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Stream: learning: questions

Topic: Geometric Product of Cubical Sets

Ruby Khondaker (she/her) (Jul 10 2025 at 09:34):

I've been looking a bit into the tensor product of cubical sets induced by the monoidal structure on the category of cubes with connection, gotten via Day convolution. I think I roughly have an idea of what these correspond to (take a k-cell and a n-cell and "multiply" them together the way you would if they were topological spaces), but I'm interested how these interact with "pasting diagrams" of cubes

Ruby Khondaker (she/her) (Jul 10 2025 at 09:35):

As far as I can tell, the right notion of "pasting diagram" for a cube is just a cuboid - so for 1-cells it's a line, for 2-cells it's a rectangle, for 3-cells a cuboid, etc etc

Ruby Khondaker (she/her) (Jul 10 2025 at 09:36):

taking the "collection of all pasting diagrams on a cubical set" i think is what the "free strict cubical omega-category monad" does? and then strict cubical omega-categories are just algebras for this - you can compose any cuboid to a cube

Ruby Khondaker (she/her) (Jul 10 2025 at 09:37):

i'm trying to figure out how this monad $T$ interacts with the geometric product - in particular getting a map from $TX \otimes TY \to T(X \otimes Y)$

Ruby Khondaker (she/her) (Jul 10 2025 at 09:38):

my geometric intuition tells me that this is just "a product of cuboids may be regarded as a cuboid of products", but i can't quite tell whether this actually works or if i've missed something subtle

Ruby Khondaker (she/her) (Jul 10 2025 at 09:39):

e.g. if you have a string of $5$ 1-cubes from $X$ and $7$ 1-cubes from $Y$, then the corresponding element in $TX \otimes TY$ is a formal product of these strings - but you should be able to view it as a $5 \times 7$ rectangle of individual products of 1-cubes from $X$ and $Y$, which i think would be an element of $T(X \otimes Y)$?