## § 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.
#### § Number of irreducible representations

Recall that the characters of irreducible representations are orthogonal. Also,
the dimension of the space of class functions is equal to the number of
conjugacy classes of the group $G$, since a class function takes on a distinct
value over each conjugacy class, so there are those many degrees of freedom.
This tells us that the number of irreducible representations is at most
the number of conjugacy classes of the group.
#### § 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