nLab Verdier-Grothendieck context

Idea

The specialization of a context of six operations $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ to the case that the “projection formula” $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$. (May 05, def. 2.12)

Properties

In a Verdier-Grothendieck context, duality intertwines $f_!$ with $f_\ast$ and $f^!$ with $f^\ast$.

(May 05, section 3, Joshua, corollary 5.4)

Related concepts

• Verdier duality, Grothendieck duality

• Wirthmüller context, Grothendieck context

References

A general abstract discussion of the axioms and their consequences is in

• Halvard Fausk, Po Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories, 11 4 (2003) 107-131 [tac:11-04, pdf]

A fairly general class of implementations is in

• Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)