[Merged by Bors] - feat(Algebra/Homology): two descriptions of the derived category as a localized category #9660
Conversation
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
… localized category
leanprover-community-mathlib4-bot
added
the
blocked-by-other-PR
leanprover-community-mathlib4-bot
removed
the
blocked-by-other-PR
|
This PR/issue depends on: |
leanprover-community-mathlib4-bot
added
the
merge-conflict
leanprover-community-mathlib4-bot
removed
the
merge-conflict
joelriou
added
the
awaiting-review
leanprover-community-mathlib4-bot
added
ready-to-merge
mathlib-bors bot
pushed a commit
that referenced
this pull request
Feb 13, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
|
Build failed (retrying...): |
mathlib-bors bot
pushed a commit
that referenced
this pull request
Feb 13, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
|
Build failed (retrying...): |
mathlib-bors bot
pushed a commit
that referenced
this pull request
Feb 13, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
|
Build failed (retrying...): |
mathlib-bors bot
pushed a commit
that referenced
this pull request
Feb 13, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
|
Pull request successfully merged into master. Build succeeded: |
mathlib-bors
bot
changed the title
riccardobrasca
pushed a commit
that referenced
this pull request
Feb 18, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
dagurtomas
pushed a commit
that referenced
this pull request
Mar 22, 2024
… localized category (#9660) In this PR, it is shown that under certain conditions on the complex shape `c`, the category `HomologicalComplexUpToQuasiIso C c` of homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian category `C` can be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
In this PR, it is shown that under certain conditions on the complex shape
c, the categoryHomologicalComplexUpToQuasiIso C cof homological complexes up to quasi-isomorphisms, which is a localization of the category of homological complexes, is also a localization of the homotopy category. In particular, in the case of cochain complexes indexed by the integers, this means that the derived category of an abelian categoryCcan be obtained either in a single step by formally inverting the quasi-isomorphisms in the category of cochain complexes, or in two steps by first passing to the homotopy category (which is a quotient category) and then formally inverting the quasi-isomorphisms in the homotopy category.