I have a quite broad question about localisations of categories:
Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $\frac{ac}{b^{-1}d^{-1}}$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.