There are various natural questions one can ask about monomorphisms and epimorphisms all of which lead to the same answer:

- What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
- What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
- A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
- If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.

**Equivalent characterizations and examples**

The first question is the easiest one to get out of the way: a morphism is an epimorphism if and only if it is right-cancellative, and the “easiest” way to cancel something on the right is to give it an inverse on the right. (We discuss epimorphisms first because we saw earlier that they behave less like surjective maps than monomorphisms behave like injective maps, so we are more interested in looking for stronger notions of epimorphism than for stronger notions of monomorphism.)

**Definition-Proposition:** The following conditions on a morphism in a category are equivalent, and a morphism satisfying the first condition (hence any of the following conditions) is a **split epimorphism**:

- has a right inverse ; that is, a map such that .
- is an
**absolute epimorphism**in the sense that for every functor , the morphism is an epimorphism. - is
**surjective on generalized points**in the sense that, for any , the induced map is surjective.

*Proof.* : let be a parallel pair of morphisms in . If , then , and since , it follows that . Hence is an epimorphism.

: surjective on generalized points is equivalent to the condition that is an epimorphism for the functor .

: let . Then the induced map is surjective, so there exists some with image . But this is precisely the statement that .

Dually, a morphism is a monomorphism if and only if it is left-cancellative, and the “easiest” way to cancel something on the left is to give it an inverse on the left. Dualizing the above proposition, we obtain the following:

**Definition-Proposition:** The following conditions on a morphism in a category are equivalent, and a morphism satisfying the first condition (hence any of the following conditions) is a **split monomorphism**:

- has a left inverse ; that is, a map such that .
- is an
**absolute monomorphism**in the sense that for every functor , the morphism is a monomorphism. - is
**surjective on generalized copoints**in the sense that, for any , the induced map is surjective.

Note that the right inverse to a split epimorphism is a split monomorphism and conversely, so every example of one also gives an example of the other.

If is a morphism with a right inverse , then we say that

- is a
**section**of , - is a
**retraction**of , and - is a
**retract**of .

*Example.* Let be a split epimorphism in . Then in particular is surjective. A right inverse to is a choice, for every , of an element of ; in other words, it is equivalent to a choice function for the family of sets , and consequently the statement that every epimorphism in is split is equivalent to the axiom of choice.

Motivated by this example, we might say that the axiom of choice holds in a category if every epimorphism is split.

On the other hand, every monomorphism in (edit: with non-empty domain!) is split with no set-theoretic assumptions: if is injective, then we can define a left inverse by defining it to be equal to whenever and equal to some fixed element otherwise.

*Example.* Let be a split epimorphism in with right inverse . Then in particular is surjective and is injective. Furthermore, the statement that is the identity implies that is an embedding and that is a quotient map; in other words, is simultaneously a quotient space and a subspace of .

Note that a split epimorphism in must in particular be a quotient map, so a generic epimorphism in cannot be split. In other words, the axiom of choice is false in ! This provides a great reason for caring about when the axiom of choice gets used in an argument; see, for example, this MO question for further discussion. Similarly, a split monomorphism must in particular be an embedding, so a generic monomorphism in cannot be split.

A large class of examples of split epimorphisms come from deformation retracts. This is the origin of the term “retraction” for and “retract” for .

Another large class of examples is the vector bundles, thought of via their projection maps . If is, for example, the tangent bundle of a smooth manifold , then a section of the projection map is precisely a vector field on . This is the origin of the term “section” for . Any vector bundle has a distinguished section, the zero section, given by sending each point to the zero vector in the vector space .

A closely related class of examples are principal bundles, which admit a section if and only if they are trivializable. The existence of nontrivial principal bundles is an important topological fact; for example, isomorphism classes of principal -bundles on a topological space are in natural correspondence with classes in upon taking Chern classes, with trivial bundles corresponding to , so if the axiom of choice were true in then second cohomology would always be trivial.

