# Functors on the category of factorisations of a function

Functors on the category of factorisations of a function
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Factorisations on category theory

Tipo de documento: Artículo - Article

Source: http://www.bdigital.unal.edu.co

## Teaser

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), ...