Let ({mathcal {F}}) be a family of self-mappings defined on K such that any map in (mathcal {F}) is a strongly ρ-continuous R-map.
Let ({mathcal {F}}) be a family of commuting self-mappings defined on K such that any map in (mathcal {F}) is a strongly ρ-continuous R-map.
vertical maps in (mathcal {C}).
For any relative category (langle mathcal {C},W rangle ), we have the opposite relative category (langle mathcal {C}^o,W^o rangle ), where (W^o) consists of maps (w in W) considered as maps in (mathcal {C}^o).
Note that since [1] is a finite ordered category, every map in (mathsf{Ind},mathcal {C}) can be represented by a filtered system (langle I,g rangle ) of maps in (mathcal {C}), and we can further assume that (I^o) is ordered.
For any relative category (langle mathcal {C},W rangle ), we denote by (widehat{W}) the minimal saturated class of maps in (mathcal {C}) that contains W, and we call (langle mathcal {C},widehat{W} rangle ) the saturation of (langle mathcal {C},W rangle ).
Then by virtue of the isomorphism (12.1), we can represent (tau ^*(f')) by a filtered colimit (mathsf{colim}_{I'}widetilde{f}') of maps in (mathcal {C}^I), and since (widetilde{f}) is compact in (mathsf{Ind},mathcal {C}^{I times [1]}), the map (widetilde{f} rightarrow tau ^*(f')) given by g, (g') factors through (widetilde{f}'(i)) for some (i' in in).
Since Fourier transforms of all maps in (mathcal {E}_{varLambda}) are supported on a circle, we may see a function in any of the (mathcal{E}_{varLambda}) as a complex-valued function on the unit circle; but the G-action depends on Λ.
end{aligned} (5.7 If c is the class (tau ) of all maps in (mathcal {C}), or if v is the class (mathsf{id}) of identity maps, we will drop it from notation, so that (Delta _vmathcal {C}= Delta ^tau _vmathcal {C}) and (Delta mathcal {C}=Delta ^tau _{mathsf{id}}mathcal {C}) (in particular, (Delta I) for a small category I is its simplicial replacement, so that our notation is consistent).
It immediately follows from the definition that any map f in (mathcal {C}) cartesian over I must be also cartesian over (I').
Therefore (i) is equivalent to saying that any map f in (mathcal {C}) cartesian over (I') and such that (pi (f)) is cartesian over I is itself cartesian over I. Let (f c' rightarrow c) be such a map.
