On Grothendieck transformations in Fulton-MacPherson's bivariant theory
type: Item
instance of: scholarly article
title: On Grothendieck transformations in Fulton-MacPherson's bivariant theory
zbMATH Open document ID: 1139.14005
author: Jean-Paul Brasselet
author: Jörg Schürmann
author: Shoji Yokura
published in: Journal of Pure and Applied Algebra
publication date: 2007-09-03T00:00:00Z
review text: 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 …
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 5186655
zbMATH Keywords: Grothendieck transformation
zbMATH Keywords: bivariant theory
zbMATH Keywords: bivariant Chern class
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.03.004
OpenAlex ID: W2088644561
Wikidata QID: Q112881980
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DOI: 10.1016/J.JPAA.2007.03.004
description: scientific article; zbMATH DE number 5186655