On Grothendieck transformations in Fulton-MacPherson's bivariant theory
Let \(\mathcal C\) be a category with a final object \( pt\), on which a fiber product is defined together with a class of independent squares and confined maps [cf. \textit{W. Fulton} and \textit{R. MacPherson}, Matematika: Novoe v Zarubezhnoj Nauke, 33. Moskva: Izdatel'stvo ``Mir'' (1983; Zbl 0526.55007)]. A bivariant theory \(\mathbb B\) on the category \(\mathcal C\) with values in the category of abelian groups is a correspondence that assigns to each morphism in \(\mathcal C\) \[ X @ >f>> Y \] an abelian group \[ {\mathbb B}(X @>f>>Y) \] equipped with the product, pushforward and pullback operations. These operations have to satisfy natural compatibility conditions [loc. cit.]. For a bivariant theory \(B_{*}(X):={\mathbb B}(X\rightarrow pt)\) is a covariant functor for confined maps, and \(B^{*}(X):= {\mathbb B}(X\rightarrow X)\) is a contravariant functor. Let \(\mathcal C\) and \({\mathcal C}^{\prime}\) be two categories as above, \(-: {\mathcal C}\rightarrow {\mathcal C}^{\prime}\) a functor preserving the basic structures, and \(\mathbb B\) and \({\mathbb B}^{\prime}\) two corresponding bivariant theories on \(\mathcal C\) and \({\mathcal C}^{\prime}\) respectively. A Grothendieck transformation: \({\gamma}: {\mathbb B}\rightarrow {\mathbb B}^{\prime}\) is a collection of group homomorphisms \[ {\gamma}_{f}: {\mathbb B}(X @>f>>Y)\longrightarrow {\mathbb B}^{\prime} ({\overline X} @>{\bar f}>>{\overline Y}) \] preserving the three basic operations. The main examples of bivariant theories are: bivariant \textbf{K} theory introduced by Fulton and MacPherson [loc. cit.] and Fulton-MacPherson bivariant \(\mathbb F\) theory of constructible functions. One also has bivariant versions of homology theories with various coefficient groups. The Grothendieck transformation \({\gamma}: {\mathbb B}\rightarrow {\mathbb B}^{\prime}\) yields natural transformations of the associated covariant and contravariant functors. The authors study the reverse question concentrating on the covariant aspect. Namely, given a natural transformation \(\tau : {\mathbb B}_{*} \rightarrow {\mathbb B}_{*}^{\prime}\) of covariant functors coming from bivariant theories \({\mathbb B}\) and \({\mathbb B}^{\prime}\) does there exist a Grothendieck transformation \({\gamma}\) of bivariant theories whose covariant part is \({\tau}?\) A typical situation is given by MacPherson's Chern class transformation \(c_{*} : F\rightarrow H_{*}\) from the covariant functor of constructible functions to the Borel-Moore homology with integer coefficients. The authors give the existence and a kind of uniqueness theorem for a suitable general Grothendieck transformation \(\gamma\) associated to a natural transformation \(c_{*} : {\mathbb F}_{*}\rightarrow {\mathbb H}_{*}\) of bivariant theories \({\mathbb F}\) and \({\mathbb H}.\) Here \(c_{*}\) has to satisfy certain conditions, but it is not required to be necessarily the MacPherson's Chern class.
scientific article; zbMATH DE number 5186655
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