# Localisation of Category

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?

• Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it. – Eric Wofsey Apr 18 at 18:13

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.

• For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains. – Eric Wofsey Apr 18 at 18:15
• @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that. – Kevin Carlson Apr 18 at 18:20
• do you know a recomendable source discussing the explicite construction – KarlPeter Apr 20 at 2:03
• @KarlPeter It's a very simple construction, simpler even than the localization of a commutative ring. But I believe it's discussed at some length in Manin's homological algebra book. – Kevin Carlson Apr 20 at 6:24