Example

Given two pointed objects $(X,x)$ and $(Y,y)$, then:

1. their product in $\mathcal{C}^{\ast/}$ is simply $(X\times Y, (x,y))$;

2. their coproduct in $\mathcal{C}^{\ast/}$ has to be computed using the second clause in prop. : since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $\mathcal{C}$, but by the pushout in $\mathcal{C}$ of the form:

   $$\array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,.$$

   This is called the wedge sum operation on pointed objects.

Generally for a set $\{X_i\}_{i \in I}$ in $\mathcal{C}^{\ast/}$

1. their product is formed in $Top$, with the new basepoint canonically induced;

2. their coproduct is formed by the colimit in $Top$ over the diagram with a basepoint adjoined, and is called the wedge sum $\vee_{i \in I} X_i$.

Example

For $X$ a CW-complex, then for every $n \in \mathbb{N}$ the quotient of its $n$-skeleton by its $(n-1)$-skeleton is the wedge sum, def. , of $n$-spheres, one for each $n$-cell of $X$:

Example

For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. , then

• $X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;

• $X_+ \wedge Y_+ \simeq (X \times Y)_+$.

Example

Let $\mathcal{C}^{\ast/} = Top^{\ast/}$ be pointed topological spaces. Then

denotes the standard interval object $I = [0,1]$, with a disjoint basepoint adjoined, def. . Now for $X$ any pointed topological space, then

is the reduced cylinder over $X$: the result of forming the ordinary cyclinder over $X$, and then identifying the interval over the basepoint of $X$ with the point.

(Generally, any construction in $\mathcal{C}$ properly adapted to pointed objects $\mathcal{C}^{\ast/}$ is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

Just like the ordinary cylinder $X\times I$ receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top$, so the reduced cyclinder receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top^{\ast/}$, which is the wedge sum from example :

Example

For $\mathcal{C} = Top_{Quillen}$, the classical model structure on topological spaces, then the model structure on pointed topological spaces induced via prop. we call the classical model structure on pointed topological spaces $Top_{Quillen}^{\ast/}$. Its homotopy category of a model category is the classical pointed homotopy theory $Ho(Top^{\ast/})$.

Example

The fibrant objects in the pointed model structure $\mathcal{C}^{\ast/}$, prop. , are those that are fibrant as objects of $\mathcal{C}$.

But the cofibrant objects in $\mathcal{C}^{\ast}$ are now those for which the basepoint inclusion is a cofibration in $X$.

For $\mathcal{C}^{\ast/} = Top^{\ast/}$, then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces with non-degenerate basepoints. Notice that the point itself is cofibrant in $Top_{Quillen}$, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.

Example

For $\mathcal{C}$ any model category, with $\mathcal{C}^{\ast/}$ its pointed model structure according to prop. , then the corresponding homotopy category is, by remark , canonically enriched in pointed sets, in that its hom-functor is of the form