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Stream: learning: questions

Topic: Cubes as chain complexes

Amar Hadzihasanovic (Jan 17 2023 at 08:21):

Take your favourite [[category of cubes]].
Unless it's a really strange one, it should admit a functor to the category of augmented chain complexes:

• You send the 1-cube to the complex with $C_1 = \mathbb{Z}$, $C_0 = \mathbb{Z} \oplus \mathbb{Z}$, where $\partial_0$ sends $1$ to $(1, -1)$, and the augmentation sends both generators of $\mathbb{Z} \oplus \mathbb{Z}$ to $1$.
• You send the $n$-cube to the $n$-fold tensor product of this complex.
• Unless it's a really strange category of cubes, there should be only one sensible way of extending to morphisms.

Furthermore, this functor should be faithful.
What I'd like to know is: Are there known characterisations of categories of cubes as subcategories of augmented chain complexes?

More specifically, I'd be interested in this:
The cubes-as-chain-complexes are free, and can be equipped with a “canonical” choice of generating basis.
I would like to characterise the category of cubes with connections as the full subcategory on those morphisms that send these basis elements either to basis elements or to 0.