-
Notifications
You must be signed in to change notification settings - Fork 71
New issue
Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.
By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.
Already on GitHub? Sign in to your account
Refactor shift, pointed identity types are loop spaces #727
Merged
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
EgbertRijke
reviewed
Sep 9, 2023
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/functoriality-loop-spaces.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
Outdated
Show resolved
Hide resolved
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Outdated suggestions cannot be applied.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.
This pr does two related things:
(i) it shows that pointed identity types are equivalent to loop spaces.
this will be useful in the Eckmann-Hilton and Hopf proof (issue #702), since we can construct an eckmann-hilton term of type$s \cdot s = s \cdot s$ , where $\Omega^2 \mathbb{S}^2$ `. Then, using this equivalence, we get a term of type $\Omega^3 \mathbb{S}^2$ . Due to the way the equivalence is defined, we will get many nice functoriality properties. In particular, the equivalence preserves path concatination. This will be helpful in the proof. It will also later be helpful when applying sypllepsis.
s
is the generator of(ii) the pr uses the above equivalenecs to replace
unshift
in the fileuniversal-property-suspensions-pointed-types