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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 26th 2012
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThe category of strict ω-categories admits two biclosed monoidal products, namely the lax tensor product and the cartesian product. I have encountered a few instances of cartesian products 'mixed in' with lax tensor products. The first thing that we might ask is whether or not they 'associate' with one another, namely, we might ask if $X\otimes (Y \times Z)\cong (X\otimes Y) \times Z$ The identity functor on ω-Cat is an (op?)lax monoidal functor from the lax tensor product monoidal structure to the cartesian monoidal structure when we equip it with the natural transformation $\mu_{X,Y}:X\otimes Y \to X\times Y$ induced by projecting the tensor product onto each factor. However, this transformation doesn't appear to have a well-behaved action in the following sense: Let's assume that the tensor product is strictly associative, which we can assume by one of the coherence theorems. Then $\mu_{X\otimes Y,Z}$ does not appear to be equal to $X\otimes \mu_{Y,Z}$. The reason why this is the case is just another Eckmann-Hilton sort of argument, namely because this would imply that $X\otimes Y = X\otimes (\ast \times Y) = (X\otimes \ast)\times Y = X\times Y$, which is something we know to be false. Even though we can't say something this strong, is it possible to say _something_ about how these products behave when they appear together?

   The category of strict ω-categories admits two biclosed monoidal products, namely the lax tensor product and the cartesian product. I have encountered a few instances of cartesian products ’mixed in’ with lax tensor products.

   The first thing that we might ask is whether or not they ’associate’ with one another, namely, we might ask if

   $X\otimes (Y \times Z)\cong (X\otimes Y) \times Z$

   The identity functor on ω-Cat is an (op?)lax monoidal functor from the lax tensor product monoidal structure to the cartesian monoidal structure when we equip it with the natural transformation $\mu_{X,Y}:X\otimes Y \to X\times Y$ induced by projecting the tensor product onto each factor. However, this transformation doesn’t appear to have a well-behaved action in the following sense:

   Let’s assume that the tensor product is strictly associative, which we can assume by one of the coherence theorems. Then $\mu_{X\otimes Y,Z}$ does not appear to be equal to $X\otimes \mu_{Y,Z}$. The reason why this is the case is just another Eckmann-Hilton sort of argument, namely because this would imply that

   $X\otimes Y = X\otimes (\ast \times Y) = (X\otimes \ast)\times Y = X\times Y$, which is something we know to be false.

   Even though we can’t say something this strong, is it possible to say something about how these products behave when they appear together?