In mathematics, a Δset S, often called a semisimplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δset is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.
Formally, a Δset is a sequence of sets
{
S
n
}
n
=
0
∞
together with maps
d
i
:
S
n
+
1
→
S
n
with i = 0,1,...,n + 1 for n ≥ 1 that satisfy
d
i
∘
d
j
=
d
j
−
1
∘
d
i
whenever i < j.
This definition generalizes the notion of a simplicial complex, where the
S
n
are the sets of nsimplices, and the d_{i} are the face maps. It is not as general as a simplicial set, since it lacks "degeneracies."
Given Δsets S and T, a map of Δsets is a collection
{
f
n
:
S
n
→
T
n
}
n
=
0
∞
such that
f
n
∘
d
i
=
d
i
∘
f
n
+
1
whenever both sides of the equation are defined. With this notion, we can define the category of Δsets, whose objects are Δsets and whose morphisms are maps of Δsets.
Each Δset has a corresponding geometric realization, defined as

S

=
(
∐
n
=
0
∞
S
n
×
Δ
n
)
/
∼
where we declare that
(
σ
,
d
i
t
)
∼
(
d
i
σ
,
t
)
for all
σ
∈
S
n
,
t
∈
Δ
n
−
1
.
Here,
Δ
n
denotes the standard nsimplex, and
d
i
:
Δ
n
−
1
→
Δ
n
is the inclusion of the ith face. The geometric realization is a topological space with the quotient topology.
The geometric realization of a Δset S has a natural filtration

S

0
⊂

S

1
⊂
⋯
⊂

S

,
where

S

N
=
(
∐
n
=
0
N
S
n
×
Δ
n
)
/
∼
is a "restricted" geometric realization.
The geometric realization of a Δset described above defines a covariant functor from the category of Δsets to the category of topological spaces. Geometric realization takes a Δset to a topological space, and carries maps of Δsets to induced continuous maps between geometric realizations (which are topological spaces).
If S is a Δset, there is an associated free abelian chain complex, denoted
(
Z
S
,
∂
)
, whose nth group is the free abelian group
(
Z
S
)
n
=
Z
⟨
S
n
⟩
,
generated by the set
S
n
, and whose nth differential is defined by
∂
n
=
d
0
−
d
1
+
d
2
−
⋯
+
(
−
1
)
n
d
n
.
This defines a covariant functor from the category of Δsets to the category of chain complexes of abelian groups. A Δset is carried to the chain complex just described, and a map of Δsets is carried to a map of chain complexes, which is defined by extending the map of Δsets in the standard way using the universal property of free abelian groups.
Given any topological space X, one can construct a Δset
s
i
n
g
(
X
)
as follows. A singular nsimplex in X is a continuous map
σ
:
Δ
n
→
X
.
Define
s
i
n
g
n
(
X
)
to be the collection of all singular nsimplicies in X, and define
d
i
:
s
i
n
g
i
+
1
(
X
)
→
s
i
n
g
i
(
X
)
by
d
i
(
σ
)
=
σ
∘
d
i
,
where again d^{i} is the ith face map. One can check that this is in fact a Δset. This defines a covariant functor from the category of topological spaces to the category of Δsets. A topological space is carried to the Δset just described, and a continuous map of spaces is carried to a map of Δsets, which is given by composing the map with the singular nsimplices.
This example illustrates the constructions described above. We can create a Δset S whose geometric realization is the unit circle
S
1
, and use it to compute the homology of this space. Thinking of
S
1
as an interval with the endpoints identified, define
S
0
=
{
v
}
,
S
1
=
{
e
}
,
with
S
n
=
∅
for all n ≥ 2. The only possible maps
d
0
,
d
1
:
S
1
→
S
0
,
are
d
0
(
e
)
=
d
1
(
e
)
=
v
.
It is simple to check that this is a Δset, and that

S

≅
S
1
. Now, the associated chain complex
(
Z
S
,
∂
)
is
0
⟶
Z
⟨
e
⟩
⟶
∂
1
Z
⟨
v
⟩
⟶
0
,
where
∂
1
(
e
)
=
d
0
(
e
)
−
d
1
(
e
)
=
v
−
v
=
0.
In fact,
∂
n
=
0
for all n. The homology of this chain complex is also simple to compute:
H
0
(
Z
S
)
=
ker
∂
0
i
m
∂
1
=
Z
⟨
v
⟩
≅
Z
,
H
1
(
Z
S
)
=
ker
∂
1
i
m
∂
2
=
Z
⟨
e
⟩
≅
Z
.
All other homology groups are clearly trivial.
One advantage of using Δsets in this way is that the resulting chain complex is generally much simpler than the singular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated.
One drawback of this method is that one must prove that the geometric realization of the Δset is actually homeomorphic to the topological space in question. This can become a computational challenge as the Δset increases in complexity.