Artículo
Abstract:
Given a 2-category A, a 2-functor A (Formula Presented) Cat and a distinguished 1-subcategory Σ ⊂ A containing all the objects, a σ-cone for F (with respect to Σ) is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit is the universal (up to isomorphism) σ-cone. The notion of σ-limit generalizes the well known notions of pseudo and lax limit. We consider the fundamental notion of σ-filtered pair (A, Σ) which generalizes the notion of 2-filtered 2-category. We give an explicit construction of σ-filtered σ-colimits of categories, a construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that σ-filtered σ-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a σ-filtered σ-colimit of exact category valued 2-functors is exact. This corollary is essential in the 2-dimensional theory of flat and pro-representable 2-functors, that we develop elsewhere. © Descotte M.E., Dubuc E.J., Szyld M., 2018.
Registro:
| Documento: | Artículo |
| Título: | A construction of certain weak colimits and an exactness property of the 2-category of categories |
| Autor: | Descotte, M.E.; Dubuc, E.J.; Szyld, M. |
| Filiación: | LaBRI, Université de Bordeaux, France IMAS, Universidad de Buenos Aires, Argentina |
| Palabras clave: | 2-category; Exactness property; Filtered; Weak colimit |
| Año: | 2018 |
| Volumen: | 33 |
| Página de inicio: | 193 |
| Página de fin: | 215 |
| Título revista: | Theory and Applications of Categories |
| Título revista abreviado: | Theory Appl. Categories |
| ISSN: | 1201561X |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte |
Referencias:
- Artin, M., Grothendieck, A., Verdier, J., SGA 4, Ch VII, (1963-64) (1972) Springer Lecture Notes in Mathematics, p. 270
- Canevali, N., (2016) 2-Filtered Bicolimits and Finite Weighted Bilimits Commute in Cat, , http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/, degree thesis, 2016
- Data, M.I., (2014) Una construcción De bicolímites 2-Filtrantes De categorías, , http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/2014, degree thesis
- Descotte, M.E., Dubuc, E.J., Szyld, M., (2017) On the Notion of Flat 2-Functors
- Dubuc, E.J., Street, R., A construction of 2-filtered bicolimits of categories (2006) Cahiers De Topologie Et Géométrie Différentielle Catégoriques, 47 (2), pp. 83-106
- Gray, J.W., Formal category theory: Adjointness for 2-categories (1974) Springer Lecture Notes in Mathematics, p. 391
- Kelly, G.M., Basic concepts of enriched category theory (1982) London Mathematical Society Lecture Note Series, 64. , Cambridge Univ. Press, New York
- Kelly, G.M., Elementary observations on 2-Categorical limits (1989) Bull. Austral. Math. Soc, 39, pp. 301-317
- Kelly, G.M., Street, R., Review of the elements of 2-categories (1974) Springer Lecture Notes in Mathematics, 420, pp. 75-103
- Kennison, J., The fundamental localic groupoid of a topos (1992) Journal of Pure and Applied Algebra, 77, pp. 67-86
- Pronk, D.A., Etendues and stacks as bicategories of fractions (1996) Compositio Mathematica, 102 (3), pp. 243-303
- Street, R., Fibrations in bicategories (1980) Cahiers De Topologie Et géométrie différentielle catégoriques, 21 (2), pp. 111-160
- Street, R., Correction to Fibrations in bicategories (1987) Cahiers De Topologie Et géométrie différentielle catégoriques, 28 (1), pp. 53-56
Citas:
---------- APA ----------
Descotte, M.E., Dubuc, E.J. & Szyld, M. (2018) . A construction of certain weak colimits and an exactness property of the 2-category of categories. Theory and Applications of Categories, 33, 193-215.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte [ ]
---------- CHICAGO ----------
Descotte, M.E., Dubuc, E.J., Szyld, M. "A construction of certain weak colimits and an exactness property of the 2-category of categories" . Theory and Applications of Categories 33 (2018) : 193-215.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte [ ]
---------- MLA ----------
Descotte, M.E., Dubuc, E.J., Szyld, M. "A construction of certain weak colimits and an exactness property of the 2-category of categories" . Theory and Applications of Categories, vol. 33, 2018, pp. 193-215.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte [ ]
---------- VANCOUVER ----------
Descotte, M.E., Dubuc, E.J., Szyld, M. A construction of certain weak colimits and an exactness property of the 2-category of categories. Theory Appl. Categories. 2018;33:193-215.Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte [ ]
