Universal property of initial topology

I'm learning some category theory and I thought I had understood universal objects but maybe I have not because I cannot write down the definition of initial topology in terms of categories.

My understanding of universal objects so far: I thought universal objects are either initial or terminal objects in a comma category $(O \downarrow F)$ where $O$ is a selection functor $\textbf{1} \to O \in C$ and $F$ is a forgetful functor into $C$. In the example of a free group, $F: \textbf{Group} \to \textbf{Set}$ and $O$ is the set that generates the free group. The objects in this category are pairs $(f, G)$ where $f$ is a (set-theoretic) map from $S \to G$.

Now I would like to do the same for the initial topology but I'm struggling to see what the objects are and also what $F$ should be. $O$ is probably going to be $X$, the space we would endow with the topology. So we probably want $F$ to be a functor $\textbf{Top} \to \textbf{Set}$. The objects are probably $(f,Y)$ where $f$ is a map $X \to Y$ (or $Y \to X$?) such that there exists a continuous map $f^\prime$ with $F(f^\prime) = f$.

Is this right? If not: what are the objects? And is the initial topology an initial or a terminal object?

Edit

I think what I wrote above is wrong: I get one commutative diagram for each $Y_i$ in the family with respect to which the initial topology is defined. This means that the objects are $(g_i, Z)$ where $g_i : Z \to Y_i$ and the comma category would be $(Y_i, F)$. Is this right?

The diagram for $(Y_i \downarrow F)$ would look like this:

X --(f_i)--> Y_i
^
|
|
(g_i)
|
|
Z


Then the question is whether the unique continuous map $\varphi$ is $Z \to X$ or $X \to Z$.

• If I'm not mistaken, en.wikipedia.org/wiki/Initial_topology#Categorical_description, is what you want. Apr 29, 2012 at 9:07
• @MartinWanvik Thank you! It might be but at the moment I don't understand yet whether this is the same as what I wrote above or not because I don't know much about category theory yet and I don't know what cones are or adjoint functors. I will have to think about this. Apr 29, 2012 at 9:48
• I'll write an answer, trying to expand a bit on the description given on the wikipedia page. Apr 29, 2012 at 9:57
• @ClarkKent: The universal property of the initial topology looks like a limit, so if it's going to be given by a limit, it has to be a terminal object! Apr 29, 2012 at 15:16
• @ZhenLin Thank you very much for pointing this out! May 3, 2012 at 10:41

(This is an expanded version of this).

First of all, the given collection of spaces $$\{ Y_j \}_{j \in J}$$ can be viewed as a functor $$Y: \mathbf{J} \to \mathbf{Top}$$, where $$\mathbf{J}$$ is simply the indexing set $$J$$ viewed as a discrete category (i.e. one whose objects are $$J$$ and containing only identity morphisms). In other words, we have $$Y(j) = Y_j$$ and $$Y(\operatorname{id}_j) = \operatorname{id}_{Y(j)}$$, which is trivially a functor. The other part of the description of the initial topology is the collection of maps $$f_j: X \to Y_j$$ where $$X$$ is a set. More formally, if $$U: \mathbf{Top} \to \mathbf{Set}$$ is the forgetful functor, then we can also write $$f_j: X \to U(Y_j)$$ (this takes place in the category $$\mathbf{Set}$$). As the article mentions, the collection $$\{f_j\}$$ can be viewed as a cone from $$X$$ to $$UY$$. Here comes the definition:

Given categories $$\mathbf{C},\mathbf{D}$$, and a functor $$F: \mathbf{D} \to \mathbf{C}$$, a cone from an object $$X$$ in $$\mathbf{C}$$ is a natural transformation $$\eta: \Delta(X) \Rightarrow F$$, where $$\Delta: \mathbf{C} \to \mathbf{C^D}$$ is the functor mapping each object $$Z$$ in $$\mathbf{C}$$ to the constant functor $$\Delta(X): \mathbf{D} \to \mathbf{C}$$ (which takes each object in $$\mathbf{D}$$ to $$Z$$ and each morphism in $$\mathbf{D}$$ to the identity on $$Z$$). If $$f: V \to W$$ in $$\mathbf{C}$$, then $$\Delta(f)$$ is the natural transformation $$\Delta(f): \Delta(V) \Rightarrow \Delta(W)$$ whose component at each object is $$f$$. For each functor $$F$$, we define the category of cones $$\mathbf{Cone}(F)$$ as the comma category $$(\Delta \downarrow F)$$.

