nLab dg-quiver

Idea

As explained in that entry, ‘quiver’ is often used to refer to a directed graph, but, as is argued at that entry, this is almost always then used to derive the free category on the that directed graph. The notion of a differential graded quiver, as studied by Lyubashenko and Manzyuk (see reference below), is an enriched analogue of directed graph, enriched over the category of differential graded modules over a commutative ring $k$. (It only uses the underlying category in fact, not the tensor structure). It is introduced precisely to define the analogue of the path category of a directed graph in this enriched context but the structure they define is in fact a linear A-infinity category.

Definition

A differential graded $k$-quiver (or simply dg-quiver) $\mathcal{Q}$ is specified by

• a small set of objects, denoted $Ob(\mathcal{Q})$;

• for each pair $(x,y)$ of objects of $\mathcal{Q}$, a chain complex (= graded $k$-module with differential of degree +1) denoted $\mathcal{Q}(x,y)$.

(More work to be done here.)

References

• V. Lyubashenko and O. Manzyuk, Free $A_\infty$-categories, Theory and Applications of Categories, Vol. 16, 2006, No. 9, pp 174-205.(pdf)

• J. P. May, Operadic categories, $A_\infty$-categories, and $n$-categories, Talk given in Morelia, Mexico, May 25, 2001 available at: (pdf)

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