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 kk. (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 kk-quiver (or simply dg-quiver) 𝒬\mathcal{Q} is specified by

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

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

(More work to be done here.)

References

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

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

Revised on January 27, 2012 17:58:11 by Eric Forgy (203.145.92.183)