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I have spelled out statement and proof of:
every reflection is a reflective localization,
reflective localizations are given by full subcategories of local objects.
I have also tried to produce decent cross-links to various entries that mention something related. But there will be more left to do.
Conversely, when is reflection onto a full subcategory of local objects also a localization in the sense of universally inverting the given morphisms?
I have to puzzle through the terminology – you’re asking if, given a category with weak equivalences $(C,W)$ such that the full subcategory $Loc_W$ of $W$-local objects in reflective, whether the reflector $C \to Loc_W$ is equivalent to the localization functor $C\to C[W^{-1}]$ that universally inverts $W$?
By property (1), it seems that this is equivalent to asking whether every functor $C\to D$ that inverts $W$ also inverts every $W$-local equivalence (morphism that is seen as an isomorphism by every $W$-local object)?
Yes, for $W$ any class of morphisms. In “Modalities in homotopy type theory” you write: “We will not be concerned here with the universal property of the localized category”. Suppose you were concerned, what could you say?
Well, it is true in general that every left adjoint $F:C\to D$ that inverts $W$ also inverts every $W$-local equivalence. For if $G$ is its right adjoint, then $F$ inverting $W$ means that $G$ takes values in $W$-local objects, which in turn implies that $F$ preserves $W$-local equivalences. Since the reflector $C\to Loc_W$ is also a left adjoint, it follows that it is the coinverter of $W$ in the 2-category of categories and left adjoints (which contains, in particular, the 2-category of locally presentable categories and left adjoints).
However, for non-adjoints I think it is false; here is a proposed counterexample. Let $P$ be a poset with binary meets but no top element, and let $C = P^\triangleright$ be $P$ with a new top element $\top$ adjoined. Take $W = Mor P \subsetneq Mor C$, i.e. every inequality $x\le y$ in $C$ where $y\neq \top$. Then for any $x\in P$ there is a $y\in P$ with $y\nleq x$ (since $P$ has no top element), but $x\wedge y \le x$, so $x$ does not invert the weak equivalence $x\wedge y \le y$; thus no element of $P$ is $W$-local. But of course $\top$ is $W$-local, so $Loc_W = \{\top\}$ is contractible and reflective. However, $C[W^{-1}] = P[W^{-1}]^\triangleright$, which is not contractible unless $P = \emptyset$; thus they are inequivalent.
Have not read the thread or page closely, so might be missing something, but is this not just a question of whether $W$ is saturated? If I recall correctly, it is an if and only if.
Hi Mike, thanks for the bit about left adjoints. I was just wondering about this being the right condition, since this is how Hovey’s book defines localization.
Richard, yes, every reflector is the localization at the morphisms it inverts, but I was hoping for a tighter condition. I suppose localization by left adjoins is useful here.
Yes, perhaps one could say that one is asking for conditions to ensure saturation. The one at saturated class of morphisms is no doubt already familiar to you.
Okay, I have recorded the statement about left-adjoint localization, here.
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