Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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34 I Fundamental Concepts<br />
universal property: given h ∈ C(B,X) and k ∈ C(C,X) such that hf = kg, there<br />
exists a unique morphism ℓ ∈ C(D,X) such that ℓ f = k and ℓg = h.<br />
We depict this situation with the commutative diagram<br />
A<br />
��<br />
C<br />
g<br />
f<br />
¯f<br />
��<br />
B<br />
¯g<br />
��<br />
��<br />
h<br />
D<br />
k<br />
The symbol at the right lower angle of the commutative square <strong>in</strong>dicates that we<br />
have a pushout diagram.<br />
As a consequence of the universal property of pushouts, a pushout of ( f ,g) is<br />
unique up to isomorphism. In fact, suppose that ( f ′ ,g ′ ) with f ′ : C → D ′ and<br />
g ′ : B → D ′ is a pushout; then by the universal property, there is a unique ℓ ′ : D ′ → D<br />
such that ℓ ′ g ′ = g and ℓ ′ f ′ = f ; hence, ℓ ′ ℓ = 1D. Similarly, we conclude that ℓℓ ′ = 1C.<br />
The morphism ℓ is an isomorphism and its <strong>in</strong>verse is ℓ ′ .<br />
We say that a category C is closed by pushouts or closed regard<strong>in</strong>g pushouts if<br />
every pair of morphisms f : A → B and g: A → C of C has a pushout. Not every<br />
category is closed by pushouts; we now prove that the category Top of topological<br />
spaces and the category Gr of groups are closed by pushouts.<br />
Let us start with Top.Wedef<strong>in</strong>ethedisjo<strong>in</strong>t union of two topological spaces B<br />
and C by tak<strong>in</strong>g a set of two po<strong>in</strong>ts, say, {i, j} and construct<strong>in</strong>g the spaces B ×{i}<br />
and C ×{j}, homeomorphic to B and C, respectively. We then def<strong>in</strong>e the union<br />
B ⊔C = B ×{i}∪C ×{j} with the <strong>in</strong>clusions<br />
ℓ<br />
��<br />
��<br />
��<br />
X<br />
ιB : B → B ⊔C , b ↦→ (b,i) ,<br />
ιC : C → B ⊔C , c ↦→ (c, j)<br />
and give B ⊔C the topology def<strong>in</strong>ed by the open sets<br />
ιB(U) ∪ ιC(V )=(U ×{i}) ∪ (V ×{j})<br />
with U open <strong>in</strong> B and V open <strong>in</strong> C; <strong>in</strong> this way, we obta<strong>in</strong> the topological space<br />
called disjo<strong>in</strong>t union of B and C. By construct<strong>in</strong>g a quotient of the disjo<strong>in</strong>t union, it<br />
is possible to prove that Top is closed regard<strong>in</strong>g pushouts.<br />
(I.2.2) Theorem. The category Top is closed by pushouts.<br />
Proof. Let f : A → B and g: A → C be any two maps <strong>in</strong> Top. In the disjo<strong>in</strong>t<br />
union B ⊔C, identify f (a) with g(a) for every a ∈ A; take the quotient set B ⊔ f ,g<br />
C obta<strong>in</strong>ed by the identifications f (a) =g(a) and then the canonic surjection