All Questions
Tagged with localization higher-category-theory
11 questions
5
votes
1
answer
290
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Pullback of localizations
In https://kerodon.net/tag/02LZ, Lurie observes that the pullback of a localization $F$ is not necessarily a localization. If all pullbacks are localizations, then he calls $F$ universally localizing ...
- 719
4
votes
0
answers
198
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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
7
votes
1
answer
186
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How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?
I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.
Let's say $C$ is a certain category, and ...
- 366
2
votes
1
answer
171
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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
8
votes
1
answer
864
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Is hammock localization a localization in the sense of Lurie?
In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...
- 2,152
9
votes
1
answer
179
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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
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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
4
votes
0
answers
105
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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
9
votes
4
answers
2k
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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
10
votes
1
answer
284
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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
3
votes
2
answers
415
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Localisation in a quasi-category
Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction.
Now consider ...
- 1,271