Say we are localizing a category $\mathcal{B}$ at the $P$-morphisms, where by $P$-morphism I mean a morphism having some property $P$.

Fix a category $\mathcal{C}$. Is it true that the category of functors from the resulting category of fractions of $\mathcal{B}$ to $\mathcal{C}$ is isomorphic to the category of functors from $\mathcal{B}$ to $\mathcal{C}$ that turn $P$-morphisms to isomorphisms?

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    $\begingroup$ That is the definition of localisation. The category of fractions is just one way of explicitly constructing it. $\endgroup$ – Zhen Lin Sep 19 '15 at 11:50

As Zhen suggested: the definition of a localization of $\mathcal C$ at $S$ is that the functor categories $[\mathcal C[S^{-1}],\mathcal D]$ are isomorphic to the categories of functors $[\mathcal C,\mathcal D]$ sending $S$ to isomorphisms, via composition with the localization functor.

  • $\begingroup$ Oh, ok, thanks! So can I still speak about the ``localization of $\mathcal{C}$ at $S$'', even if the morphisms of $S$ do not satisfy the axioms of a multiplicative system (say, they don't compose)? If I understand what you are saying, we still get a category $\mathcal{C}[S^{-1}]$ but in the case I describe above we just don't know what it looks like - in other words, we can not explicitly construct it. Correct? $\endgroup$ – BenBarrows Sep 20 '15 at 10:39
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    $\begingroup$ You can explicitly construct it, to the extent that the general localization construction is explicit, by localizaing at the multiplicative system generated by $S$, in the same way as happens in rings. $\endgroup$ – Kevin Carlson Sep 20 '15 at 15:39

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