Quadratic functors on pointed categories
Manfred Hartl (LAMAV), Christine Vespa (IRMA)

TL;DR
This paper characterizes quadratic polynomial functors from pointed categories to abelian groups, establishing an equivalence with algebraic data called quadratic modules, extending known results to broader categories.
Contribution
It introduces a functorial equivalence between quadratic functors and quadratic modules in general pointed categories, broadening the scope of previous specific cases.
Findings
Quadratic functors correspond to quadratic modules involving cross-effects.
The equivalence simplifies when the source object is a cogroup.
Extends results from groups and modules to more general pointed categories.
Abstract
We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups , and whose source category is an arbitrary category with null object such that all objects are colimits of copies of a generating object which is small and regular projective; this includes all pointed algebraic varieties. More specifically, we are interested in such quadratic functors from to which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if is Mal'cev and Barr exact. A functorial equivalence is established between such functors and certain minimal algebraic data which we call quadratic -modules: these involve the values on of the cross-effects of and certain structure maps generalizing the second Hopf invariant and the Whitehead product. Applying this general result to the case where…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
