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Stream: theory: category theory

Topic: commutative diagrams of 2-cells

Matteo Capucci (he/him) (May 21 2022 at 11:15):

I'm trying to understand the definition of the intercategory of double span in a double category, as described here.
It's all fine until it says that cubical cells are 'commutative diagrams of 2-cells', such as this:
image.png

Matteo Capucci (he/him) (May 21 2022 at 11:16):

This should be a diagram in a double category $\mathbb A$. Arrows with a dot are vertical and arrows without a dot are horizontal. Squares are 2-cells.

Matteo Capucci (he/him) (May 21 2022 at 11:17):

How am I suppose to compose the 2-cells?

Amar Hadzihasanovic (May 21 2022 at 12:47):

$\alpha_0 ; \phi_0 = \phi_2 ; \beta_0$ and $\alpha_1 ; \phi_1 = \phi_2 ; \beta_1$, where $;$ is horizontal composition?

Amar Hadzihasanovic (May 21 2022 at 12:48):

Those are the only possible composites in that diagram.

Amar Hadzihasanovic (May 21 2022 at 12:48):

The two “cubes” have no top and bottom faces (it took me a moment to figure that out).

Matteo Capucci (he/him) (May 22 2022 at 16:05):

Oooooh, I see

Matteo Capucci (he/him) (May 22 2022 at 16:05):

I thought all the faces were filled