# Is there a coproduct in the category of path connected spaces?

Well, first of all, does the coproduct exist in the category of path-connected spaces, and if not how would you prove it? If it does exist what is it and how do you find it? The usual coproduct of topological spaces is disjoint union, but it is not path-connected.

As a follow up question, given a full subcategory of a category in which products/coproducts do exist, what is the strategy for finding out if that category also has products/coproducts?

They don't exist. For instance, suppose there existed a coproduct $Y=X\coprod X$, where $X$ is a point. Then there would be a unique map $f:Y\to [0,1]$ sending the first copy of $X$ to $0$ and the second copy of $X$ to $1$. Since $Y$ is path-connected, $f$ must be surjective. But now let $g:[0,1]\to[0,1]$ be any non-identity map that sends $0$ to $0$ and $1$ to $1$. Then $gf$ is another map $Y\to[0,1]$ sending the first copy of $X$ to $0$ and the second copy of $X$ to $1$, and $gf\neq f$ since $f$ is surjective. This is a contradiction.
• The exact same argument works for connected spaces (you can still conclude that $f$ is surjective from $Y$ being connected). – Eric Wofsey Dec 19 '18 at 21:07