Do localizations of categories always preserve initial objects and terminal objects? If so, is this recorded?
Hello everyone, This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training! My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way. However, inspecting our arguments, they seem to work in much greater generality, leading to: CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}]. Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument. Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects. Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this. Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim. Regards Yemon -- Dr. Y. Choi School of Mathematical Sciences Lancaster University Bailrigg, Lancaster Lancashire LA1 4YF
HI Yemon perhaps https://link.springer.com/article/10.1007/BF00872988 will help? Localisation being a coinverter in Cat (a type of weighted colimit), and a terminal/initial object being an adjunction involving the terminal category (left or right depending on initial or terminal), you have a general setup of shape: C ---> C[W^{-1}] ^ ^ || | v | 1 <-------- where the double vertical arrows represent the adjunction, and the two-headed arrow represents a) the universal map to the category 1, and b) the composite 1 -> C -> C[W^{-1}] . Then you have shown by hand that that latter pair of functors is also an adjunction, and this is the generality I expect the "morally correct" answer to live in . This setup is immediately attackable in terms of Cat-enriched tech, I think. hope that helps, David David Michael Roberts E: droberts.65537@gmail.com W: https://thehighergeometer.wordpress.com
On 16 Aug 2026, at 7:01 AM, Yemon Choi via Categories <categories-list@categories.org.au> wrote:
CAUTION: External email. Only click on links or open attachments from trusted senders.
Hello everyone,
This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training!
My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way.
However, inspecting our arguments, they seem to work in much greater generality, leading to:
CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}].
Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument.
Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects.
Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this.
Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim.
Regards Yemon
-- Dr. Y. Choi School of Mathematical Sciences Lancaster University Bailrigg, Lancaster Lancashire LA1 4YF _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
I'm not sure I understand David's argument, in particular I don't see a "two-headed arrow" apart from the left vertical double arrow. Maybe the diagram got messed up in transmission? Here is another argument, which I propose as an answer to Q2: The crucial observation is that localization functors are *co-fully-faithful* (aka *absolutely dense*), ie precomposition with localization functors is fully faithful. This is follows directly from Lemma 1.2 on page 7 of Gabriel-Zisman. Let F : C --> D be co-fully-faithful, and 1 terminal in C. F1 is terminal in D iff the identity functor of D admits a cocone with vertex F1, ie a natural transformation id_D --> c_F1 to the functor with constant value F1, and furthermore the F1-component of this natural transformation is the identity arrow of F1 (see here <https://ncatlab.org/nlab/show/initial+object#cone>, which I learned from here <https://homotopytypetheory.org/2018/11/26/impredicative-encodings-part-3/>). Now since F is co-fully-faithful this is equivalent to the existence of a natural transformation from F to the constant-F1-functor on C whose component at 1 is id_F1, and such a natural transformation is obtained by whiskering the transformation id_C --> c_1 with F. Actually the argument can be further decomposed: one can show that (a) every co-fully-faithful functor is *final*, and (b) every final functor preserves terminal objects. To show the two claims, one has to use the following equivalent reformulation of finality: A functor F : C --> D is final iff for every functor G : D --> E and object e in E, the canonical function (1) cocone(G,e) --> cocone (GF,e) is a bijection. This condition is easily seen to be an instance of the co-fully-faithfulness of F and obviously implies that G and GF have the same colimit whenever one exists, which is a classical definition of finality. Conversely, from the characterization of finality in terms of connectness of the comma categories d/F one can show the bijectivity of the maps (1). Best, Jonas On Sun, 16 Aug 2026 at 00:23, David Roberts via Categories < categories-list@categories.org.au> wrote:
HI Yemon
perhaps https://link.springer.com/article/10.1007/BF00872988 will help?
Localisation being a coinverter in Cat (a type of weighted colimit), and a terminal/initial object being an adjunction involving the terminal category (left or right depending on initial or terminal), you have a general setup of shape:
C ---> C[W^{-1}] ^ ^ || | v | 1 <--------
where the double vertical arrows represent the adjunction, and the two-headed arrow represents a) the universal map to the category 1, and b) the composite 1 -> C -> C[W^{-1}] . Then you have shown by hand that that latter pair of functors is also an adjunction, and this is the generality I expect the "morally correct" answer to live in .
This setup is immediately attackable in terms of Cat-enriched tech, I think.
hope that helps,
David
David Michael Roberts
E: droberts.65537@gmail.com W: https://thehighergeometer.wordpress.com
On 16 Aug 2026, at 7:01 AM, Yemon Choi via Categories < categories-list@categories.org.au> wrote:
CAUTION: External email. Only click on links or open attachments from trusted senders.
Hello everyone,
This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training!
My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way.
However, inspecting our arguments, they seem to work in much greater generality, leading to:
CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}].
Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument.
Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects.
Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this.
Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim.
Regards Yemon
-- Dr. Y. Choi School of Mathematical Sciences Lancaster University Bailrigg, Lancaster Lancashire LA1 4YF _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
Hi, yes, sorry, the formatting was broken. It should have been a diagonal arrow between 1 and the localisation, but I made it a mirrored L shape instead, and it was meant to be a cheap ascii replacement for second pair of (anti-)parallel arrows. I was hoping to place the question more precisely in the context of the 2-categorical setup that Yemon hinted at, and the setting of the paper I linked, so that 'category with structure' was in this case 'category with terminal object' i.e. equipped with an appropriate adjunction to 1. Apologies for the garbled mess, and not actually answering properly! All the best, David Michael Roberts E: droberts.65537@gmail.com W: https://thehighergeometer.wordpress.com
On 19 Aug 2026, at 10:20 AM, Jonas Frey <jonas743@gmail.com> wrote:
I'm not sure I understand David's argument, in particular I don't see a "two-headed arrow" apart from the left vertical double arrow. Maybe the diagram got messed up in transmission?
