Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups

If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z is Map∗(EK+, Z) K , the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X is often given by (HG,X), where HG,X is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum HG,X is just “a profinite version” of Map∗(EK+, Z): at each stage of its construction, HG,X replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map∗(EK+, Z) (up to a certain natural identification).