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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.

Let ${\mathrm{Map}}_{\mathrm{sp}}(D,E)$ be the space of point-set maps of topological spectra. Its underlying set is the first set in the comparison. A member is a compatible collection $D_j\to E_j$, rather than independently chosen maps at the levels.

Because the source is cellular and the target is an Omega-spectrum, taking spectrum homotopy classes gives all maps in the stable homotopy category:Thus the first set maps onto the second. Two point-set maps have the same stable class precisely when they are spectrum-homotopic, using the cellular cylinder and fibrant target. The homotopy must commute with the structure maps throughout; unrelated levelwise homotopies are not the definition.

A spectrum homotopy between compatible maps $D\to E$ is a map from the cylinder spectrum $D\wedge [0,1{]}_{+}$ restricting to them at the endpoints. Thus the homotopies at all levels must themselves respect the structure maps.

The stable homotopy category is obtained from topological spectra by inverting stable equivalences of spectra. Its morphisms are abelian groups; for a cellular source and fibrant target they are spectrum homotopy classes of point-set maps of topological spectra.