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Martin Sleziak
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Defining properties of categories out of an indicial category

$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it.

Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, then the category $\Diag(D, S)$ is called the category of arrows of $S$, denoted $\Fl(S)$.

Definition 2: Let $\Delta^1$ be the category associated with the totally ordered set of two elements [0, 1]. So the category of arrows of $S$, denoted $\Fl(S)$ is the category $\Hom(\Delta^1, S)$.

The relationship between the two is evident, but I would like to know the motivation for choosing each one of them.

Question: When defining properties of categories with an indicial category when is useful to work with the category $\Delta^n$ and when with abstract diagrams.