We refer to Quillen's notion of a category <i>C</i> furnished with
a <em>model structure</em>
{<i>W</i>,&nbsp;<i>F</i>,&nbsp;<i>co</i><i>F</i>} (weak equivalences,
fibrations, cofibrations), and the construction of the
localisation <i>C</i>[<i>W</i><sup>&#8722;1</sup>] at <i>W</i> as the
homotopy category <i>H</i><i>o</i>(<i>C</i>,&nbsp;<i>W</i>), that is,
the quotient by the congruence determined by homotopies in the
sets <i>C</i>(<i>X</i>,&nbsp;<i>Y</i>) of morphisms of <i>C</i>.

<P>
In this talk I will construct the
2-localisation <i>C</i>[<i>W</i><sup>&#8722;<i>e</i></sup>]
at <i>W</i> as the homotopy
2-category <i>H</i><i>o</i><sup>~</sup>(<i>C</i>,&nbsp;<i>W</i>), a
2-category with the same arrows of <i>C</i>, and where the homotopies
determine the 2-cells, instead of a congruence.

<P>
A novel feature is the introduction of a generalisation of cylinder
objects which allows the development of Quillen's theory of homotopies
and the construction of the homotopy 2-category for an arbitrary
category <i>C</i> and a single arbitrary class <i>W</i>
&sub; <i>C</i>.  There is a
2-functor <i>H</i><i>o</i><sup>~</sup>(<i>C</i>,&nbsp;<i>W</i>)
&#8594; <sup></sup> <i>C</i>[<i>W</i><sup>&#8722;<i>e</i></sup>],
which is an isomorphism  if <i>W</i> is <em>split generated</em> (an
arrow <i>f</i> is split if there exists <i>g</i> such that <i>fg</i>
= <i>id</i> or <i>gf</i> = <i>id</i>) and satisfies the 
<em>3 for 2 property</em>.

<P>
When <i>W</i> is the class of weak equivalences of a model category,
and <i>C</i> is the category of fibrant-cofibrant objects, taking the
set of connected components of the hom categories we obtain Quillen's
results.

<P>
This is a particular case of a joint work with Martin Szyld and Emilia
Descotte, where we have developed a 2-dimensional version of a model
category and its homotopy category. It corresponds to the case where
the model bicategory is the trivial model bicategory determined by a
model category.