Now, it is easy to see that the collection $$f_j: X \to UY(j)$$ can be viewed as a cone from $$X$$ to $$UY$$, i.e an object in $$\mathbf{Cone}(UY)$$. In other words, it defines a natural transformation $$f: \Delta(X) \Rightarrow UY$$ (here $$\Delta(X)$$ is a functor on the category $$\mathbf{J}$$, described previously). Similarly, a collection of continuous maps $$g_j: Z \to Y_j$$ defines an object $$\mathbf{Cone}(Y)$$. Applying the forgetful functor $$U: \mathbf{Top} \to \mathbf{Set}$$, we get a forgetful functor $$U': \mathbf{Cone}(Y) \to \mathbf{Cone}(UY)$$.

Now we are in a position to describe the universal property of the initial topology: as mentioned, we start with an object $$f_j: X \to UY$$ in $$\mathbf{Cone}(UY)$$. We will write this as $$(X,f)$$, where $$f = \{f_j: X \to Y_j\}.$$ Now, a morphism of cones $$\tau: (Z,g) \Rightarrow (X,f)$$ is simply a morphism $$\tau: Z \to X$$ such that $$f_j \circ \tau = g_j$$. Imposing the initial topology (coming from the maps $$f_j$$) on $$X$$, we get an object $$I(X,f)$$ in $$\mathbf{Cone}(Y)$$. Moreover, the identity map defines a morphism $$\varepsilon: U'(I(X,f)) \Rightarrow (X,f)$$ (in $$\mathbf{Cone}(UY)$$), which is universal in the following sense: whenever $$\eta: U'(Z,g) \Rightarrow (X,f)$$ is another morphism in $$\mathbf{Cone}(UY)$$, there exists a unique morphism $$\xi: (Z,g) \Rightarrow (X,f)$$ such that $$\eta = \varepsilon \circ \xi$$. In other words, if we are given

a topological space $$Z$$, a collection of continuous maps $$g_j: Z \to Y_j$$, and a map (a morphism in $$\mathbf{Set}$$) $$\eta: Z \to X$$ such that $$f_j \circ \eta = g_j$$,

then

there exists a (unique) continuous map $$\xi: Z \to X$$, satisfying $$f_j \circ \xi = g_j$$ for all $$j \in J$$, such that $$\varepsilon \circ \xi = \eta$$.

Of course, since $$\varepsilon$$ is secretly the identity map on $$X$$, this implies that $$\xi = \eta$$ (as set functions, or more accurately, $$U(\xi) = \eta$$), so all this is saying is that $$\xi$$ is continuous if $$f_j \circ \eta = f_j \circ \xi$$ is continuous for all $$j \in J$$ ($$f_j \circ \xi = g_j$$, which is the original collection of continuous maps specified above). The converse is of course immediate, since a composite of continuous maps is again continuous. So this highly elaborate categorical language is only saying that

a map $$\eta: Z \to X$$ is continuous if and only if $$f_j \circ \eta$$ is continuous for all $$j \in J$$.

This also implies that imposing the initial topology defines a functor $$I: \mathbf{Cone}(UY) \to \mathbf{Cone}(Y)$$, and this is right adjoint to the functor $$U'$$ (and, in fact any right adjoint functor can be described by a similar universal property), but that is another story.

• Thank you very much. I will take some time to read it! May 3, 2012 at 10:41
• So, is the initial topology an initial object or not?? Mar 22, 2014 at 17:09
• @AlexanderFrei $\big( I(X,f), \epsilon \big)$ is a terminal "arrow", i.e. a terminal object in the comma category $(U' \downarrow X)$. That's funny actually... (initial in the opposed category I guess) Jul 6, 2017 at 22:38
• Just to let you know I used your beautiful answer to expand the wiki article on the initial topology. Thanks a lot. :) May 6, 2019 at 3:47
• @Noix07 I'm reading Riehl's book and she says: The most basic formulation of a universal property is to say that a particular object deﬁnes an initial or terminal object in its ambient category, is she talking about being an initial or terminal object in the comma category? Apr 9, 2021 at 2:34