Quadratic categories and square rings

Baues, Hans–Joachim; Hartl, Manfred; Pirashvili, Teimuraz

doi:10.1016/s0022-4049(97)00130-8

Cited by 17 publications

(33 citation statements)

References 14 publications

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“…A theory of cogroups is a category T with zero object V and finite sums such that each object X has the structure of a cogroup given by the maps +: X Ä X 6 X, &: X Ä X satisfying the usual identities. A morphism f : X Ä Y is linear if the diagram One can check that this notion of a quadratic category coincides with the one in [3] where quadratic categories are studied. For any theory of cogroups T one obtains canonically an additive category T ad and a quadratic category T quad together with quotient functors:…”

Section: Linear and Quadratic Functors On Theories Of Cogroupsmentioning

confidence: 94%

“…For example a square ring (as defined in [3]) is a monoid in the category of square groups (compare (8.10)); this generalizes the classical notion of ring being a monoid in the monoidal category of abelian groups. The many examples of square rings in [3] yield examples of square groups. Moreover Z nil above is also a square ring which, in fact, is the initial object in the category of square rings.…”

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confidence: 96%

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Quadratic Endofunctors of the Category of Groups

Baues

Pirashvili

1999

Advances in Mathematics

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Let Gr be the category of groups. In this paper we study functors F: Gr Ä Gr which preserve cokernels and filtered colimits. The functor F is linear if the mapis an isomorphism where X 6 Y Ä Y is the sum in the category of groups and r 1 =X 6 Y Ä X, r 2 : X 6 Y Ä Y are the retractions. Moreover F is quadratic if F (X | Y)=kernel(Fr 1 , Fr 2 ) as a bifunctor is linear in X and Y.Our main result shows that the monoidal category of such quadratic endofunctors of Gr is equivalent to the monoidal category of square groups; see (3.10) and (8.9). Here a square group is a diagramwhere M ee is an abelian group, M e is a group of nilpotency degree 2, P is a homomorphism and H is a quadratic function with properties as in (3.5). We show that the quadratic endofunctor F of Gr can be described by a quadratic tensor productwhere M is a square group. A similar result for quadratic endofunctors of the category Ab of abelian groups was obtained in [2]. In (3.10) we specify the square groups corresponding to quadratic functors Ab Ä Gr and Gr Ä Ab respectively. The category of linear endofunctors of Gr which preserve cokernels and filtered colimits is equivalent to the category of abelian groups. In fact such linear endofunctors L of Gr have a factorization L: Gr w Ä ab Ab ww Ä A Ab/Gr Article ID aima.1998.1784, available online at http:ÂÂwww.idealibrary.com on

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Section: Linear and Quadratic Functors On Theories Of Cogroupsmentioning

confidence: 94%

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confidence: 96%

Quadratic Endofunctors of the Category of Groups

Baues

Pirashvili

1999

Advances in Mathematics

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“…The sum of two homomorphisms ϕ + ψ is the unique one such that (ϕ + ψ)(e) = ϕ(e) + ψ(e) for any e ∈ E. In this way morphism sets in nil are nil(2)-groups. The sum and the composition of morphisms in nil satisfy the following rules (see [10]):…”

Section: 3mentioning

confidence: 99%

Suspensions of crossed and quadratic complexes, co-H-structures and applications

Muro

2004

Trans. Amer. Math. Soc.

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“…with H(r) = r 2 and P = 0; many other examples are discussed in [9,10]. A quadratic Z-module M is a square group for which H is linear and HP H = 2H; see (3.1).…”

Section: Square-homologymentioning

confidence: 99%

Quadratic homology

Baues

1999

Trans. Amer. Math. Soc.

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Abstract. We describe axioms for a 'quadratic homology theory' which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group. IntroductionThe development of algebraic topology was profoundly affected by the notion of homology. Originally homology had coefficients in abelian groups and EilenbergSteenrod described the axioms of such an ordinary homology theory. Somewhat later important examples of generalized homology theories were found which led to the notion of homology with coefficients in a spectrum. The spectrum-homology can also be described as a homology theory satisfying all Eilenberg-Steenrod axioms except the dimension axiom concerning the value of the homology on spheres. In fact this value characterizes a homology theory in the sense that a natural transformation of homology theories which is an isomorphism on spheres is also an isomorphism on all finite CW-complexes.In his recent work on the calculus of homotopy functors Goodwillie observed that a spectrum is equivalent to a linear homotopy functor and spectrum-homology of a space X can be equivalently described by the homotopy groups.where L is a linear homotopy functor. We therefore call spectrum-homology also a linear homology theory. The non-linear homology theories are then obtained by the homotopy groupswhere D is any homotopy functor. Hence we again generalize homology by choosing now homotopy functors as coefficients. Clearly this is a very far reaching generalization since in particular homotopy groups of a space are the homology groups with coefficients in the identity functor. It is an old problem to find axioms which characterize the theory of homotopy groups in a similar way as ordinary homology theory is characterized by the Eilenberg-Steenrod axioms. The identity functor is

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Quadratic categories and square rings

Cited by 17 publications

References 14 publications

Quadratic Endofunctors of the Category of Groups

Quadratic Endofunctors of the Category of Groups

Suspensions of crossed and quadratic complexes, co-H-structures and applications

Quadratic homology

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