We generalize Marshall Cohen’s notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let
f
:
K
→
L
f:K \to L
be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in
K
K
. (3) The inclusion of
L
L
into the mapping cylinder of
f
f
is collared. (4) The mapping cylinder triad
(
C
f
,
K
,
L
)
({C_f},K,L)
is homeomorphic to the product triad
(
K
×
I
;
K
×
1
,
K
×
0
)
(K \times I;K \times 1,K \times 0)
rel
K
=
K
×
1
K = K \times 1
. Condition (2) is slightly weaker than
f
−
1
{f^{ - 1}}
(point) is homogeneously collapsible in
K
K
. Condition (4) when stated more precisely implies
f
f
is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of “attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter". Second, we classify when two maps from
X
X
to
Y
Y
have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation
f
∼
g
f \sim g
, where
f
,
g
:
X
→
Y
f,g:X \to Y
means
f
=
g
r
f = gr
for
r
:
X
→
X
r:X \to X
some transverse cellular map.