Re #38: I have no objections to the new phrasing, which is also more detailed.

Not all sources are equally relevant, though: the books of Gabriel–Zisman and Borceux are widely cited (over 2000 citations each on Google Scholar), the paper of Fritz not so much. If there is a more widely cited source, it would be beneficial to add it.

Perhaps the confusion that you pointed out stems from the fact that in the commutative case, a “fraction” is an expression of the form $s^{-1}x=x s^{-1}$, whereas in the noncommutative case, there are several options: $s^{-1}x$, $x t^{-1}$, $s^{-1}x t^{-1}$, and the most general $s_1^{-1}x_1s_2^{-1}x_2s_3^{-1}\cdots$. The latter expression can also be called a (noncommutative) fraction, I would say.

]]>I have touched the Idea-section (here) and the first few in the list of references (here)

]]>I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available.

No, this is incorrect.

If only there were “correct” and “incorrect” on matters of terminology.

In my experience, Hurkyl’s #33 is an accurate reflection of common use of terminology.

Here is another example:

- Tobias Fritz,
*Categories of Fractions Revisited*, Morfismos**15**2 (2011) 19-38 [arXiv:0803.2587, morfismos:vol15-n2-3]

who writes:

]]>Under certain conditions, the localization of a category with respect to a class of morphisms can be described in terms of “formal fractions”. If this construction is possible, the resulting localization is a category of fractions.

Renamed. Will now create a disambiguation page.

]]>Re #34:

I created the articles localization of a category and quasicategory of fractions.

]]>I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available.

No, this is incorrect. The article cites (in the lead) definitions of a category of fractions in Gabriel–Zisman and Borceux, both of which do not assume any calculus of fractions. In a category of fractions $C[W^{-1}]$, morphisms are formal compositions of morphisms in $C$ and formal inverses of morphisms in $W$, i.e., “fractions”.

When a calculus of fractions is available, the category of fractions can be computed using fractions of a very specific type, e.g., $W^{-1}C$, $CW^{-1}$, or $W^{-1}CW^{-1}$.

Cisinski, Kan, and Riehl use “localization” to mean “category of fractions”.

Lurie, and lots of people following Lurie, use “localization” to mean “reflective localization”.

Lurie’s usage actually seems to be prevalent in the literature using quasicategories, notwithstanding the examples of Cisinski, Kan, and Riehl.

]]>I think it would be better to rename this page “localization of a category”, and have “localization” be a disambiguation page. A quick search of the nLab shows 58 different articles which are “localizations” of some sort or other:

]]>I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available. “Localization” is the term I always hear; flipping through my references folder, some examples

- Cisinski,
*Higher Categories and Homotopical Algebra*, calls it localization (in the context of $\infty$-category theory via quasicategories) - Lurie,
*Higher Topos Theory*, uses “localization” specifically for reflective localization, but remarks in Warning 5.2.7.3 that some authors use it in the sense here. (in the context of $\infty$-category theory via quasicategories) - Dwyer, Hirschhorn, Kan, Smith,
*Homotopy Limit Functors on Model Categories and Homotopical Categories*, calls it “localization”, in the context of 1-category theory - Barwick, Kan,
*Relative Categories: Another Model for the Homotopy Theory of Homotopy Theories*, calls it “localization”, in the context of 1-category theory - Riehl,
*Categorical Homotopy Theory*, calls it “localization”, in the context of 1-category theory. Although somewhat informally; her language is oriented towards calling $C \mapsto C[W^{-1}]$ a “localization functor”

I don’t think the phrase “category of fractions” actually appears in any of the references I have. (but admittedly, I don’t collect references for basic 1-category theory)

]]>Adjusted the lead-in to make it clear that “category of fractions” is a more standard terminology.

I propose renaming this entry to “category of fractions”, since this is a more standard terminology, and “localization” is ambiguous in this context and has other meanings, especially in quasicategories. “Localization” could be made into a disambiguation page then.

]]>added pointer to Ex. 4.15 in

- Henning Krause,
*Localization theory for triangulated categories*, in proceedings of*Workshop on Triangulated Categories, Leeds 2006*[arXiv:0806.1324]

as an example of a localization of a locally small category which is not itself locally small

]]>added pointer to:

- William Dwyer,
*Localizations*, in:*Axiomatic, enriched and motivic homotopy theory*, NATO Sci. Ser. II**131**, Kluwer Acad. Publ. (2004) 3-28 [doi:10.1007/978-94-007-0948-5]

Removed the following comment, made by me some time ago, but never followed up

]]>David Roberts: This could probably be described as the fundamental category of 2-dimensional simplicial complex with the directed space structure coming from the 1-skeleton, which will be the path category above. In that case, we could/should probably leave out the paths of zero length.

Thanks! I added a redirect to make replacement axiom go to the right place.

]]>Added some “foundational” remarks on the construction of localizations of large categories, involving especially Scott’s trick (a page we don’t have yet).

]]>Yep!

]]>Yes, we should also have an article on Scott’s trick. (I can remember the trick itself, but never who it’s attributed to. According to Wikipedia the Scott is Dana?)

]]>Thanks; yes, I think that’s probably detailed enough. I can get started on that later.

We should also have a little article on the trick at the end, aka “Scott’s trick”.

]]>Fair enough; it does require a bit of trickery. Yes, a large category is something defined by class formulas. If I have a class formula specifying the morphisms and weak equivalences in a category, I can define a new class formula specifying zigzags of such morphisms with backwards-pointing weak equivalences, and another one specifying sequences of generating equalities between such zigzags. Now we need to take the “quotient” of the first proper class by the second proper-class equivalence relation, which we can’t do directly because the “equivalence classes” may be proper classes and hence cannot be elements of another class, but we can use an axiom-of-foundation trick and consider the sub-sets of each equivalence proper-class consisting of all elements of least rank therein.

Is that detailed enough?

]]>I learned a new word from #21. :-)

It might be worth spelling out the details behind that claim, because I think a lot of people would believe that the construction is foundationally tricky. When you refer to a large category in ZF, I guess you mean not a formal object but something defined by a class formula, and then the construction is also given by a class formula?

]]>I don’t think it depends on foundations. I’m pretty sure that even in ZF you can perform that construction (suitably frobnicated) on a large category and get a large (non-locally-small) category as output.

]]>Best, Alexander ]]>

Okay, thanks. Maybe I’ll find the time to add in something less telegraphic.

]]>I added a sentence.

]]>@Zhen Lin/Mike

Ah, that’s much better :-)

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