Homotopy theory of bicomplexes

Abstract

We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are detected by the E 2 -term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.

Model category structures and spectral sequences

“…The homotopy theory of bicomplexes has recently been studied by Muro and Roitzheim in [17], by considering the total weak equivalences as well as the equivalences given after taking horizontal and vertical cohomology. This second class of equivalences corresponds to E 1 in our setting.…”

Derived universal Massey products

“…The source is the vertical homology of Ã, which is a minimal bicomplex O-algebra by construction. Moreover, it is k-projective by [16,Proposition 2.4] and [21,Theorem 4.1]. Hence ρ ′ is a horizontal resolution.…”

“…An E 2 -equivalence is a morphism which induces an isomorphism between the E 2 pages of the spectral sequences associated to the increasing filtration by the horizontal degree (the first spectral sequence in the sense of [17,Theorem 2.15]) of the source and target bicomplexes. The category bCh of bicomplexes is closed symmetric monoidal with the usual tensor product of complexes and the obvious assignation of horizontal and vertical degrees, compare [21,Definition 2.7]. The symmetry constraint uses the Koszul sign rule with respect to the total degree.…”

“…If ρ is another horizontal resolution then ρ is a trivial fibration in the Cartan-Eilenberg semi-model structure by [21,Theorem 4.1] and Remark 2.3. Therefore, there is a lift…”

Model category structures on multicomplexes

“…The words listed in (3) above form a basis for the ∞-disk; see [11,Definition 5.4] for an explicit description of the ∞-disk for multicomplexes concentrated in the right half-plane. For n finite, the words listed in (3) are not necessarily distinct or nonzero, so do not form a basis for the n-disk.…”