I am trying to find any references to the following construction that relates to zigzag categories (https://ncatlab.org/nlab/show/zigzag+category).
Let C be a category. Let Z(C) be the category with objects $|Z(C)| = |C|$ and morphisms $A \rightarrow B$ in Z(C) given by diagrams that look like: $\{ A \leftarrow A_1 \rightarrow A_2 \leftarrow \ldots \rightarrow B\}$ formed from morphisms in C.
More precisely, a morphism $f : A \rightarrow B$ in Z(C) is a zigzag from $A$ to $B$ defined by a sequence $t$ of size $n$ of morphisms in $C$ paired with their polarity $\{+, -\}$, where for $n>0$:
- $t_1 = (f_1 : A \rightarrow A_1, +)$ or $t_1 = (f_1 : A \leftarrow A_1, -)$ (for some $A_1$ and $f_1$ in C);
- $t_n = (f_n : A_{n-1} \rightarrow B, +)$ or $t_n = (f_n : A_{n-1} \leftarrow B, -)$ (for some $A_{n-1}$ and $f_n$ in C);
- for all $1 \leq i < n$ then $trg(t_i) = src(t_{i+1})$ where
- $trg(g : A \rightarrow B, +) = B,$ $trg(g : A \leftarrow B, -) = B$
- $src(f : A \rightarrow B, +) = A,$ $src(f : A \leftarrow B, -) = A$
Identity morphisms in Z(C) are then given by the empty zigzag of size $0$ and composition is defined by zigzag concatenation (essentially concatenating lists).
Does anyone know of such a construction already in the literature?