Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: epimorphism

Bottom of Page

1 to 16 of 16

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2010

    felt like adding a handful of basic properties to epimorphism

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeOct 28th 2020

    Noted that any functor that preserves pushouts preserves epimorphisms.

    diff, v30, current

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeDec 10th 2020
    Below the proof of Proposition 4.7, it is noted that

    > For example, the unique morphism to the terminal object is not always an epimorphism (though this is true in Set)

    I believe the unique function from the empty set to a singleton is not surjective, so the claim is false in Set as well.

    - Aniruddh Agarwal <me@anrddh.me>
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 10th 2020

    Thanks for the alert.

    This statement was introduced in rev 31. Maybe what was really meant was the converse statement, that an epimorphism implies a global section only in well-pointed toposes.

    For the moment I have just removed the corresponding half-sentence. But everyone please feel invited to (re-)expand on this point.

    (NB, the paragraph in question is here)

    diff, v36, current

  5. It said that any morphism from an initial object is an epimorphism, which is not true. I changed it to say that any morphism TO an initial object is an epimorphism, which is the proof actually shows anyway.

    Jonathan Beardsley

    diff, v38, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2021

    Gave the (counter-)examples of monos that are epi but not iso their own sub-section (here) and added mentioning of the example of dense subspace inclusions in Hausdorff spaces.

    (Also cleaned up the previous material, as the example ℤ↪ℚ!A = \sum_{E: V} (E \to A}\mathbb{Z} \hookrightarrow \mathbb{Q} had been mentioned twice.)

    diff, v41, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeMar 18th 2021

    Just occured to me, an elementary question, which functors are epimorphisms in the 1-category of small categories ? It is clear that all localization functors belong to this class.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 18th 2021

    Try this: Epimorphic functors and this: Generalized congruences – Epimorphisms in Cat (both by the same people, I strongly suspect there is a lot of overlap, but I haven’t checked.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeMar 19th 2021

    Oh, thanks David this is quite helpful! There is no complete answer there, and I did read in detail the second (earlier, 1999) listed paper around 2006 or so, and it flashed across my vague memory when I was yesterday asking the question, but did not look at it as I felt it was not what I was looking for. Now I see it does have much of relevant reasoning on most important classes of epimorphisms and factorizations of interest. I will have some use of it (in fact it is also relevant for some of my interest in so called iterated localizations and 2-categorical analogues).

  10. Corrected the type of the canonical map i 1i_1 in the proof that the coproduct of epimorphisms is an epimorphism: the map was given as i 1:x 1→x 1+x 2i_1 : x_1 \to x_1 + x_2 but it has to be i 1:y 1→y 1+y 2i_1 : y_1 \to y_1 + y_2 for g∘i 1∘f 1g \circ i_1 \circ f_1 to make sense when f 1:x 1→y 1f_1 : x_1 \to y_1 and g:y 1+y 2→zg : y_1 + y_2 \to z.

    • Jussi Piitulainen

    Anonymous

    diff, v43, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 22nd 2023

    Added:

    • G. M. Kelly, Monomorphisms, Epimorphisms, and Pullbacks, Journal of the Australian Mathematical Society 9.1-2 (1969), pp. 124–142. doi:10.1017/s1446788700005693](https://doi.org/10.1017/s1446788700005693).

    diff, v46, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v48, current

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeNov 11th 2023

    Extended and started more precise the discussion of an epimorphism of rings. Still under construction though (written in a bus, and will be checked and improved later).

    Added an example of epimorphism of sheaves of sets over a topological space.

    diff, v50, current

    • CommentRowNumber14.
    • CommentAuthorʇɐ
    • CommentTimeJan 1st 2024

    Changed “strong epimorphism = an epimorphism right orthogonal to monomorphisms” to left orthogonal (to match strong epimorphism).

    diff, v52, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeJan 23rd 2024

    Added simpler example of a ring epimorphism that’s not surjective. Reordered text so that “But beware that the converse fails:” comes after the statement it’s supposed to, namely “Every isomorphism is both an epimorphism and a monomorphism.”

    diff, v53, current

  16. Change “every other object” into “every object” in the definition of an epimorphism.

    ymat

    diff, v55, current

1 to 16 of 16

  1.