Here is another argument, which I propose as an answer to Q2:
The crucial observation is that localization functors are *co-fully-faithful* (aka *absolutely dense*), ie precomposition with localization functors is fully faithful. This is follows directly from Lemma 1.2 on page 7 of Gabriel-Zisman.
Let F : C --> D be co-fully-faithful, and 1 terminal in C. F1 is terminal in D iff the identity functor of D admits a cocone with vertex F1, ie a natural transformation id_D --> c_F1 to the functor with constant value F1, and furthermore the F1-component of this natural transformation is the identity arrow of F1 (see here <https://ncatlab.org/nlab/show/initial+object#cone>, which I learned from here <https://homotopytypetheory.org/2018/11/26/impredicative-encodings-part-3/>). Now since F is co-fully-faithful this is equivalent to the existence of a natural transformation from F to the constant-F1-functor on C whose component at 1 is id_F1, and such a natural transformation is obtained by whiskering the transformation id_C --> c_1 with F.
Actually the argument can be further decomposed: one can show that (a) every co-fully-faithful functor is *final*, and (b) every final functor preserves terminal objects. To show the two claims, one has to use the following equivalent reformulation of finality: A functor F : C --> D is final iff for every functor G : D --> E and object e in E, the canonical function (1) cocone(G,e) --> cocone (GF,e) is a bijection. This condition is easily seen to be an instance of the co-fully-faithfulness of F and obviously implies that G and GF have the same colimit whenever one exists, which is a classical definition of finality. Conversely, from the characterization of finality in terms of connectness of the comma categories d/F one can show the bijectivity of the maps (1).
Best,
Jonas
On Sun, 16 Aug 2026 at 00:23, David Roberts via Categories <categories-list@categories.org.au <mailto:categories-list@categories.org.au>> wrote:
HI Yemon
perhaps https://link.springer.com/article/10.1007/BF00872988 will help?
Localisation being a coinverter in Cat (a type of weighted colimit), and a terminal/initial object being an adjunction involving the terminal category (left or right depending on initial or terminal), you have a general setup of shape:
C ---> C[W^{-1}] ^ ^ || | v | 1 <--------
where the double vertical arrows represent the adjunction, and the two-headed arrow represents a) the universal map to the category 1, and b) the composite 1 -> C -> C[W^{-1}] . Then you have shown by hand that that latter pair of functors is also an adjunction, and this is the generality I expect the "morally correct" answer to live in .
This setup is immediately attackable in terms of Cat-enriched tech, I think.
hope that helps,
David
David Michael Roberts
E: droberts.65537@gmail.com <mailto:droberts.65537@gmail.com> W: https://thehighergeometer.wordpress.com <https://thehighergeometer.wordpress.com/>
On 16 Aug 2026, at 7:01 AM, Yemon Choi via Categories <categories-list@categories.org.au <mailto:categories-list@categories.org.au>> wrote:
CAUTION: External email. Only click on links or open attachments from trusted senders.
Hello everyone,
This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training!
My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way.
However, inspecting our arguments, they seem to work in much greater generality, leading to:
CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}].
Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument.
Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects.
Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this.
Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim.
Regards Yemon
-- Dr. Y. Choi School of Mathematical Sciences Lancaster University Bailrigg, Lancaster Lancashire LA1 4YF _______________________________________________ Categories mailing list -- categories-list@categories.org.au <mailto:categories-list@categories.org.au> To unsubscribe send an email to categories-list-leave@categories.org.au <mailto:categories-list-leave@categories.org.au>
_______________________________________________ Categories mailing list -- categories-list@categories.org.au <mailto:categories-list@categories.org.au> To unsubscribe send an email to categories-list-leave@categories.org.au <mailto:categories-list-leave@categories.org.au>
Dear Yemon, there is a direct argument: the localization functor RelCat -> Cat preserves the terminal object. Moreover, it is not just a functor, but a 2-functor so it preserves adjunctions. In particular if C -> 1 is a left/right adjoint, then so is C[W^{-1}] -> 1. In fact, the localization functor preserves all finite products, which similarly implies that if C has finite products/coproducts, then so does C[W^{-1}]. This can be checked by direct computation or more conceptually by using cartesian closedness of RelCat and Cat or the fact that the right adjoint Cat -> RelCat is an exponential ideal. See: https://mathoverflow.net/a/44155 https://nforum.ncatlab.org/discussion/4769/connected-components-preserve-fin... https://ncatlab.org/nlab/show/exponential+ideal Best, Karol On 2026-08-15 23:31, Yemon Choi via Categories wrote:
Hello everyone,
This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training!
My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way.
However, inspecting our arguments, they seem to work in much greater generality, leading to:
CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}].
Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument.
Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects.
Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this.
Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim.
Regards Yemon
participants (4)
-
David Roberts -
Jonas Frey -
Karol Szumiło -
Yemon Choi