Applying the universal object definition to topological spaces(especially product topology space).

Question

The universal property of product topological space is as follows:

Let $Z,~Y_\alpha(\alpha\in A)$ be topological spaces. For every $g_\alpha: Z\to Y_\alpha$ continuous, there exists exactly one $g:Z\to \prod_{\alpha\in A}Y_\alpha$ such that $p_\alpha\circ g=g_\alpha$ for every $\alpha\in A$.

And the definition of universal/couniversal object in category theory is:

So applying such definition in the previous context, is it to say that $\prod_{\alpha\in A}Y_\alpha$ is a couniversal object in the topological space category?

Answer 1

No, it is not. A one-point space would be terminal: given any space $X$, there is one and only one continuous map $X \to \{*\}$, namely the constant map. There will typically be MANY maps of the form $f: Z \to \prod_\alpha Y_\alpha$, given only $Z$ and the $Y_\alpha$. The product is the unique "receiver" of a space $Z$ subject to satisfying the conditions you state, given the input maps $Z \to Y_\alpha$. This is very different from the terminal definition.

For example, how many continuous maps are there $f: \mathbb{R} \to \mathbb{R} \times \mathbb{R}$? Clearly, infinitely many. So, the euclidean plane is not a terminal object. Now, how many maps are there $f: \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ that are continuous and also satisfy $f(x) = (\sin x, x^2)$? Clearly only one. That's the difference: the second question contains much more data/constraints on $f$. This is the universal property of the product of spaces.

To be fair, products and terminal objects are special cases of "limits" in category theory. Disjoint unions and initial objects are special cases of "colimits." So yes, terminal objects and products are of the same "flavor" but the input data is very different.

Answer 2

The product space is (a component of) a terminal object, but not in the category $\mathbf{Top}$. For a set $I$, let $\mathbf{Top}^I$ be the category of $I$-indexed families of topological spaces, and let $\Delta:\mathbf{Top}\to\mathbf{Top}^I$ be the functor sending the space $A$ to the family that takes the value $A$ at every $i\in I$. Then the product of a family of spaces $(X_i)_{i\in I}$ exists if and only if there is a terminal object in the comma category $(\Delta\downarrow X)$, where $X$ is the family $(X_i)_{i\in I}$ considered as an object of $\mathbf{Top}^I$. If you unpack what such an object actually is, this is just the product space and projections all packaged together as a single object.

This is a common pattern with the paradigm cases of "objects with universal properties": the objects themselves are rarely terminal or initial in their categories, but they are key pieces of an initial or terminal object in some related category in which that object is packaged with some other data. This isn't the definition of universal property -- generally, "universal property" is an informal phrase -- but it is a nice deep fact about the various things we call universal properties/universal constructions.