# Definition of category of pointed topological spaces

I have a technical question about the definition of the category of pointed topological spaces $$\mathbf{Top}_*$$. Every book that I read about the topic defines an arrow between two pointed spaces $$(X,x_0)$$ and $$(Y,y_0)$$ as a continous function that maps $$x_0$$ into $$y_0$$. That is to say that the set of arrows between $$(X,x_0)$$ and $$(Y,y_0)$$ is:

$$\text{Hom}_{\mathbf{Top}_*}((X,x_0),(Y,y_0)):=\{F:X\to Y \ | \ F(x_0)=y_0 \text{ and } F \text{ is continous}\}$$

But this implies that the class of arrows of $$\mathbf{Top}_*$$ is: $$\text{Hom}_{\mathbf{Top}_*}:=\{\text{continous functions between non-empty topological spaces}\}$$ According to this definition an arrow can have more than one (co)domain. If: $$f:X\to Y\in \text{Hom}_{\mathbf{Top}_*},$$ who are $$\text{dom}(f)$$ and $$\text{cod}(f)$$?. Shouldn't an arrow contain some information about the chosen domain base-point?

• I don't understand your sentence "But this implies..." I don't see how it does at all. Commented Mar 9, 2022 at 14:50
• You've defined your Hom incorrectly, the actual class of arrows is the proper subclass of yours which preserve basepoints, but your actual question still makes sense. It's just a technical thing - you can just assume that what the basepoints are is actually part of the data for each arrow and that clears it up. So technically an arrow could be an ordered triple with a basepoint-preserving map and two basepoints. That technicality is just glossed over and not said out loud because it would get annoying very quickly. Commented Mar 9, 2022 at 14:55
• @FrancescoScavella Yes, but you have to remember what "object" and "morphism" really mean. This is essentially Christian's point. Commented Mar 9, 2022 at 15:01
• @Christian We could also omit the codomain basepoint since it's univocally determined by the domain basepoint and by the function, right? Commented Mar 9, 2022 at 15:01
• I guess I'll be the contrarian here. I don't think ignoring the basepoint of the codomain is a great idea, even though it is implied by the function. That codomain could also end up as the domain of another function that you would like to compose against, and you need the basepoint there. Commented Mar 9, 2022 at 15:11

This is "just" a technical subtlety. You're correctly observing that a single continuous map $$F\colon X\rightarrow Y$$ with $$|X|\ge2$$ would be both a morphism $$(X,x)\rightarrow(Y,F(x))$$ and $$(X,x^{\prime})\rightarrow(Y,F(x^{\prime}))$$ for $$x,x^{\prime}\in X$$ two distinct elements, so domain and codomain of a morphism are not well-defined. This type of deficiency occurs fairly generally and is harmless, the fix is simply to make $$\mathrm{Hom}((X,x_0),(Y,y_0))$$ the set of triples $$((X,x_0),F,(Y,y_0))$$, where $$F\colon X\rightarrow Y$$ is a continuous function such that $$F(x_0)=y_0$$. This remembers domain and codomain by design.

(In fact, a similar issue occurs in the category $$\mathbf{Set}$$, because set theorists define a function $$X\rightarrow Y$$ to be a certain type of subset of $$X\times Y$$, the graph, which is equally a subset of $$X\times f(X)$$, so functions in the strict set-theoretic sense don't have a codomain either, and the fix is just the same - we make functions tuples $$(\text{graph},\text{codomain})$$ instead. Taking that into account and going back into the original context, $$F$$ remembers $$Y$$ and $$y_0=F(x_0)$$ is implied, so technically including the codomain is not necessary and you may instead use tuples $$((X,x_0),F)$$. Even more efficiently, $$F$$ remembers $$X$$, so we really could just use tuples $$(x_0,F)$$ with $$x_0\in\mathrm{dom}(F)$$ instead.)

Maybe it bears repeating that a standard "categorical" way of describing "pointed sets" (with or without additional structure) is to fix a one-element set $$X=\{x_o\}$$, and have pointed sets be sets $$S$$ with choice of set-hom $$X\to S$$... and homs of pointed sets be set-maps that commute with the special maps of $$X$$ to sets.

