Categories with dependent arrows

We present an abstract, categorical formulation of dependent functions in a fundamental manner and independently from the Sigma-construction. For that, we define first the notion of a category with family-arrows, or a $\text{f}$-category. A $(\text{f},\mathrm{\Sigma })$-category is a $\text{f}$-category with Sigma-objects, where a $(\text{f},\mathrm{\Sigma })$-category with a terminal object is exactly a type-category of Pitts, or a category with attributes of Cartmell. We introduce categories with dependent arrows, or $\text{di}$-categories, and we show that every $(\text{f},\mathrm{\Sigma })$-category is a $\text{di}$-category in a canonical way. The notion of a Sigma-object in a $\text{di}$-Category is affected by the existence of dependent arrows, and we show that every $(\text{f},\mathrm{\Sigma })$-category is a $(\text{di},\mathrm{\Sigma })$-category in a canonical way.