Structure sheaves of definable additive categories

Prest, Mike and Rajani, Ravi (2008) Structure sheaves of definable additive categories. [MIMS Preprint]

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Abstract

2-equivalences are described between the category of small abelian categories with exact functors, the category of definable additive categories with functors which commute with products and direct limits and the category of locally coherent Grothendieck categories with "coherent" morphisms. There is a comparison, for definable additive categories, between the presheaf of finite-type localisations and the presheaf of localisations of associated functor categories. The image of the free abelian category in Mod-R is described and related to special bases of the Ziegler and rep-Zariski spectra restricted to the set of indecomposable injectives. In the coherent case there is a particularly nice form (which is essentially elimination of imaginaries in the model-theoretic sense).

Item Type:	MIMS Preprint
Uncontrolled Keywords:	definable category, abelian category, functor category, locally coherent category, 2-category, exact functor, finite-type localisation, pure-injective, injective, free abelian category, Gabriel-Zariski spectrum, rep-Zariski spectrum, Ziegler spectrum, presheaf, elimination of imaginaries, pp formula
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations MSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebras MSC 2010, the AMS's Mathematics Subject Classification > 18 Category theory; homological algebra
Depositing User:	Professor Mike Prest
Date Deposited:	20 Oct 2009
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1339

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• Structure sheaves of definable additive categories. (deposited 21 Nov 2008)
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