The data is said to be a cofibration ( like an inclusion )
iff given any homotopy , and a map
downstairs such that ,
we can extend into . We see that this is simply
the HEP (homotopy extension property), where we have a homotopy of subspace
, and a starting homotopy of , which can be extended to a full homotopy.
A --gA[t]--> X
§ Lemma: Cofibration is always inclusion (Hatcher)
The pushout intuitively glues to along 's subspace . For this
interpretation, let us say that is a subspace of (ie, is an
injection). Then the result of the pushout is a space where we identify
with . The pushout in Set is
where we generate an equivalence relation from . In
groups, the pushout is amalgamated free product.
A <-i- P -β-> B
f :: C -> A
g :: C -> B
inl :: Pushout A B C f g
inr :: Pushout A B C f g
glue :: Π(c: C) inl (f(c)) = inr(g(c))
Suspension can "add homotopies". Example,
1 <- A -> 1
S1 = Susp(2).
We want to show that
is a cofibration if is a cofibration.
Reference: F. Faviona, more on HITs
A --f--> P
B -----> B Uf P