This is yet more on "algebraic objects in functional analysis".

Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable category representing functors from that category to CompHaus.

An obvious such functor is the spectrum of a unital $C^*$-algebra. This seems to imply that $\mathbb{C}$ is a *compact Hausdorff object* in the category of unital $C^*$-algebras. So:

Question 1: Is this right?

Followed by the obvious:

Question 2: Are there any other interesting "Compact Hausdorff" objects in other categories?

Similarly, $C^\ast$-algebras is algebraic, and whilst Banach spaces isn't algebraic then it embeds in an algebraic theory (of totally convex spaces). Again, to any compact Hausdorff space one can assign its $C^\ast$-algebra of continuous functions to $\mathbb{C}$. This suggests that $\mathbb{C}$ is a "$C^\ast$-algebra" object in CompHaus - except that $\mathbb{C}$ is not a compact Hausdorff space. However, we have a way out due to the way that $C^\ast$-algebras are algebraic: it's the unit ball that we should be thinking of and this is continuous functions to the closed unit disc in $\mathbb{C}$, which is compact Hausdorff. Thus $\{z \in \mathbb{C} : |z| \le 1\}$ seems to be a $C^\ast$-algebra object in Compact Hausdorff spaces. Again:

Question 3: Is this right?

and

Question 4: Are there any other interesting "$C^\ast$-algebra" objects in other categories?

and

Question 5: Are there any "Banach space" objects (or "totally convex space" objects) floating around anywhere?

nonexistentnLab page! $\endgroup$ – Reid Barton Feb 10 '10 at 20:37