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A homotopy colimit of spaces is the homotopy-invariant replacement for a colimit of spaces. For a filtered diagram of CW complexes with subcomplex inclusions, the union is a model: the inclusions are cofibrations, and every sphere map and homotopy has image in a finite subcomplex. A suspension spectrum carries this construction to the corresponding homotopy colimit of spectra.

A homotopy colimit of spectra is the derived colimit of a diagram of topological spectra. For a sequential diagram $T_0\to T_1\to \cdots$, it is the cofiber of spectra of $1-s:{\bigvee }_j{T}_j\to {\bigvee }_j{T}_j$, where $s$ uses the bonding maps into the next summands.

The Yoneda lemma, applied to the representing Omega-spectrum spaces in each degree, makes the comparison more explicit. PutThe structure map of $D$ supplies $T_j\to T_{j+1}$, and $D$ is the homotopy colimit of spectra of this sequence. The inverse-tower transition on maps into $E$ is precomposition with $T_j\to T_{j+1}$. Under the suspension spectrum adjunction it takes $[D_{j+1},E_{j+1}{]}_{\ast }$ to $[D_j,E_j{]}_{\ast }$ by taking the adjoint through $D_j\to \Omega D_{j+1}$ and the Omega-spectrum identification $E_j\simeq \Omega E_{j+1}$.

Consequently the maps of represented cohomology theories are exactly the compatible degreewise operations:Only nonnegative degrees need be listed, since the suspension compatibility recovers the negative degrees. The Milnor exact sequence for maps of spectra gives the sharper descriptionThus $\mathrm{HPh}(D,E)\cong {\lim }_j^1[\Sigma D_j,E_j{]}_{\ast }$, with bonding maps coming from the same telescope. For an inverse tower $A_j$ with maps $r_j:A_{j+1}\to A_j$, $\lim A_j$ is the kernel and ${\lim }^1{A}_j$ is the cokernel ofThe first comparison quotients by spectrum homotopy; the second quotients by hyperphantom maps. These are different identifications.