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?
 A: 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.)
A: 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.
A: 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:

*

*Objects: $A, B, C, \ldots$

*Arrows: $f, g, h, \ldots $

*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 $B = \operatorname{cod}(f)$.

*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 $g$.

*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

*

*a class $\mathcal O$, whose members are called $\mathbf A$-objects,

*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"],

*for each $\mathbf A$-object $A$, a morphism $A \stackrel{id_A}{\to} A$, called the  $\mathbf A$-identity on $A$,

*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.
