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Erschienen in:
Derived Equivalences Induced by Good Silting Complexes Kapitel lesen Erstes Kapitel lesen

2024 | OriginalPaper | Buchkapitel

Definable Categories and Monoidal Categories

verfasst von : Mike Prest

Erschienen in: Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods

Verlag: Springer Nature Switzerland

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Abstract

Definable categories are axiomatisable additive categories. They appear as definable subcategories of module categories, equivalently as the categories of exact functors on some small abelian category. We give an exposition of their structure and their model theory from an essentially intrinsic point of view. We recall the anti-equivalence between definable categories and small abelian categories and we describe a monoidal version of this due to Wagstaffe.

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Vorheriges Kapitel An Abelian Ambient Category for Behaviors in Algebraic Systems Theory
Nächstes Kapitel Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions
Fußnoten
1
The notion of definable category has been extended beyond the additive case, see [15, 16], but, in this paper, all our categories are additive so we don’t usually say “definable additive category”.
 
2
The category \(\textrm{fun}({\mathcal {D}})\) can be defined as the category of functors (additive, as always in this paper) from \({\mathcal {D}}\) to \(\textbf{Ab}\) which commute with direct products and directed colimits, see Sect. 4. It also has a model-theoretic definition as the category of interpretation functors from \({\mathcal {D}}\) to \(\textbf{Ab}\), that is, functors given by pp-definable sorts, see [23, Chap. 25].
 
3
It sits as the finitely generated projectives within \(({\mathcal {A}}, \textbf{Ab}) = {\mathcal {A}}{\text {-}}\mathrm{{Mod}}\) which is another possible context to use.
 
4
Or arbitrary finitely generated projective objects if we were using \({\mathcal {A}}{\text {-}}\mathrm{{Mod}}\), rather than \(\textrm{Ind}({\mathcal {A}})\), as the context.
 
5
Note that the equivalence relation \(\sim \) might vary with k since the elements \(a^k\) might belong to different sorts.
 
6
The condition “\(T=T^{\aleph _0}\)” which appears in the hypotheses of many results in [20] says that we ignore the exact sizes and look only at which pp-pairs are open and which are closed.
 
7
This terminology refers to their model-theoretic definition which turns out to be equivalent to the stated algebraic preservation properties, see [23, Chap. 2].
 
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Metadaten
Titel
Definable Categories and Monoidal Categories
verfasst von
Mike Prest
Copyright-Jahr
2024
Buch
Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods

Print ISBN: 978-3-031-53062-3

Electronic ISBN: 978-3-031-53063-0

Copyright-Jahr: 2024

DOI
https://doi.org/10.1007/978-3-031-53063-0_6