So each publish it will likely be published in a single category, as well as for each publish I wish to obtain the published in category permalink, just the URL without markup. How do i do this?

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Well the publish is published inside a category, right? Well I have to reveal that category, although not the title from the category, the url for your category.

I'd use get_the_category(), that will return a range of objects, one for every category. Inside your situation, you will get an assortment that contains one object back, since you have only one category. Then use get_category_link() to turn the ID from the category right into a URL.

So, if you are in The Loop, I'd make a move such as this (NB: untested!)

$categories = get_the_category();
$url = get_category_link($categories[0]->cat_ID);

Clearly, you will want to make certain this does not fall on its backside if you will find no groups designated, for instance.

This is comparable to what WordPress does should you incorporate a category inside your permalink structure -- it'll obtain the category using the cheapest number ID and employ that within the permalink.