  We study the Deligne interpolation categories
$\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ for $t\in \mathbb{C}$, first
introduced by F. Knop. These categories interpolate the categories of finite
dimensional complex representations of the finite general linear group
$GL_n(\mathbb{F}_q)$. We describe the morphism spaces in this category via
generators and relations. We show that the generating object of this category
(an analogue of the representation $\mathbb{C}\mathbb{F}_q^n$ of
$GL_n(\mathbb{F}_q)$) carries the structure of a Frobenius algebra with a
compatible $\mathbb{F}_q$-linear structure; we call such objects
$\mathbb{F}_q$-linear Frobenius spaces, and show that
$\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ is the universal symmetric
monoidal category generated by such an $\mathbb{F}_q$-linear Frobenius space of
categorical dimension $t$. In the second part of the paper, we prove a similar
universal property for a category of representations of
$GL_{\infty}(\mathbb{F}_q)$.
