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

Abstract

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