# Cubical Structures for Higher-Dimensional Type Theories

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1 Cubicl Structures for Higher-Dimensionl Type Theories Ed Morehouse October 30, / 36

2 A higher-dimensionl type theory depends on notion of higher-dimensionl bstrct spces. Mny choices: globulr, simplicil, cubicl, opotopic, etc. We wnt bstrct spces with good topologicl properties s well s good combintoril nd computtionl properties. Ltely, we hve been thinking bout cubicl structure. 2 / 36

3 The Cubicl Perspective Severl cubicl structures hve been proposed s bsis for models of higher-dimensionl type theory. We survey some of their fetures. 3 / 36

4 Abstrct Cubes A cube ctegory is symmetric monoidl ctegory with distinguished object, the bstrct intervl, I. In cube ctegory,, for ech n N, we hve n bstrct n-dimensionl cube, [n] I I. n 4 / 36

5 0-Dimensionl Cube (point) [0] 5 / 36

6 1-Dimensionl Cube (intervl) [1] 6 / 36

7 2-Dimensionl Cube (squre) [2] 7 / 36

8 3-Dimensionl Cube (cube) [3] 8 / 36

9 n-dimensionl Cube??? [n] 9 / 36

10 Cubiness We seek n equtionl presenttion of cubes so we cn describe cubes of ny dimension nd the reltionships between them. 10 / 36

11 Cubicl Sets A cubicl set is preshef on cube ctegory: b c d f h g A B [0] [1] [2] The cubes we re interested in reside in the fibers, sorted by dimension. Mps between bstrct cubes determine contrvrint functions describing reltionships between cubes. 11 / 36

12 Boundry Mps An bstrct intervl hs two distinguishble boundry points. This gives us notion of pth., + ([0] [1]) f b b [0] [1] 12 / 36

13 Boundry Mps An bstrct intervl hs two distinguishble boundry points. This gives us notion of pth., + ([0] [1]) f b b [0] [1] + 13 / 36

14 Degenercies Represent the ide of trivil pth: ε ([1] [0]) [1] ε [0] 14 / 36

15 Boundry-Degenercy Lws i ε = id([0]) ε [0] [1] [0] i id 15 / 36

16 (, ε) genertors, +, ε reltions ε = = + ε 16 / 36

17 Digonl Mps Represent the ide of pth cutting through the middle of squre: ([1] [2]) A d b A c d [1] [2] 17 / 36

18 (, ) So fr, the digonl is under-specified: we don t sy how to cut through the middle of squre. But there is still something tht we know for certin: its boundry. genertor reltions i = i i 18 / 36

19 Symmetricl Digonls If the digonl cuts through the squre in stright line then we get more lws: digonl-digonl lw ( [1]) = ([1] ) represents cutting through the middle of 3-cube. 19 / 36

20 Symmetricl Digonls If the digonl cuts through the squre in stright line then we get more lws: digonl-digonl lw ( [1]) = ([1] ) represents cutting through the middle of 3-cube. 20 / 36

21 Symmetricl Digonls If the digonl cuts through the squre in stright line then we get more lws: digonl-digonl lw ( [1]) = ([1] ) represents cutting through the middle of 3-cube. 21 / 36

22 Symmetricl Digonls Also, putting the intervl in the digonl of the squre nd then squishing the squre bck into the intervl long either dimension is identity: digonl-degenercy lws (ε [1]) = id([1]) = ([1] ε) 22 / 36

23 (ε, ) You my recognize these s the comonoid lws: reltions = ε = = ε If we extend this comonoid structure nturlly to ll [n], then the monoidl structure becomes crtesin. 23 / 36

24 Crtesin Cubicl Sets Crtesin cubicl sets hve severl good properties, eg: It is strict test ctegory (hs the right homotopy theory ). Contexts of dimension vribles behve structurlly (dmit exchnge, wekening nd contrction). 24 / 36

25 Reversls Represent the ide of following pth bckwrds: ρ ([1] [1]) b!f f b [1] ρ [1] 25 / 36

26 (, ε, ρ) The theory (, ε) plus: genertor ρ reltions ρ ρ =, ρ ε = ε ρ = +, + ρ = 26 / 36

27 Connections Represent nother kind of degenercy, identifying djcent, rther thn opposite, sides of n bstrct cube. They collpse squre to n intervl, like folding pper fn: γ ([2] [1]) f b f f b [2] γ [1] 27 / 36

28 (, ε, γ) The theory (, ε) plus: genertor γ reltions (γ, + ) forms monoid: γ γ = γ γ + = = + γ γ 28 / 36

29 (, ε, γ) The theory (, ε) plus: genertor γ reltions is n bsorbing element (zero) for this monoid: = γ ε = γ 29 / 36

30 (, ε, γ) The theory (, ε) plus: genertor γ reltions ε is morphism for this monoid structure: γ ε = ε ε (plus boundry-degenercy lw from before) 30 / 36

31 Connections nd Reversls Using reversl, we get three more connections, one for folding the squre t ech of its corners: γ ++ (f) f f γ + (f) f b b!f f b b!f γ + (f)!f!f γ (f) 31 / 36

32 Composition Fncier structures, such s cubicl groupoids, extend cubicl sets with composition structure. E.g. f + x g: f g b c E.g. A + x B: b c A B d e f (lws vilble but elided) 32 / 36

33 Subdivision In some cses, we my be ble to subdivide cubes in cnonicl wy. E.g. pdding: f b b = A = 33 / 36

34 Box Filling Cubicl sets with the box-filling property re clled Kn : e f h f A h g g Kn cubicl sets tht re uniform with respect to degenercies re importnt for interpreting higher-dimensionl type theories. 34 / 36

35 Constructive Box Filling With reversls nd connections, we cn constructively fill pdded boxes in cubicl set. A right djoint to subdivision then lets us fill boxes in the fibrnt replcement of cubicl set. 35 / 36

36 Thnks! / 36

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