All Questions
Tagged with localization ct.category-theory
49 questions
13
votes
2
answers
665
views
Categories in which isomorphism of stalks does not imply isomorphism of sheaves
Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams.
For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
- 15.9k
5
votes
1
answer
184
views
Commutativity of pairs of reflective localizations
Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two
two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$-local}: i_1$ and $L_2: C \overset{\...
- 719
1
vote
2
answers
220
views
Presentable categories as colimits of finitely presentable categories
I am trying to understand the relationship betweeen compactly generated presentable categories, also called finitely presentable categories, and general presentable categories (which I have less ...
- 719
6
votes
0
answers
180
views
Does the localization functor $\mathcal{C}\to S^{-1}\mathcal{C}$ preserve finite colimits when $\mathcal{C}$ is not small? (size issues in proof)
$\def\colim{\operatorname{colim}}
\def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a left multiplicative system. The ...
4
votes
0
answers
198
views
Localizations that are endofunctors
Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...
- 41
8
votes
1
answer
439
views
Model categories as a tool to resolve size issues for localizing categories
I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
- 5,966
3
votes
1
answer
224
views
Kernels and cokernels in a quotient of an abelian category
I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
- 323
5
votes
1
answer
235
views
Hammock localization and free adjoints
The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal
{W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical ...
- 42.4k
6
votes
2
answers
390
views
Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category
When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the ...
- 5,966
3
votes
0
answers
96
views
Cohn's localization for rings with enough idempotents
I am in the following situation: I have a non-unitary (associative) ring $R$ with enough idempotents or, if you prefer, a small pre-additive category. Actually, I even know that $R$ is right coherent (...
- 2,452
7
votes
2
answers
593
views
Overloading of the word "local" in category theory
The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the ...
- 511
2
votes
1
answer
171
views
Are hammock localizations locally truncated?
Let us take a relative category $(\mathcal{C},\mathcal{W})$, and consider its hammock localization $L_H \mathcal{C}$. It seems to me that for every two objects $X,Y \in \mathcal{C}$ the mapping ...
- 1,071
7
votes
1
answer
303
views
Localisation of categories, but instead of "isomorphisms" I want "morphisms with right inverse". Construction via calculus of fractions possible?
Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ ...
- 696
12
votes
1
answer
430
views
Can the category of S-local objects be reflective but not a localization by S?
This is cross-posted from MSE (and substantially re-written) after receiving no answers.
Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of ...
- 295
9
votes
1
answer
179
views
Morphisms of hammocks in the simplicial localization
Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$.
In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined to ...
- 457
8
votes
1
answer
394
views
When is an $\infty$-categorical localization left exact?
Let $L: \mathcal C^\to_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, $L$ is left exact iff the class of $L$-local morphisms is stable under base ...
- 64k
1
vote
1
answer
145
views
Elementary example of right localization of functor
I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.
$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\...
- 2,368
7
votes
0
answers
159
views
When do zigzags of weak equivalences detect isomorphisms in the localization?
The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question.
...
- 4,459
4
votes
0
answers
95
views
A variation of the hammock localization
Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...
- 4,459
4
votes
0
answers
105
views
Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?
Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
- 9,019
4
votes
0
answers
175
views
Do we have criteria of strict localization of a Grothendieck category?
Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
- 9,019
23
votes
2
answers
2k
views
What is the correct definition of localisation of a category?
Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the ...
- 453
7
votes
2
answers
374
views
When a localization of a category is (non-)reflective?
Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...
- 4,459
9
votes
4
answers
2k
views
Localization of $\infty$-categories
In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
- 3,019
5
votes
1
answer
277
views
Factorization of Gabriel-Zisman localization construction?
My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$.
The localization ...
- 595
7
votes
3
answers
1k
views
Are localization functors always essentially surjective?
Let $\mathcal{C}$ be a category and $\mathcal{W} \subseteq \text{Arr}(\mathcal{C})$ a set (or class) of arrows.
