We define a family of categories related to the category of finite sets and injective functions. We show that the k-linearizations of these categories are Koszul, where k is a field of characteristic 0, using the language of directed graded k-linear categories. We also describe their quadratic dual categories in special cases.

The category $VI^n$ is the product category $\underbrace{VI \times \cdots \times VI}_{n~\text{copies}}$ and a $VI^n$-module is a functor from the category $VI^n$ to the category of $R$-modules. In this thesis, we will prove any finitely generated $VI^n$-module has finite regularity along with many interesting results, among them is the analogue of the Shift Theorem.