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

Question

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to the category $\textbf{Top}^{*n}$ of topological spaces with an ordered set of $n$ distinct basepoints and continuous maps between them which preserve the ordered basepoint set.

Let $X$ be a topological space and let $A=\{x_1,\ldots,x_n\}\subseteq X$ be it's baspoint set for some finite natural number $n$.

I'm particularly interested in the homotopy of these spaces and so I feel it is probably natural to consider the relative fundamental groupoid $\pi_1(X,A)$ of these spaces. I don't know too much about the fundamental groupoid of a space though other than basic definitions. I wonder if someone could outline what properties we gain and lose by considering the category $\textbf{Top}^{*n}$ instead of the category $\textbf{Top}^*$. For instance, this category still has an initial object which is just the finite set of $n$ elements with the discrete topology however, I don't believe $\textbf{Top}^{*n}$ has terminal objects. This category still has a coproduct given by identifying the basepoint sets of two objects element-wise with respect to their order (a kind of wedge product at $n$ points) but it's not clear that we still have products. How might these properties change if we loosen our category to include maps which possibly permute the ordering of the basepoint set?

Also how does the fundamental groupoid functor $\pi_1(.,A)$, which takes an object in $\textbf{Top}^{*n}$ to its fundamental groupoid $\textbf{Top}^{*n}$ relative to its basepoint set $A$, differ from the usual fundamental group functor. If there is a functor which takes an object in $\textbf{Top}^{*n}$ to $\textbf{Top}^*$, does this induce a functor $\textbf{Grpd}_n\rightarrow \textbf{Grp}$ from the category of groupoids with $n$ objects to the category of groups (possibly via classifying spaces?)?

Hopefully the questions I've asked are closely related to each other enough that I don't need to break this in to multiple questions. If anyone thinks that would be wise though, please feel free to make a comment and I shall consider doing so.

Answer 1

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