Subsection 4.1.2. (0P2L) · ScalingStacks

4.1.2. Markov $2$ -traces

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 $s\in S$ .

One can ask in addition that the functors $T_{S,s,\pm}$ are invertible. On the other hand, one can get a more general definition by dropping the functoriality of ${\mathcal{C}}$ and by requiring the existence of functors $D_{S,s,\pm}:{\mathcal{C}}_{S\setminus s}\to{\mathcal{C}}_{S}$ such that $M_{S}(\gamma_{S\setminus s}(?)\cdot F_{s}^{\pm 1})\simeq D_{S,s,\pm}M_{S\setminus s}(?)$ .

0P1Y

Remark 4.3. Let $\bar{{\mathcal{C}}}=\operatorname{colim}\nolimits{\mathcal{C}}$ and assume there are endofunctors $T_{\pm}$ of $\bar{{\mathcal{C}}}$ which restrict to $T_{S,s,\pm}$ for any $S$ and $s\in S$ . Replacing ${\mathcal{C}}_{(W,S)}$ by $\bar{{\mathcal{C}}}$ , one can construct from a Markov $2$ -trace another one taking value in the constant category $\bar{{\mathcal{C}}}$ , and with fixed endofunctors $T_{\pm}$ .

The first “trace” condition, once formulated in the appropriate homotopical setting, leads to a universal solution (“abelianization”) that can be described explicitly [BeNa]. It would be interesting to find a suitable formulation of the second condition in this setting leading to a universal solution. It would also be interesting to study functoriality with respect to cobordism in type $A$ , along the lines of [KhTh] and [ElKr].

Raphaël Rouquier

Original source: arXiv:1203.5065v1

4.1.2. Markov 22-traces

4.1.2. Markov $2$ -traces