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An indexed prespectrum over $U$ consists of based spaces $E(V)$ indexed by finite-dimensional $V\subset U$, with unital associative maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for inclusions $V\subset W$. The orthogonal differences make iterated suspension coordinates compose consistently.

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.

Write $f_x$ for the linear isometry specified at $x$. Choose a cofinal sequence $V_i$ from the indexing family $I$, and write the target part of the supplied subordinate flag for a family of linear isometries as $W_i$. After a cofinal enlargement it hasBoth flags exhaust their universe for spectra. Compactness of the parameter space ensures that a finite target stage can contain all images of each fixed source stage.

Let ${\xi }_i$ be the vector bundle on $X$ with fiberHere $\ominus$ is the orthogonal complement in the indicated ambient subspace. Its rank is $\dim W_i-\dim V_i$. Let $\mathrm{Th}({\xi }_i)$ be its Thom space, the disk bundle modulo its sphere bundle, so that every fiber is compactified and all its points at infinity become the single basepoint. For the zero bundle this convention gives $\mathrm{Th}(0_X)=X_{+}$. Define a target indexed prespectrum on the flag byThe smash product here is of based spaces, not yet the derived smash product of spectra.

To specify all structure maps, for $i\le j$ use the continuous fiberwise orthogonal isomorphismBoth sides identify with the complement of $f_x(V_i)$ in $W_j$. In particular this is the isometry obtained by orthogonally projecting onto $f_x(V_j\ominus V_i)$ and its complement; it is not an assertion that the displayed summands coincide separately. Pass to fiberwise one-point compactifications, identify the last summand with $V_j\ominus V_i$ using $f_x^{-1}$, and apply the source structure map. This definesMore explicitly, decompose $z+w\in W_j\ominus f_x(V_i)$, where $z\in W_j\ominus W_i$ and $w\in ({\xi }_i{)}_x$, as $w^{\prime }+f_x(v)$ with $w^{\prime }\in ({\xi }_j{)}_x$ and $v\in V_j\ominus V_i$. Send its pair with $e\in E(V_i)$ to the pair $(w^{\prime },{\sigma }_{V_i,V_j}(v,e))$ over $x$. The notation for finite vectors extends to the compactifications, sending infinities and basepoints to the basepoint. Orthogonal decomposition and associativity of the source structure maps prove the prespectrum identities.

Extend the indexed prespectrum from the cofinal flag to the target indexing family and apply spectrification $L$. The point-set definition isOn a map of topological spectra $u:E\to E^{\prime }$ it is induced at stage $i$ by $1_{\mathrm{Th}({\xi }_i)}\wedge u(V_i)$. Thus the prescribed isometry family, indexing family and flag have all been used. A common cofinal refinement identifies the resulting objects in the stable homotopy category; the point-set model retains the chosen flag. If $f$ is a constant standard inclusion, the complement bundles are trivial and this recovers the usual reindexed $X_{+}\wedge E$.

In the genuine indexed point-set model, spectrification $L$ is the left adjoint to the forgetful functor from indexed spectra to indexed prespectra. An indexed spectrum has adjoint structure maps $E(V)\to {\Omega }^{W\ominus V}E(W)$ that are homeomorphisms. Applying $L$ to a cellular indexed prespectrum supplies the associated genuine topological spectrum.