There are (at least) two notions of localization of $\mathcal{C}$ with respect to $\...
- 1,704
4
votes
0
answers
216
views
Checking a monad is idempotent
I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...
- 13.5k
10
votes
1
answer
336
views
How to identify localization of categories?
Let $C, D$ be categories with finite limits, $F:C\to D$ be a essentially surjective functor that commutes with finite limits, and let $S$ be the set of morphisms of $C$ that become isomorphisms in $D$ ...
- 629
3
votes
2
answers
691
views
Localization of a symmetric monoidal category is monoidal when the morphisms being inverted are closed under tensor product
In the answer to question Localization of symmetric monoidal category, it was mentioned that '' Assuming that the tensor product of two morphisms in $S$ is again in $S$, the localised category should ...
- 131
6
votes
0
answers
2k
views
Verdier Quotient a quotient?
This question seems trivial, so hopefully it will be resolved quickly.
As pointed out in this question on quotient categories and localization, the two constructions are sometimes related, but in ...
- 2,516
13
votes
3
answers
946
views
Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products
A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
- 1,484
4
votes
1
answer
695
views
What kinds of limits does localization of commutative rings reflect?
Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
- 10.5k
12
votes
2
answers
829
views
Reflective Localizations vs. categories of local objects
Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by ...
- 1,704
10
votes
1
answer
763
views
Example of a saturated class of morphisms which is not _obviously_ saturated?
By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...
- 64k
3
votes
2
answers
756
views
Localization of symmetric monoidal category
Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T \...
- 1,838
10
votes
1
answer
284
views
Localizing 2-categories about a single morphism
This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...
- 15.5k
5
votes
0
answers
225
views
Weak equivalences of left Bousfield localizations
Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1.
If necessary, the model structures can be assumed to be simplicial, ...
- 37.8k
4
votes
1
answer
307
views
Localisation of inclusion functors
Let $\mathcal C$ be a category and suppose $\cal B \subseteq C$ is a full subcategory. Let $i \colon \mathcal B \longrightarrow \cal C$ denote the inclusion functor. Suppose that $S \subseteq \...
- 351
9
votes
1
answer
589
views
Is the localisation of a product of categories the product of the localisation?
Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) \...
- 351
10
votes
2
answers
824
views
Is there a notion of a “model category which admits left Bousfield localization?”
At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...
- 30.3k
3
votes
0
answers
246
views
The multiplicative system in a symmetric monoidal category
Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...
- 9,019
4
votes
2
answers
403
views
Is it possible to define the notion of a localization of a category without reference to a set of morphisms, $S$?
Let $\mathcal{C}$ Be A Category and $S$ a class of morphisms (let us call these weak equivalences) of $\mathcal{C}$. One often defines the localization of $\mathcal{C}$ with respect to $S$ is the ...
- 1,325
11
votes
2
answers
1k
views
Localization of a symmetric monoidal category at a single morphism
Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property
$$\mathrm{Hom}_{\...
- 63.1k
13
votes
2
answers
3k
views
Elements in a localization - category theoretic approach
This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...
- 63.1k
3
votes
1
answer
317
views
Controlling Reflective Subcategories and Localizations
Localizations are an extremely important part of modern homotopy theory. Both the category of spaces an spectra have a plethora of interesting localizations: at a fixed prime, rational, with respect ...
- 1,484
13
votes
4
answers
2k
views
Localizing an arbitrary additive category
Under which conditions localizing an additive category by some class S of morphisms yields and additive category? It seems easy to define certain addition on morphisms if we fix their representatives ...
- 16.9k
9
votes
4
answers
2k
views
is localization of category of categories equivalent to |Cat|
It might be a stupid question.
Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories
Take multiplicative system S={category ...
- 5,515
0
votes
3
answers
2k
views
Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
9
votes
4
answers
3k
views
Localization(s) of Categories
I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that ...
- 411