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This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization).

Let $C$ be a category, let $W \in Mor(C)$ be a collection of morphisms, let $L_WC$ be the reflective localization of $C$. Are $C$ and and $L_WC$ equivalent categories? The inclusion functor $L_WC \hookrightarrow C$ is fully faithful and $L_WC$ has the same objects as $C$, what am I missing?

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$C$ and $L_W C$ having the same objects is a red herring and an artifact of the construction of $L_W C$. In particular, the "inclusion functor" won't be the identity on objects. Being fully faithful, it embeds $L_W C$ into $C$ in the categorical sense: it establishes an equivalence between $L_W C$ and a full subcategory of $C$, in this case the subcategory of $W$-local objects (proof).

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    $\begingroup$ How can the inclusion functor not be the identity on objects? $\endgroup$
    – The K
    Aug 15, 2015 at 4:38
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    $\begingroup$ Sometimes functors are not the obvious ones. $\endgroup$
    – Zhen Lin
    Aug 15, 2015 at 7:52
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    $\begingroup$ @TheK: inclusion was your choice of words. It's not inappropriate but it can be confusing. The domain and the image of a fully faithful functor are categorically the same (equivalent), which is why embedding is in the context an appropriate name for such functors and why nlab's $\hookrightarrow$ is an appropriate notation. Note that what we call inclusions in the rest of the mathematics often don't restrict to identities either: the way I was taught $ℕ \hookrightarrow ℤ$ certainly doesn't, and the natural number $1 = \{\{\}\}$ is different from the integer $1 = \{(n+1, n) : n ∈ ℕ\}$. $\endgroup$
    – user54748
    Aug 15, 2015 at 13:31
  • $\begingroup$ I think I see where the confusion was now, going to check out that question you linked to, thank you both. $\endgroup$
    – The K
    Aug 16, 2015 at 17:02

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