The fact that split epimorphisms are absolute can be used to show the nonexistence of certain maps in . For example, proving the Brouwer fixed point theorem amounts to proving the nonexistence of a retraction of the ball onto its boundary, the sphere , and this can be shown using the fact that such a retraction induces a retraction of the homology onto the homology . But the former is trivial and the latter is not, so the induced map on homology cannot be surjective (hence cannot be an epimorphism, hence cannot be a retraction).

*Example.* Let be a split epimorphism in with right inverse . Then in particular is surjective and is injective, so this data determines a short exact sequence

where is the kernel of . Because has a section , we know furthermore that is both a quotient group and a subgroup of , and in fact the existence of is equivalent to being a semidirect product .

Many epimorphisms of groups, even abelian groups, are not split; the simplest example is the quotient map . Thus the axiom of choice is also false in .

Similarly, many monomorphisms of groups, even abelian groups, are not split; the simplest example is the inclusion . (This example can be thought of as obtained from the above example by Pontrjagin duality.)

**Isomorphisms**

In familiar algebraic categories like and , a morphism which is both injective and surjective on underlying sets is already an isomorphism. Unfortunately, the more general statement that a morphism which is both a monomorphism and an epimorphism is an isomorphism is usually false.

*Example.* Any morphism in a poset is a monomorphism and an epimorphism in which is not an isomorphism if .

*Example.* Any localization of an integral domain is a monomorphism and an epimorphism in which is not an isomorphism if is nontrivial.

*Example.* Any inclusion of a dense subspace into a space is a monomorphism and an epimorphism in the category of Hausdorff topological spaces which is not an isomorphism if the dense subspace is nontrivial.

A salvage of this statement is the following.

**Proposition:** The following conditions on a morphism are equivalent.

- is an isomorphism.
- is bijective on generalized points.
- is a monomorphism and a split epimorphism.

*Proof.* : clear.

: clear.

: let be a monomorphism with a right inverse . Then , hence , hence (by left cancellability) .

Dually, we have the following.

**Proposition:** The following conditions on a morphism are equivalent.

- is an isomorphism.
- is bijective on generalized copoints.
- is a split monomorphism and an epimorphism.

This is not an ideal salvage, however; it is not a strong enough statement to recover the fact that a morphism in which is both a monomorphism and an epimorphism is an isomorphism in the sense that we can’t assume that a monomorphism or an epimorphism in is split.

on October 1, 2012 at 12:58 pm |Zhen LinThere are other kinds of mono + epi that imply invertibility. The keyword is orthogonality: so, for example, because regular epimorphisms are always orthogonal to monomorphisms, any morphism that is both a mono and a regular epi must be an isomorphism. The nLab page on epimorphisms has a good discussion.

on October 1, 2012 at 1:10 pm |Qiaochu YuanYes, I’m covering that result in the next post, but I didn’t want to jinx it by actually saying anything about there being a next post.

on October 1, 2012 at 1:47 pm |Zhen LinOh, good. Will you talk about orthogonal factorisation systems in general then? I’ve been meaning to learn about that, especially in connection with homotopical abstract nonsense…

on October 1, 2012 at 2:07 pm |Qiaochu YuanI haven’t decided what to say about factorization systems yet…

on October 1, 2012 at 7:31 pm |Owen BieselJust to nitpick, not every monomorphism in Set is split. The domain has to be inhabited (i.e. nonempty) for your argument to go through, and in particular the only split monomorphism out of ø is the identity function.

on October 1, 2012 at 7:41 pm |Qiaochu YuanThanks for the correction!

on October 15, 2012 at 3:44 pm |Aaron Mazel-GeeNice post. When you say that a category does or doesn’t admit Choice, are you saying something topos-theoretic or are you just analogizing with the category of sets?

on October 15, 2012 at 5:03 pm |Qiaochu YuanI’m just saying that epimorphisms are or aren’t all split. The nLab article has various comments on this.

