§ 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.

§ 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 GG, 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 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