The tensor trick #
A special contraction of (M, d) onto (N, d') induces a special contraction of the reduced
tensor coalgebras Tᶜ(M) = ⨁_{n ≥ 1} M^{⊗ n} onto Tᶜ(N). The endomorphisms on words are the
letterwise extensions of d and d', the degree-one graded coderivations
ReducedTensorWords.gradedCoderiv G (d ∘ letter) 1 whose only Taylor component is d on single
letters; on a word they apply d to one letter at a time, with the Koszul sign of the letters it
passes. When d and d' square to zero, these extensions are differentials. The inclusion and
projection act letterwise, and the homotopy is
H = ∑_j τ^{⊗ j} ⊗ h ⊗ (i p)^{⊗ (n - j - 1)}
on words of length n, where τ = InternalGrading.koszulTwist G 1 is the Koszul sign of moving
the odd map h past a letter. Cross terms of d H + H d cancel because d and h are odd and
i p commutes with d and with τ, while the diagonal terms telescope to 1 - (i p)^{⊗ n}.
The side conditions of the letters give those of the words.
This is the input of homological transfer along a contraction of an A∞ algebra onto a
retract such as its cohomology: the bar differential of the algebra is the letterwise extension
of its unary operation plus a perturbation which shortens words, so the perturbation lemma
applies to the contraction of bar constructions produced here.
Main definitions #
TauCeti.LinearSpecialContraction.reducedTensorWordsHomotopy: the homotopyHon reduced tensor words.TauCeti.LinearSpecialContraction.reducedTensorWords: the induced special contraction of the reduced tensor coalgebras.
Main results #
TauCeti.LinearSpecialContraction.reducedTensorWordsHomotopy_of: the value ofHon an arbitrary homogeneous tensor word.TauCeti.LinearSpecialContraction.reducedTensorWordsHomotopy_of_tprod: the value ofHon a pure tensor word.TauCeti.LinearSpecialContraction.deconcatenation_comp_reducedTensorWordsHomotopy:His a coderivation homotopy,Δ H = (H ⊗ i p + τ ⊗ H) Δ.TauCeti.LinearSpecialContraction.map_koszulTwist_comp_reducedTensorWordsHomotopy:Hanticommutes with the letterwise Koszul twist.
References #
- V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential homological algebra II, Illinois Journal of Mathematics 35 (1991), 357--373.
- J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Mathematische Zeitschrift 207 (1991), 245--280.
Families of letter maps acting around one slot #
The contraction identity on words of one length #
Operators on reduced tensor words acting length by length #
The tensor-trick homotopy on reduced tensor words: on words of length n it is
∑_j τ^{⊗ j} ⊗ h ⊗ (i p)^{⊗ (n - j - 1)}, where τ is the Koszul twist of parameter one for
the grading G.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The tensor-trick homotopy on an arbitrary word of length n, expanded as the sum of its
single-slot actions.
The tensor-trick homotopy on a pure tensor word: the sum over positions j of the word with
the letters before j Koszul-twisted, h applied at j, and i p applied after j.
On a single letter the tensor-trick homotopy is the homotopy of the contraction.
On a two-letter word the tensor-trick homotopy is h ⊗ i p + τ ⊗ h, where τ is the Koszul
twist of parameter one.
The tensor-trick homotopy lowers the total degree of words by one when the homotopy of the
contraction has degree -1 and its inclusion and projection have degree zero.
The tensor-trick homotopy and deconcatenation #
The tensor-trick homotopy is a coderivation homotopy. Cutting H z either cuts to the
right of the letter carrying h, where the letters already carry i p, or cuts to its left,
where the letters passed by h carry the Koszul twist:
Δ H = (H ⊗ (i p)) Δ + (τ ⊗ H) Δ.
The tensor-trick homotopy anticommutes with the letterwise Koszul twist: it moves one odd map
h past the letters it acts on.
The tensor trick. A special contraction of (M, dM) onto (N, dN) by graded maps of the
expected degrees induces a special contraction of the reduced tensor coalgebras, whose
endomorphisms are the letterwise extensions of dM and dN with the Koszul signs of G and H.
When dM and dN square to zero, these extensions are differentials. The inclusion and
projection act letterwise, and the homotopy is reducedTensorWordsHomotopy.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inclusion of the tensor-trick contraction is the letterwise inclusion.
The projection of the tensor-trick contraction is the letterwise projection.
The homotopy of the tensor-trick contraction is reducedTensorWordsHomotopy.