@article{Bespalov_2019, title={Categories: Between Cubes and Globes. Sketch I}, volume={64}, url={https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019525}, DOI={10.15407/ujpe64.12.1125}, abstractNote={<p>For a finite partially ordered set I, we define an abstract polytope P<sub>I</sub> 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 ♦<sub>P</sub> with finite lower subsets in P as objects. This category ♦<sub>P</sub> = ♦<sub>P</sub><sup>+</sup>♦<sub>P</sub><sup>-</sup>&nbsp;is factorized into a product of two wide subcategories ♦<sub>P</sub><sup>+</sup>&nbsp;of faces and ♦<sub>P</sub><sup>-</sup>&nbsp;of degenerations. One can imagine a degeneration from I to J ⊂ I as a projection of an abstract polytope P<sub>I</sub> to the subspace spanned by J. Morphisms in ♦<sub>P</sub><sup>+</sup>&nbsp;with fixed target I are identified with faces of P<sub>I</sub> . The composition in ♦<sub>P</sub> admits the natural geometric interpretation. On the category&nbsp;♦<sub>I</sub> of presheaves on ♦<sub>I</sub> , 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.</p>}, number={12}, journal={Ukrainian Journal of Physics}, author={Bespalov, Y.}, year={2019}, month={Dec.}, pages={1125} }