simplicialTensor(T, Options => {Degeneracy => false, TopDegree => null})This function computes the simplicial tensor product of a list T, which can consist of simplicial modules or complexes. It caches direct sum indices using tensorwithComponents so that the user can easily access components of the resulting face/degeneracy maps on particular direct summands of the tensor product.
The simplicial tensor product of complexes is built as the Dold-Kan extension of the tensor product functor on the category of R-modules. In general, this complex looks quite different from the standard tensor product of complexes. In fact, the simplicial tensor product is always homotopy equivalent to the classically defined tensor product.
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Here is how to access specific components of the face maps for a tensor product of simplicial modules:
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One reason for using the simplicial tensor product is that its components as a simplicial module are more canonically built, and thus naturally extend functorial maps on the category of R-modules to the category of chain complexes.
We can see actually see this in an example. The exterior power functor admits a canonical comultiplication map $$\bigwedge^{i+j} \to \bigwedge^i \otimes \bigwedge^j.$$ This means that for any chain complex $C$ there is a canonical inclusion of complexes $$\bigwedge^{i+j} C \hookrightarrow \bigwedge^i C \otimes \bigwedge^j C.$$ Constructing this inclusion directly using the naive definition of the exterior power functor on complexes would be extremely unnatural, but filtering through the simplicial category makes this a very easy task:
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Just for sake of illustration, let us see how the above example changes in the modular setting; note that the only characteristic for which the inclusion $$\bigwedge^3 \hookrightarrow \bigwedge^2 \otimes \bigwedge^1$$ is not canonically split is for characteristic 3, so we should expect the Dold-Kan extension of the Schur functor $\mathbb{S}^{(2,1)} (K)$ to look different in characteristic 3. (the resulting minimization of $\mathbb{S}^{(2,1)} (K)$ should not have coefficients, for instance).
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Note that the above method of computing the Schur functor $\mathbb{S}^{(2,1)} (K)$ is significantly faster than the schurMap command.
This method is also implemented functorially, so it can be applied to simplicial maps and complex maps.
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The user should remember to prune the output upon normalizing a simplicial tensor product.
The object simplicialTensor is a method function with options.
The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:3737:0.