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 $\mathcal{C}$ be a $\lambda$-accessible additive category. We characterise the additive functors $C\to \mathrm{A}b$ 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 $\mathcal{C}$ 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 $\mathcal{C}$, it is sufficient to consider \emph{finitary} pp formulas in the language of right ${\mathrm{Pres}}_{\lambda }\mathcal{C}$-modules, where ${\mathrm{Pres}}_{\lambda }\mathcal{C}$ is the category of $\lambda$-presented objects of $\mathcal{C}$, with the caveat that these pp formulas are interpreted among right ${\mathrm{Pres}}_{\lambda }\mathcal{C}$-modules which preserve $\lambda$-small products. In particular, for an additive category $\mathcal{R}$ with $\lambda$-small products (e.g. $\mathcal{R}={\mathrm{Pres}}_{\lambda }{\mathcal{C}}^{\mathrm{o}p}$ for $\mathcal{C}$ a $\lambda$-presented additive category), the $\lambda$-accessible functors $\mathcal{N}\to \mathrm{A}b$ which preserve products are precisely the finitely accessible functors $\mathcal{R}\mathrm{M}od\to \mathrm{A}b$ which preserve products, restricted to $\mathcal{N}$, where $\mathcal{N}\subseteq \mathcal{R}\mathrm{M}od$ is the category of left $\mathcal{R}$-modules which preserve $\lambda$-small products.