A construction of certain weak colimits and an exactness property of the 2-category of categories
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...
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2018
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte http://hdl.handle.net/20.500.12110/paper_1201561X_v33_n_p193_Descotte |
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paper:paper_1201561X_v33_n_p193_Descotte2023-06-08T16:10:00Z A construction of certain weak colimits and an exactness property of the 2-category of categories 2-category Exactness property Filtered Weak colimit 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. 2018 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte http://hdl.handle.net/20.500.12110/paper_1201561X_v33_n_p193_Descotte |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
2-category Exactness property Filtered Weak colimit |
| spellingShingle |
2-category Exactness property Filtered Weak colimit A construction of certain weak colimits and an exactness property of the 2-category of categories |
| topic_facet |
2-category Exactness property Filtered Weak colimit |
| description |
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. |
| title |
A construction of certain weak colimits and an exactness property of the 2-category of categories |
| title_short |
A construction of certain weak colimits and an exactness property of the 2-category of categories |
| title_full |
A construction of certain weak colimits and an exactness property of the 2-category of categories |
| title_fullStr |
A construction of certain weak colimits and an exactness property of the 2-category of categories |
| title_full_unstemmed |
A construction of certain weak colimits and an exactness property of the 2-category of categories |
| title_sort |
construction of certain weak colimits and an exactness property of the 2-category of categories |
| publishDate |
2018 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1201561X_v33_n_p193_Descotte http://hdl.handle.net/20.500.12110/paper_1201561X_v33_n_p193_Descotte |
| _version_ |
1768543959021256704 |