commutative, associative S-algebras not necessarily augmented.
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Kuhn, N.J. (2003). The McCord Model for the Tensor Product of a Space and a Commutative Ring Spectrum. In: Arone, G., Hubbuck, J., Levi, R., Weiss, M. (eds) Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics, vol 215. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7863-0_13
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