The homotopy theory of cyclotomic spectra
Let \(\mathbb{T}\) be the circle group of unit complex numbers. For orthogonal \(\mathbb{T}\)-representations \(V\) and \(W\) we let \(\mathcal{I}_\mathbb{T}\,(V, W)\) denote the Thom space of the bundle whose elements consist of the pairs \((f, x)\) of linear isometries \(f : V\to W\) and points \(x\) in \(f(V)^\bot\). This construction of Thom spaces allows us to define a category \(\mathcal{I}_\mathbb{T}\) enriched in based \(\mathbb{T}\)-spaces where objects are the orthogonal \(\mathbb{T}\)-representations. By an orthogonal \(\mathbb{T}\)-spectrum is meant an \(\mathbb{T}\)-equivariant functor from this \(\mathcal{I}_\mathbb{T}\) to based \(\mathbb{T}\)-spaces. Then we call a \(p\)-cyclotomic spectrum the orthogonal \(\mathbb{T}\)-spectrum \(X\) together with a map of \(\mathbb{T}\)-spectra \(\rho_p^*\Phi^{C_p}X \to X\) which is an \(\mathcal{F}_p\)-equivalence, i.e. which induces an isomorphism on \(\pi^{C_{p^n}}_*\) for all \(n \geq 0\), where \(\Phi^{C_p}\) denotes the geometric fixed point functor and \(\rho_p\) denotes the isomorphism \(\mathbb{T}\cong \mathbb{T}/C_p\) induced by \(p\)-th root map. In this paper the authors introduce a new model category called a \(model^\ast \, category\). It is defined as a category that has a model structure in the sense of [\textit{M. Hovey}, Model categories. Mathematical Surveys and Monographs. 63. Providence, RI: American Mathematical Society (1999; Zbl 0909.55001)] and admits finite coproducts and products, pushouts over cofibrations and pullbacks over fibrations. Based on this new notion, three results about the \(p\)-cyclotomic spectra are presented. The first result can be roughly stated as follows: The category of \(p\)-cyclotomic spectra is a model* category with weak equivalences of the orthogonal spectra and with \(\mathcal{F}_p\)-fibrations of the orthogonal \(\mathbb{T}\)-spectra. The second one states that the homotopy category of \(p\)-cyclotomic spectra becomes a triangulated category. The last one is a result related to topological cyclic homology \(TC(-; p)\), which states that letting \(X\) be a \(p\)-cyclotomic spectrum, the \(p\)-completion of \(TC(X; p)\) can be identified as the derived spectrum of maps from the sphere spectrum \(S\) to \(X\).
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