Add extremal monomorphisms and extremal epimorphisms - #320
Open
ScriptRaccoon wants to merge 7 commits into
Open
Conversation
ScriptRaccoon
force-pushed
the
extremal-monos
branch
2 times, most recently
from
August 5, 2026 10:55
96b8750 to
95e4ee5
Compare
ScriptRaccoon
force-pushed
the
extremal-monos
branch
3 times, most recently
from
August 6, 2026 17:06
4d16cc5 to
b70e35c
Compare
- notation for morphisms is $m$ - notation for epimorphisms is $p$ - add more related properties - fix some grammar - add small remarks in the definitions - remove redundant implications for morphisms
ScriptRaccoon
force-pushed
the
extremal-monos
branch
from
August 6, 2026 17:09
b70e35c to
296cdcf
Compare
Owner
Author
|
@dschepler I think I am finished here. Can you please review this PR? (Probably you are still busy with the PR on total categories, which means that, of course, this here can wait.) |
This file contains hidden or 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
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
1 participant
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Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be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 fills another gap from #267 by adding extremal monomorphisms and extremal epimorphisms.
Some basic results are added as well (e.g. that every strong mono is extremal, and that the converse holds when pushouts exist).
The "forked commutative square"
Since every consistent morphism combination (or its dual) should have at least one witness, this PR also adds an example of a non-strong extremal monomorphism. Apparently, such examples are not so easy to find "in nature", which is why this PR also adds a new category to the database: the "forked commutative square", which is generated by the graph
and the evident relations. Here,$m$ is an extremal monomorphism, but not a strong one. All properties except one have been decided for this category: it is not clear whether cosifted limits exist. (Of course, this can be determined, but I would like to skip this for the moment.) However, all properties of $m$ have been decided.
Special morphism rules
Moreover, three special morphism rules have been added:
This made it easier to classify the special morphisms in the forked commutative square, but it also means that several isomorphism assignments in examples of gaunt categories could be removed.
Graphical overview
This PR also adds a content page that includes a graphical overview of the relationships between the various types of epimorphisms and monomorphisms.
Maybe this is better suitable for the nLab, and I tried to add this to the page on epimorphisms, but the nLab keeps throwing errors at me. So let's keep it here for now. Of course, it should be adjusted as soon we add more types of epimorphisms or monomorphisms (such as descent morphisms).