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In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the amalgamation of the underlying quiver of $C$ and $W^{op}$ with the objects of $W$, tautologously inverting $W$---is represented by a span $(~\overset{w}{\leftarrow}~\overset{\varphi}{\rightarrow}~)$ with $w \in W$ and $\varphi \in C$. Further, composition by exchange of `interior' cospan for span is associative up to the equivalence. And best of all, the general equivalence relation may be substituted for a much simpler one: $$ (~\overset{w}{\leftarrow}~\overset{\varphi}{\rightarrow}~) \sim (~\overset{v}{\leftarrow}~\overset{\psi}{\rightarrow}~) \iff \exists s,t \in C:~ ws=vt \in W,~ \varphi s=\psi t $$ This is all pretty cool and gives a much more tractable presentation of the localization. However, insofar as these $\sim$-classes of spans between $c$ and $d$ are expressible as: $$ C[W^{-1}](c,d) \cong \operatorname{colim}\limits_{(\circ \overset{w}{\to} c)\in W} C(\circ,d) $$ where the colimit is taken over the full subcategory of $C/c$ whose objects are morphisms from $W$, I don't see how the localization can fail to be (locally) small. $\operatorname{Set}$ is cocomplete and the overcategory $C/c$ is small.

Would appreciate someone pointing me toward my error(s)?

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  • $\begingroup$ Some motivation for the question: on the relevant nLab page mentions a result guaranteeing local smallness provided: - $W$ admits a calculus of right fractions - $C$ admits small filtered colimits - $W/c$ is cofinally small In context of the question, this seems unnecessary. $\endgroup$
    – Tyler Bryson
    Commented Jan 17, 2019 at 19:59
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    $\begingroup$ Where did you read that it wasn't small? I think everything is okay when C is small. The problem is that if C is large but locally small, the localization isn't locally small. Usually when you perform localization, it's done with respect to large locally small categories. The way that you escape smallness is that the zig-zags can generally range over all objects of C, so you get a locally large thing because C is large. $\endgroup$
    – Harry Gindi
    Commented Jan 17, 2019 at 20:00
  • $\begingroup$ There are lots of warnings scattered across the nLab pages about local smallness in localizations...including explicit warnings about smallness on both the localization page and the calculus of fractions page. Just doing my due diligence in case I am missing something serious. $\endgroup$
    – Tyler Bryson
    Commented Jan 17, 2019 at 20:05
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    $\begingroup$ That's because when performing localization, the motivating examples are large and come from homotopy theory. $\endgroup$
    – Harry Gindi
    Commented Jan 17, 2019 at 20:07

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If $C$ is small, $C[W^{-1}]$ is again small, even without a calculus of fractions. The problem in general is that when $C$ is a locally small large category, the zig-zag sequences can range over finite sequences of objects of $C$, which form a large set.

Essentially, the problem is that the hom-sets in the localization before taking quotients can, without additional assumptions, have the cardinality of the set of objects, and there may be sets of morphisms even after quotienting with cardinality equal to the cardinality of C.

This is a problem when C is large, since it means that the hom-sets can be large.

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  • $\begingroup$ Excellent. I've worrying over this for the past hour thinking I'd missed some essential point. $\endgroup$
    – Tyler Bryson
    Commented Jan 17, 2019 at 20:14

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