Functors from the infinitary model theory of modules and the Auslander-Gruson-Jensen 2-functor

Abstract

We define the notion of a $\lambda$-definable category, a generalisation of the notion of definable category from the model theory of modules. Let ${\cal C}$ be a $\lambda$-accessible additive category. We characterise the additive functors ${\cal} C\to{\mathrm Ab}$ which preserve $\lambda$-directed colimits and products, by showing that they are the finitely presented functors determined by a morphism between $\lambda$-presented objects (the same result appears, for the case $\lambda=\omega$, in \cite{prest2011}, but we give a proof for any infinite regular cardinal $\lambda$). We remark that \cite{arb} shows that every $\lambda$-definable subcategory of ${\cal C}$ is the class of zeroes of some set of such functors, thus obtaining a $\lambda$-ary generalisation of the finitary ($\lambda = \omega$) result from the finitary model theory of modules. We show that, to analyse the $\lambda$-ary model theory of a locally $\lambda$-presentable additive category ${\cal C}$, it is sufficient to consider \emph{finitary} pp formulas in the language of right ${\mathrm{Pres}_\lambda}{\cal C}$-modules, where ${\mathrm{Pres}_\lambda}{\cal C}$ is the category of $\lambda$-presented objects of ${\cal C}$, with the caveat that these pp formulas are interpreted among right ${\mathrm{Pres}_\lambda}{\cal C}$-modules which preserve $\lambda$-small products. In particular, for an additive category ${\cal R}$ with $\lambda$-small products (e.g. ${\cal R}={\mathrm{Pres}_\lambda}{\cal C}^{\mathrm op}$ for ${\cal C}$ a $\lambda$-presented additive category), the $\lambda$-accessible functors ${\cal N}\to{\mathrm Ab}$ which preserve products are precisely the finitely accessible functors ${\cal R}{\mathrm Mod}\to{\mathrm Ab}$ which preserve products, restricted to ${\cal N}$, where ${\cal N}\subseteq{\cal R}{\mathrm Mod}$ is the category of left ${\cal R}$-modules which preserve $\lambda$-small products.