ScalingStacks

Definition 4.2. Let ${\mathcal{C}}:{\mathcal{F}}\to{\mathcal{C}}{at}$ be a functor.

A Markov $2$-trace on ${\mathcal{F}}$ (relative to ${\mathcal{C}}$) is the data of functors $M_{(W,S)}:{\mathcal{B}}_{(W,S)}\to{\mathcal{C}}_{(W,S)}$ such that the following holds

• •

  $M_{S}(?_{1}\cdot?_{2})\simeq M_{S}(?_{2}\cdot?_{1})$ as functors ${\mathcal{B}}_{S}\times{\mathcal{B}}_{S}\to{\mathcal{C}}_{S}$

• •

  $M_{S}(\gamma_{S\setminus s}(?)\cdot F_{s}^{\pm 1})\simeq T_{S,s,\pm}{\mathcal{C}}(i_{s})M_{S\setminus s}(?)$ as functors ${\mathcal{B}}_{S\setminus s}\to{\mathcal{C}}_{S}$, for some endofunctors $T_{S,s,\pm}$ of ${\mathcal{C}}_{S}$, for all $s\in S$.