## § 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.

#### § 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.
-- | 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))

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