Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins
In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of...
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MDPI AG
2022-02-01
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| author | José Manuel García-Calcines Luis Javier Hernández-Paricio María Teresa Rivas-Rodríguez |
| author_facet | José Manuel García-Calcines Luis Javier Hernández-Paricio María Teresa Rivas-Rodríguez |
| author_sort | José Manuel García-Calcines |
| collection | DOAJ |
| description | In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set <i>X</i> its cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></semantics></math></inline-formula> induces the sequential cardinal that associates to each augmented semi-simplicial finite set <i>X</i> an augmented sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula> of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets. |
| first_indexed | 2024-03-09T21:30:58Z |
| format | Article |
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| language | English |
| last_indexed | 2024-03-09T21:30:58Z |
| publishDate | 2022-02-01 |
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| series | Mathematics |
| spelling | doaj.art-e39d1746f41b43db90857c5a1ac0d7cf2023-11-23T20:57:06ZengMDPI AGMathematics2227-73902022-02-0110459010.3390/math10040590Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and JoinsJosé Manuel García-Calcines0Luis Javier Hernández-Paricio1María Teresa Rivas-Rodríguez2Department de Matemáticas, Estadística e I.O., Facultad de Ciencias (Sección de Matemáticas), Campus de Anchieta, University of La Laguna, 38200 La Laguna, SpainDepartment de Matemáticas y Computación, Facultad de Ciencia y Tecnología, University of La Rioja, 26006 Logroño, SpainDepartment de Matemáticas y Computación, Facultad de Ciencia y Tecnología, University of La Rioja, 26006 Logroño, SpainIn this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set <i>X</i> its cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow></semantics></math></inline-formula> induces the sequential cardinal that associates to each augmented semi-simplicial finite set <i>X</i> an augmented sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula> of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets.https://www.mdpi.com/2227-7390/10/4/590augmented semi-simplicial setaugmented integer sequencemonoidal categorysimplicial combinatorics |
| spellingShingle | José Manuel García-Calcines Luis Javier Hernández-Paricio María Teresa Rivas-Rodríguez Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins Mathematics augmented semi-simplicial set augmented integer sequence monoidal category simplicial combinatorics |
| title | Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins |
| title_full | Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins |
| title_fullStr | Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins |
| title_full_unstemmed | Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins |
| title_short | Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins |
| title_sort | augmented simplicial combinatorics through category theory cones suspensions and joins |
| topic | augmented semi-simplicial set augmented integer sequence monoidal category simplicial combinatorics |
| url | https://www.mdpi.com/2227-7390/10/4/590 |
| work_keys_str_mv | AT josemanuelgarciacalcines augmentedsimplicialcombinatoricsthroughcategorytheoryconessuspensionsandjoins AT luisjavierhernandezparicio augmentedsimplicialcombinatoricsthroughcategorytheoryconessuspensionsandjoins AT mariateresarivasrodriguez augmentedsimplicialcombinatoricsthroughcategorytheoryconessuspensionsandjoins |