Construct a canonical composition-compatible transfer map

Establish whether the functor from the category of regular closed-immersion factorizations of a finite locally free map to the space of null-homotopies of the induced filtered transfer composite admits a colimit, thereby constructing a filtered transfer map that is canonically functorial under composition of finite locally free maps.

Background

The paper constructs filtered transfer maps for regular finite locally free maps by factoring each map through a regular closed immersion into affine or projective space. The construction is functorial under base change, but composition is established only up to a non-canonical homotopy.

The authors consider the category of all such regular closed-immersion factorizations and obtain, for each factorization, a null-homotopy of the composite from the filtered object on the source to the quotient of the filtered object on the target. A colimit of this diagram would provide a stronger, canonical construction whose transfer maps are strictly functorial under composition. The authors explicitly note that the indexing category is not filtered, so existence of the required colimit is unresolved.