§ Cofibration

A --gA[t]--> X
|           ^
i           |
|           |
v           |
B >-gB[0]---*
The data (A,B,i)(A, B, i) is said to be a cofibration ( ii like an inclusion ABA \rightarrow B) iff given any homotopy gA[t]:[0,1]×AXgA[t]: [0, 1] \times A \rightarrow X, and a map downstairs gB[0]:BXgB[0]: B \rightarrow X such that gB[0]i=gA[t](0)gB[0] \circ i = gA[t](0), we can extend gB[0]gB[0] into gB[t]gB[t]. We see that this is simply the HEP (homotopy extension property), where we have a homotopy of subspace AA, and a starting homotopy of BB, which can be extended to a full homotopy.

§ Lemma: Cofibration is always inclusion (Hatcher)

§ Pushouts

A <-i- P -β-> B
The pushout intuitively glues BB to AA along AA's subspace PP. For this interpretation, let us say that PP is a subspace of AA (ie, ii is an injection). Then the result of the pushout is a space where we identify β(p)B\beta(p) \in B with pAp \in A. The pushout in Set is AB/A \cup B/ \sim where we generate an equivalence relation from i(p)β(p)i(p) \sim \beta(p). In groups, the pushout is amalgamated free product.
-- | HoTT defn
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))
1 <- A -> 1
Suspension can "add homotopies". Example, S1 = Susp(2).
A --f--> P
|       |
|i      |i'
v       v
B -----> B Uf P
We want to show that PiBfXP \xrightarrow{i'} B \cup_f X is a cofibration if AiBA \xrightarrow{i} B is a cofibration. Reference: F. Faviona, more on HITs