## § Cofibration

```
A --gA[t]--> X
| ^
i |
| |
v |
B >-gB[0]---*
```

The data $(A, B, i)$ is said to be a cofibration ( $i$ like an inclusion $A \rightarrow B$)
iff given any homotopy $gA[t]: [0, 1] \times A \rightarrow X$, and a map
downstairs $gB[0]: B \rightarrow X$ such that $gB[0] \circ i = gA[t](0)$,
we can extend $gB[0]$ into $gB[t]$. We see that this is simply
the HEP (homotopy extension property), where we have a homotopy of subspace
$A$, and a starting homotopy of $B$, which can be extended to a full homotopy.
#### § Lemma: Cofibration is always inclusion (Hatcher)

#### § Pushouts

```
A <-i- P -β-> B
```

The pushout intuitively glues $B$ to $A$ along $A$'s subspace $P$. For this
interpretation, let us say that $P$ is a subspace of $A$ (ie, $i$ is an
injection). Then the result of the pushout is a space where we identify
$\beta(p) \in B$ with $p \in A$. The pushout in Set is $A \cup B/ \sim$
where we generate an equivalence relation from $i(p) \sim \beta(p)$. In
groups, the pushout is amalgamated free product.
```
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:
```
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 $P \xrightarrow{i'} B \cup_f X$
is a cofibration if $A \xrightarrow{i} B$ is a cofibration.
Reference: F. Faviona, more on HITs