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Stream: learning: questions

Topic: Tessellation and category theory

Asad Saeeduddin (Sep 09 2020 at 05:35):

Is there some way to formalize the concept of "interlockable surfaces" using category theory? Like, a bunch of stackable blocks where arbitrary blocks stack smoothly is a monoid, more sophisticated surfaces where only certain blocks fit with others is a category, "two dimensional" tiling puzzles like jigsaws are product categories, the ability to "laterally" compose the surfaces of blocks corresponds to a monoidal structure, etc.

Cole Comfort (Sep 09 2020 at 06:09):

Categories where you can tile squares are called double categories.

Nathaniel Virgo (Sep 09 2020 at 06:21):

In 1D it's easy - let the objects be the set of different edge patterns that a tile can have. Then we can define a tile as a morphism $A\xrightarrow{f} B$ with the edge pattern $B$ on its right edge, and the inverse of the pattern $A$ on its left edge. Then if we have tiles $A\xrightarrow{f} B$ and $B\xrightarrow{g} C$ we can stack them together to get $A\xrightarrow{f;g} C$, which is a stack of two tiles with the inverse of pattern $A$ on its left and pattern $C$ on its right. You want to be able to form all possible such stacks recursively, which means you're looking at the free category generated by your set of tiles. Identity morphisms are "empty stacks."

In 2D I guess it's probably a free double category. It's not a 2-category or a monoidal category as you might think, because those allow you to do the kinds of "elevator moves" that you can do in string diagrams, which correspond to bending and stretching the tiles, and moving edges past each other, in a way that's not very consistent with them being tiles.

Asad Saeeduddin (Sep 15 2020 at 04:10):

@Nathaniel Virgo Thanks! That gives me a bunch of stuff to chew on. Unfortunately I don't understand double categories yet, but I've just started reading about internal categories and hopefully i can move from there to double categories