Complexes of topological representations #
A chain map of complexes of monoid representations preserves differential kernels on invariant vectors. This fact applies to arbitrary complex shapes, independently of any chosen resolution or cohomology theory.
theorem
TopRep.d_mapInvariants_f_apply
{k : Type u}
[Ring k]
[TopologicalSpace k]
{G : Type v}
[Monoid G]
{ι : Type u_1}
{s : ComplexShape ι}
{J J' : HomologicalComplex (TopRep k G) s}
(f : J ⟶ J')
{i j : ι}
(z : ↥(J.X i).ρ.invariants)
(hz : (Hom.hom (J.d i j)) ↑z = 0)
:
A chain map carries an invariant element killed by a differential to an invariant element killed by the corresponding differential.