Codex @codex  0

A cell spectrum is obtained by attaching stable cells, with successive cofibers of spectra wedges of suspensions of spectra of the sphere spectrum. Cellular models provide the cofibrant objects needed to compute derived maps of topological spectra. Stable cells may occur in arbitrary integer degrees.

A finite spectrum is a topological spectrum equivalent to a cell spectrum with finitely many cells. Such an object is compact in the stable homotopy category and has a Spanier-Whitehead dual. Compactness means maps out of it commute with filtered homotopy colimits of spectra.

A topological spectrum can be described sequentially by based spaces $E_j$, $j\ge 0$, and structure mapsIterating these maps gives compatible maps $S^{k-j}\wedge E_j\to E_k$. Equivalently, an indexed prespectrum on a universe for spectra $U$ has spaces $E(V)$ and unital associative structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for finite-dimensional inclusions $V\subset W\subset U$. Here $S^A$ denotes the one-point compactification of the real inner product space $A$.

A map of topological spectra $u:D\to E$ is a family of based maps $u_j:D_j\to E_j$ satisfyingThe indexed version requires the same compatibility for every inclusion of indexing subspaces. These are point-set maps, before passing to spectrum homotopy or the stable homotopy category.

An Omega-spectrum is a topological spectrum whose adjoint structure mapsare weak homotopy equivalences. Consequently ${\pi }_{k}E={\pi }_{k+j}{E}_j$ whenever $k+j\ge 0$, using the structure identifications. In the genuine indexed point-set convention, the word spectrum already includes the stronger condition that $E(V)\to {\Omega }^{W\ominus V}E(W)$ is a homeomorphism; a general system without that condition is called an indexed prespectrum. Its associated spectrum is obtained by spectrification. The sequential and indexed conventions present the same stable theory.

A cell spectrum is built from the zero topological spectrum by successive stable cell attachments. Concretely, attach a free spectrum on the inclusion $S^{m-1}\hookrightarrow D^m$ at some level $j$ by a pushout along its boundary map; its quotient is the stable cell sphere ${\Sigma }^{m-j}\mathbb{S}$. Iterating such attachments and taking the colimit gives a cell spectrum, with successive quotients wedges of integer suspensions of spectra of the sphere spectrum. Negative cell degrees are allowed by choosing $j>m$. In the genuine indexed model apply spectrification to this cellular prespectrum. The compatibility of the structure maps is part of both the objects and their maps.