Yes, given $$f:X\to S$$ and given an arbitrary set-hom $$\varphi:S\to T$$, the composition $$\varphi\circ f$$ is a map $$X\to T$$, giving $$T$$ a unique pointed-set structure so that $$\varphi:S\to T$$ is a pointed-set hom.

Your problem does not occur if the "correct" definition of a category is used. There are essentially two approaches.

Definition 1 (see Steve Awodey, Category Theory or Emily Riehl, Category theory in context)

A category consists of the following data:

1. Objects: $$A, B, C, \ldots$$
2. Arrows: $$f, g, h, \ldots$$
3. For each arrow $$f$$, there are given objects $$\operatorname{dom}(f), \operatorname{cod}(f)$$ called the domain and codomain of $$f$$. We write $$f : A → B$$ to indicate that $$A = \operatorname{dom}(f)$$ and $$B = \operatorname{cod}(f)$$.
4. Given arrows $$f : A → B$$ and $$g : B → C$$, that is, with $$\operatorname{cod}(f) = \operatorname{dom}(g)$$, there is given an arrow $$g \circ f : A → C$$ called the composite of $$f$$ and $$g$$.
5. For each object $$A$$, there is given an arrow $$1_A : A → A$$ called the identity arrow of $$A$$.

These data are required to satisfy the following laws:

(a) Associativity: $$h \circ (g \circ f)=(h \circ g) \circ f$$ for all $$f : A → B$$, $$g : B → C$$, $$h : C → D$$.

(b) Unit: $$f \circ 1_A = f = 1_B \circ f$$ for all $$f : A → B$$.

Definition 2 (see Jiří Adámek, Horst Herrlich and George E. Strecker, Abstract and Concrete Categories. The Joy of Cats)

A category is a quadruple $$\mathbf A = (\mathcal O, \hom, id, \circ)$$ consisting of

1. a class $$\mathcal O$$, whose members are called $$\mathbf A$$-objects,
2. for each pair $$(A, B)$$ of $$\mathbf A$$-objects, a set $$\hom(A, B)$$, whose members are called $$\mathbf A$$-morphisms from $$A$$ to $$B$$ [the statement "$$f ∈ \hom(A, B)$$" is expressed more graphically by using arrows; e.g., by statements such as "$$f : A → B$$ is a morphism" or "$$A \stackrel{f}{\to} B$$ is a morphism"],
3. for each $$\mathbf A$$-object $$A$$, a morphism $$A \stackrel{id_A}{\to} A$$, called the $$\mathbf A$$-identity on $$A$$,
4. a composition law associating with each $$\mathbf A$$-morphism $$A \stackrel{f}{\to} B$$ and each $$\mathbf A$$-morphism $$B \stackrel{g}{\to} C$$ an $$\mathbf A$$-morphism $$A \stackrel{g \circ f}{\to} C$$ , called the composite of $$f$$ and $$g$$,

subject to the following conditions:

(a) composition is associative.

(b) $$\mathbf A$$-identities act as identities with respect to composition.

(c) the sets $$\hom(A, B)$$ are pairwise disjoint.

What are the differences between these definitions?

• In a category in the sense of definition 1 ("d1-category") we are given the (global) class $$\mathcal A$$ of arrows and two functions $$\operatorname{dom}, \operatorname{cod}$$ from $$\mathcal A$$ to the class of objects $$\mathcal O$$. The "local" arrow classes $$\hom(A,B)$$ are defined by taking all arrows $$f$$ with $$\operatorname{dom}(f) = A$$ and $$\operatorname{cod}(f) = B$$. Clearly $$\mathcal A$$ is the disjoint union of the $$\hom(A,B)$$ (each arrow $$f$$ belongs to $$\hom(\operatorname{dom}(f), \operatorname{cod}(f))$$ and to no other class $$\hom(A,B)$$).
The $$\hom(A,B)$$ are not required to be sets, they may be proper classes. This gives more flexibility. If all $$\hom(A,B)$$ are sets, then the category is called locally small, and we could include this requirement into definition 1. Anyway, it is just a technical point. Most categories occurring in practice are locally small.

