nLab cell spectrum

Definition

Definition

A cell spectrum is a topological sequential spectrum $X$ realized as the colimit over a sequence of spectra $\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots$ such that there are morphisms

$$j_n \;\colon\; \left( \underset{i \in I_n}{\sqcup} \Sigma^\infty S^{q_n} \right) \longrightarrow X_n$$

with $X_{n+1}= Cone(j_n)$ (the mapping cone).

A cell spectrum is a CW-spectrum if each attaching map $\Sigma^\infty S^{q_n}\to X_n$ factors through a $X_k \to X_n$ with $k \lt q$.

(e.g. Lewis-May-Steinberger 86, def. 5.1, def. 52)

Definition

A CW-spectrum, def. , is called a finite spectrum (or countable spectrum, etc.) if it has finitely many cells (countably many cells) according to def. .

Related concepts

• cell complex

References

Discussion in the generality of equivariant spectra is in

• L. Gaunce Lewis, Peter May, and M. Steinberger (with contributions by J.E. McClure), section I.5 of Equivariant stable homotopy theory Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)