on October 17, 2012 at 10:37 am |Aaron Mazel-GeeGotcha, thanks for the reference. So I guess the answer is no: there’s a weaker *internal* notion of “every epi has a section” in a topos, whereas this is an notion of the axiom of choice for any category (including a topos) based on the logic of Set.

on November 1, 2012 at 10:23 pm |PTBThis blog shows great mathematical maturity, on which you are to be congratulated. However, having just taught a few years at a major first class UK university, and knowing that nobody in my second year class could write the truth table for “if a then b else c”, or multiply 0x5 by 0xffbc, I’m perhaps too easily impressed by what should be more normal levels of math ability!

But even granted that I’m easily impressed, this blog goes waaaay beyond where I was as an undergraduate! I spent my first year trying to crack Fermat’s Last Theorem and discovering for myself group theory and transfinite ordinal algebras. Actually _reading_ other texts would never have occurred to me! Perhaps I was too impatient to read around the subject, as I should have and you apparently do. I don’t know how you get the time! As undergraduates we had time, but were too busy partying, and as graduates, we were snowed under with the impossible task of assimilating umpteen different specialist disciplines to a level of competitive excellence. To see an undergraduate (even fresh graduate) talking about category theory and toposes, homologies etc with such aplomb is quite extraordinary. The emphasis seems to be on pure topics with a lean towards algebra and its generalizations, but then I’d fall over backwards to see physics, control theory, martingales, etc, covered in this context.

Now that I need a bit of category theory for a paper and have been refreshing my memories of thirty years past (I recall chasing Saunders MacLane round the streets of Amsterdam as he searched for his favourite restaurant), I find myself getting very useful insight from your blog! It provides really good perspective! Thank you. Your strengths seem to be in discerning structure – I hope that leads to a great future.

on November 1, 2012 at 10:45 pm |Qiaochu YuanThank you! I blame the internet – it helps people like me get exposed to mathematics we wouldn’t otherwise get to see until graduate school or thereabouts. I have written a few posts about physics (in some sense) if you’d like to scroll down and possibly back a page, and plan on writing more in the future.

on November 3, 2012 at 9:13 pm |Regular and effective monomorphisms and epimorphisms « Annoying Precision[…] sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective […]

on March 27, 2013 at 6:17 am |FredThanks for your excellent and enlightening post! Small typo: in the first condition of the definition of split monomorphism, it should read id_a rather than id_b

on March 27, 2013 at 9:21 am |Qiaochu YuanYou’re welcome! I think it’s correct as written.

on April 1, 2013 at 4:03 pm |Connected objects and a reconstruction theorem | Annoying Precision[…] is connected; then WLOG the identity map factors through . It follows that the inclusion map is a split epimorphism. However, by assumption, the map on underlying sets is an injection in , and since faithful […]

on November 29, 2014 at 3:18 pm |Topological Diophantine equations | Annoying Precision[…] that if a morphism has a section, or equivalently a right inverse, then it is called a split epimorphism, and in particular it is an epimorphism. Recall also the following two equivalent alternative […]

on March 28, 2015 at 5:11 pm |Projective objects | Annoying Precision[…] all projective modules. Next we need the following observation. Recall that an object is a retract of an object if there are maps such that (so is a split epimorphism and is a split […]

on December 26, 2015 at 7:09 pm |Harrison SmithIs it really true that every monomorphism in Set always has a retract? It is not clear to me that we can always pick some element of X. For example, in SDG, letting D be the set of square-zero elements of R, my (naive) understanding is that it is not possible to pick any specific x∈D. Of course this is all resolved through excluded middle, but that doesn’t make it any more clear that every monomorphism splits; it just kills my example.

on December 26, 2015 at 7:49 pm |Qiaochu YuanYes, as long as the domain is nonempty. I gave a construction in the post. Your SDG example is irrelevant to what happens in ; that takes place in a different category.