Categories: Between Cubes and Globes. Sketch I

  • Y. Bespalov Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: category theory

Abstract

For a finite partially ordered set I, we define an abstract polytope PI which is a cube or a globe in the cases of discrete or linear poset, respectively. For a poset P, we have built a small category ♦P with finite lower subsets in P as objects. This category ♦P = ♦P+P- is factorized into a product of two wide subcategories ♦P+ of faces and ♦P- of degenerations. One can imagine a degeneration from I to J ⊂ I as a projection of an abstract polytope PI to the subspace spanned by J. Morphisms in ♦P+ with fixed target I are identified with faces of PI . The composition in ♦P admits the natural geometric interpretation. On the category ♦I of presheaves on ♦I , we construct a monad of free category in two steps: for a terminal presheaf, the free category is obtained via a generalized nerve construction; in the general case, the cells of a nerve are colored by elements of the initial presheaf. Strict P-fold categories are defined as algebras over this monad. All constructions are functorial in P. The usual theory of globular and cubical higher categories can be translated in a natural way into our general context.

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Published
2019-12-09
How to Cite
Bespalov, Y. (2019). Categories: Between Cubes and Globes. Sketch I. Ukrainian Journal of Physics, 64(12), 1125. https://doi.org/10.15407/ujpe64.12.1125
Section
Fields and elementary particles