A class of exactness properties characterized via left Kan extensions |
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| Authors: | Pierre-Alain Jacqmin |
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| Affiliation: | 1. Department of Mathematics, University of Oregon, Eugene, OR, USA;2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada;3. Department of Pure Mathematics, University of Waterloo & Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada;1. Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan;2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel;3. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany;1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China;2. Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA;3. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China;1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic and Department of Theoretical Computer Science, Technical University Braunschweig, Germany;2. Department of Mathematics and Statistics, Masaryk University, Faculty of Sciences, Kotlá?ská 2, 611 37 Brno, Czech Republic;1. School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia;2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China |
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| Abstract: | ![]()
We consider a general class of exactness properties on a finitely complete category, all of which can be expressed as the condition that a certain morphism in a diagram is a strong epimorphism. For each such exactness property, we characterize finitely bicomplete categories having the property by restricting the condition to those diagrams built from only one object in the category via a left Kan extension. In the regular context, this generalizes the theory of approximate co-operations introduced by D. Bourn and Z. Janelidze. As an application, we deduce from this a characterization of (essentially) algebraic categories satisfying such a given exactness property. The pointed version of these exactness properties is also studied. |
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| Keywords: | Exactness property Kan extension Strong epimorphism Approximate operation Essentially algebraic category Mal'tsev condition |
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