• In a category in the sense of definition 2 ("d2-category") we are given all (local) arrow sets $$\hom(A,B)$$. The global arrow class $$\mathcal A$$ can be defined as the union of all $$\hom(A,B)$$. Condition (c) provides the functions $$\operatorname{dom}, \operatorname{cod} : \mathcal A \to \mathcal O$$.

This shows that a d2-category can be regarded as a d1-category in a natural way, and each locally small d1-category can be regarded as a d2-category in a natural way.

Whatever our preferences may be, an essential ingredient in the above definitions of a category is that each arrow $$f$$ has a unique domain object $$\operatorname{dom}(f)$$ and a unique codomain object $$\operatorname{cod}(f)$$. An absolute no go is that an arrow $$f$$ has multiple domain or codomain objects (or in other words, that the $$\hom(A,B)$$ are not pairwise disjoint).

Let us now come to the definition of $$\mathbf{Top}_*$$ given in your question. The $$\hom$$-sets are given by $$\text{hom}_{\mathbf{Top}_*}((X,x_0),(Y,y_0)) =\{f:X\to Y \mid f(x_0)=y_0 \text{ and } f \text{ is continuous}\}$$. At first glance it seems that we have defined a d2-category - but it is not true. Unfortunately the $$\hom$$-sets are not pairwise disjoint, thus we do not get a category. This leads us to ask

What happens if we omit requirement (c) in in definition 2 ?

Let us call a quadruple $$\mathbf A = (\mathcal O, \hom, id, \circ)$$ satisfying the weakened requirements an improper d2-category. There exist improper d2-categories like $$\mathbf{Top}_*$$ which are no d2-categories.

The danger with an improper d2-category is that one is tempted to define the class of all arrows as the union of all $$\hom(A,B)$$ - but this is inappropriate. If the $$\hom$$-sets are not pairwise disjoint, there do not exist unique objects $$\operatorname{dom}(f), \operatorname{cod}(f)$$ if $$f$$ belongs to more than one $$\hom$$-set. In other words, we cannot expect that the union of all $$\hom(A,B)$$ gives us "global arrows" containing information about domain and codomain. We only have "local arrows" for each pair of objects $$A,B$$ (given by the sets $$\hom(A,B)$$).

In the improper d2-category $$\mathbf{Top}_*$$ the union of all $$\hom$$-sets agrees in fact with $$\text{hom}_{\mathbf{Top}}$$ = class of global arrows in $$\mathbf{Top}$$, but this class has nothing to do with global arrows in $$\mathbf{Top}_*$$.

However, the devil's advocat argues that global arrows are not really needed. In fact, the composition law in 4. says more or less explictly that given objects $$A,B,C$$ and local arrows $$f \in \hom(A,B)$$ and $$g \in \hom(B,C)$$, we get a local arrow $$g \circ f \in \hom(A,C)$$. If requirements (a) and (b) are satisfied, everything works nicely. This is a real benefit of using local arrows. Definition 1 does not have this flexibility because it is based on global arrows; without the functions $$\operatorname{dom}, \operatorname{cod}$$ we cannot define composition of arrows.

As Thorgott writes in his answer, any improper d2-category can be easily transformed into a d2-category: Simply replace each morphism $$f \in \hom(A,B)$$ by the triple $$(A, f, B)$$. This results in a genuine d2-category. Clearly the functor $$\Phi$$ which is the identity on objects and takes $$f \in \hom(A,B)$$ to $$(A, f, B)$$ is an isomorphism of (improper) d2-categories.

Doing so for $$\mathbf{Top}_*$$ resolves your problem.

By the way, the same problem occurs with $$\mathbf{Top}$$. The $$\hom$$-sets are given by $$\text{hom}_{\mathbf{Top}}(X,Y) =$$ set of functions between the sets $$X$$ and $$Y$$ which are continuous with respect to the topologies on $$X$$ and $$Y$$. They are subsets of the $$\hom$$-sets $$\text{hom}_{\mathbf{Set}}(X,Y)$$ in the category of sets and are not pairwise disjoint.