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Factorisations on category theory

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Boletı́n de Matemáticas Nueva Serie, Volumen XI No.
1 (2004), pp.
51–68 FUNCTORS ON THE CATEGORY OF FACTORISATIONS OF A FUNCTION DAVID MOND (*) A la memoria de Jairo Charris, mi profesor 1.
Introduction: a determinantal factorisation of D4 Let us begin with a simple example.
Consider the 2-variable family of matrices   x2 0 0 M (x1 , x2 ) =  0 x1 x2  0 x2 x1 Its determinant is the well known D4 singularity, f (x1 , x2 ) = x21 x2 − x32 .
So M det (C2 , 0) −→ Mat3 (C) −→ C is a factorisation of D4 .
A deformation of M induces a deformation of its determinant.
The deformation theory of D4 is very well known: a miniversal deformation is given by F (x1 , x2 , u1 , u2 , u3 , u4 ) = x21 x2 − x32 u1 x22 u2 x2 u3 x1 u4 . However, deformations of M induce only a subset of the possible deformations of D4 ; it turns out that every deformation of D4 obtained by deforming M is parametrised-right-equivalent to one induced from the deformation F1 (x1 , x2 , v1 , .
, v7 ) = x21 x2 − x32 v1 x21 − (v1 v6 )x22 v5 x1 x2 (v1 v5 − v3 v7 − v2 v4 )x1 (v2 v7 − v1 v6 v3 v4 )x2 (v2 v6 − v3 v5 )v7 . (1) (*) David Mond.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, England E-mail: mond@maths.warwick.ac.uk . 51 52 DAVID MOND Deformation of M within the (smaller) class of symmetric matrices leads to an even more restricted class of deformations of its determinant, all of which are parametrised-right-equivalent to deformations induced from the deformation F2 (x1 , x2 , w1 , .
, w4 ) = x21 x2 −x32 w1 (x21 − x22 ) (w1 w4 − w22 − w32 )x1 2w2 w3 x2 w4 x1 x2 − w32 w4 . (2) These assertions are proved by introducing suitable notions of equivalence, Kdet s and Kdet , for matrix and symmetric matrix families like M - both of which are, in fact, liftings, of right equivalence of determinants.
These notions, due principally to Jim Damon (though see [3] for the precise version given below), ...






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