Dinversion in topological and algebraic varieties

(a) For a variety of algebras V and its associated topological variety Ṽ, determine those varieties in which dinversion preserves normal maps for which dinversion in Ṽ also preserves normal maps; (b) determine those di-exact varieties for which Ṽ is also di-exact.

Background

Topological enrichments of algebraic varieties often weaken exactness properties. The questions probe how dinversion and di-exactness transfer from an algebraic variety V to its topological counterpart Ṽ.