Категорії: між кубічними та сферичними
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...
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| Дата: | 2019 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Publishing house "Academperiodika"
2019
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| Онлайн доступ: | https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019525 |
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| Назва журналу: | Ukrainian Journal of Physics |
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Ukrainian Journal of Physics| id |
ujp2-article-2019525 |
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ujp2-article-20195252019-12-20T15:19:59Z Categories: Between Cubes and Globes. Sketch I Категорії: між кубічними та сферичними Bespalov, Y. category theory - 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. Вивчаються багатовимiрнi категорiї, форма клiтин яких залежить вiд частково-впорядкованої множини. ♦ Publishing house "Academperiodika" 2019-12-09 Article Article Original Research Article (peer-reviewed) Оригінальна дослідницька стаття (з незалежним рецензуванням) application/pdf https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019525 10.15407/ujpe64.12.1125 Ukrainian Journal of Physics; Vol. 64 No. 12 (2019); 1125 Український фізичний журнал; Том 64 № 12 (2019); 1125 2071-0194 2071-0186 10.15407/ujpe64.12 en https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019525/1523 Copyright (c) 2019 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine |
| institution |
Ukrainian Journal of Physics |
| collection |
OJS |
| language |
English |
| topic |
category theory - |
| spellingShingle |
category theory - Bespalov, Y. Категорії: між кубічними та сферичними |
| topic_facet |
category theory - |
| format |
Article |
| author |
Bespalov, Y. |
| author_facet |
Bespalov, Y. |
| author_sort |
Bespalov, Y. |
| title |
Категорії: між кубічними та сферичними |
| title_short |
Категорії: між кубічними та сферичними |
| title_full |
Категорії: між кубічними та сферичними |
| title_fullStr |
Категорії: між кубічними та сферичними |
| title_full_unstemmed |
Категорії: між кубічними та сферичними |
| title_sort |
категорії: між кубічними та сферичними |
| title_alt |
Categories: Between Cubes and Globes. Sketch I |
| description |
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. |
| publisher |
Publishing house "Academperiodika" |
| publishDate |
2019 |
| url |
https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019525 |
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