Existence of Pre-Additive Topological categories Recall the notion of pre-addtive category and that of topological category (see Chapter 21 of The Joy of Cats). Well, my questions are:

*

*Does there exist an example of pre-additive topological category?


*If the answer to the previous question is affirmative, can you provide some reference on the general study of topological pre-additive categories, please?
Please, when answering, provide exhaustive point-by-point answers.
P.S. I'm extremely grateful to the user that provided in comments an answer concerning the fact that the category of binary relations is semiaddtive but not pre-additive.

If $U : C \to \text{Set}$ is a topological concrete category over $\text{Set}$ then in particular $U$ must have both a left and right adjoint (the "discrete topology" and "indiscrete topology") and so must preserve both limits and colimits; moreover (as stated by the nLab) $C$ is complete and cocomplete.  But if $C$ is a pre-additive category which has either all limits or all colimits then it must have a zero object (an object which is both initial and terminal), and $U$ applied to this object must be both the initial and terminal object of $\text{Set}$, which is impossible. So we conclude that no topological category over $\text{Set}$ can be pre-additive.
The natural question here is to ask for pre-additive topological categories over $\text{Ab}$ or similar, and here there are examples like topological abelian groups.
