Cubically thin homotopy
edit
Let
be squares in
with common vertices.
- A {\it cubically thin homotopy}

between
and
is a cube
such that
#
is a homotopy between
and
i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',}
#
is rel. vertices of
i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2 \partial^{+}_2 (U),\enskip \partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2 \partial^{+}_2 (U)}
are constant,
#
- the faces
are thin for
.
- The square
is {\it cubically}
-{\it equivalent} to
denoted
if there is a cubically
thin homotopy between
and
This definition enables one to construct
, by defining a
relation of cubically thin homotopy on the set
of squares.
- ↑
K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space,
Applied Cat. Structures , 8 (2000): 209-234.
- ↑
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space, {\it Theory and Applications of Categories} 10 ,(2002): 71-93.