  We introduce (partially) ordered Grothendieck categories and apply results on
their structure to the study of categories of representations of the Mackey Lie
algebra of infinite matrices $\mathfrak{gl}^M\left(V,V_*\right)$. Here
$\mathfrak{gl}^M\left(V,V_*\right)$ is the Lie algebra of endomorphisms of a
nondegenerate pairing of countably infinite-dimensional vector spaces
$V_*\otimes V\to\mathbb{K}$, where $\mathbb{K}$ is the base field. Tensor
representations of $\mathfrak{gl}^M\left(V,V_*\right)$ are defined as arbitrary
subquotients of finite direct sums of tensor products $(V^*)^{\otimes m}\otimes
(V_*)^{\otimes n}\otimes V^{\otimes p}$ where $V^*$ denotes the algebraic dual
of $V$. The category $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ which
they comprise, extends a category
$\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ previously studied in [4,
12,17], and our main result is that
$\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ is a finite-length, Koszul
self-dual, tensor category with a certain universal property that makes it into
a "categorified algebra" defined by means of a handful of generators and
relations. This result uses essentially the general properties of ordered
Grothendieck categories, which yield also simpler proofs of some facts about
the category $\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ established in
[12]. Finally, we discuss the extension of
$\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ by the algebraic dual
$(V_*)^*$ of $V_*$.
