Properties of the Category of topological spaces with $n$ basepoints.

I will discuss the categorical properties. Let more generally $B$ a topological space (of generic base points) and denote by $\mathsf{Top}_B$ the full subcategory of the slice category $B \downarrow \mathsf{Top}$ whose objects are injective continuous maps $B \to X$. The forgetful functor $B \downarrow \mathsf{Top} \to \mathsf{Top}$ creates all limits, in fact it is monad with corresponding monad $B + (-)$ on $\mathsf{Top}$. This monad also preserves directed colimits, so that the forgetful functor also creates directed colimits.

It is not hard to see that $\mathsf{Top}_B \subseteq B \downarrow \mathsf{Top}$ is stable under non-empty products (i.e. excluding the terminal object), equalizers, as well as under directed colimits: For example, if $(B \to X_i)_{i \in I}$ is a non-empty family of objects in $\mathsf{Top}_B$ and $(B \to \prod_i X_i)_{i \in I}$ is their product in the slice category, then $B \to \prod_i X_i$ is injective since the composition with some projection $\mathrm{pr_i}$ (which exists since $I \neq \emptyset$) gives the injective map $B \to X_i$. Hence this is also the product in $\mathsf{Top}_B$. Equalizers are easy to handle, because they are injective, and for directed colimits just use that two elements are equal iff they are equal at some stage.

The forgetful functor $\mathsf{Top}_B \to \mathsf{Top}$ has a left adjoint, sending $X$ to $B \hookrightarrow B+X$. In particular it preserves all limits. Therefore the underlying space of an terminal object has just one point, which implies that $B$ is just a point, and we get $\mathsf{Top}_*$ which is complete and cocomplete. If $B$ is more than just a point, there is no terminal object.

Coproducts in $B \downarrow \mathsf{Top}$ are pushouts in $\mathsf{Top}$ over $B$. One can check that $\mathsf{Top}_B$ is closed under them, using the explicit construction of pushouts. This also includes the initial object. I am pretty sure that coequalizers don't exist, but I don't have an example right now (of course it is not enough to see that the forgetful functor doesn't create them).

There is a smash product on $B \downarrow \mathsf{Top}$ given by $X \wedge Y = (X \times Y) / (b,y) \sim (x,b')$, where $b$ and $b'$ run through all base points. But $\mathsf{Top}_B$ is not closed under it when $B$ is more than just point, because all the base points become identified.

Summary:

• $\mathsf{Top}_B$ has non-empty limits, directed colimits, coproducts
• $\mathsf{Top}_B$ has no terminal object, coequalizers, smash products