Split epimorphisms and split monomorphisms

October 1, 2012 by Qiaochu Yuan

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 $f : a \to b$ 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 $f : a \to b$ in a category $C$ are equivalent, and a morphism satisfying the first condition (hence any of the following conditions) is a split epimorphism:

$f$ has a right inverse $g : b \to a$ ; that is, a map such that $f \circ g = \text{id}_b$ .
$f$ is an absolute epimorphism in the sense that for every functor $F : C \to D$ , the morphism $F(f) : F(a) \to F(b)$ is an epimorphism.
$f$ is surjective on generalized points in the sense that, for any $c$ , the induced map $\text{Hom}(c, f) : \text{Hom}(c, a) \to \text{Hom}(c, b)$ is surjective.

Proof. $1 \Rightarrow 2$ : let $h_1, h_2 : F(b) \to c$ be a parallel pair of morphisms in $D$ . If $h_1 \circ F(f) = h_2 \circ F(f)$ , then $h_1 \circ F(f) \circ F(g) = h_2 \circ F(f) \circ F(g)$ , and since $F(f) \circ F(g) = \text{id}_{F(b)}$ , it follows that $h_1 = h_2$ . Hence $F(f)$ is an epimorphism.

$2 \Rightarrow 3$ : surjective on generalized points is equivalent to the condition that $F(f)$ is an epimorphism for the functor $F(-) = \text{Hom}(c, -) : C \to \text{Set}$ .

$3 \Rightarrow 1$ : let $c = b$ . Then the induced map $\text{Hom}(b, f) : \text{Hom}(b, a) \to \text{Hom}(b, b)$ is surjective, so there exists some $g \in \text{Hom}(b, a)$ with image $\text{id}_b \in \text{Hom}(b, b)$ . But this is precisely the statement that $f \circ g = \text{id}_b$ . $\Box$

Dually, a morphism $f : a \to b$ 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:

$f$ has a left inverse $g : b \to a$ ; that is, a map such that $g \circ f = \text{id}_b$ .
$f$ is an absolute monomorphism in the sense that for every functor $F : C \to D$ , the morphism $F(f) : F(a) \to F(b)$ is a monomorphism.
$f$ is surjective on generalized copoints in the sense that, for any $c$ , the induced map $\text{Hom}(f, c) : \text{Hom}(b, c) \to \text{Hom}(a, c)$ 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 $f : a \to b$ is a morphism with a right inverse $g$ , then we say that

$g$ is a section of $f$ ,
$f$ is a retraction of $g$ , and
$b$ is a retract of $a$ .

Example. Let $f : X \to Y$ be a split epimorphism in $\text{Set}$ . Then in particular $f$ is surjective. A right inverse to $f$ is a choice, for every $y \in Y$ , of an element of $f^{-1}(y)$ ; in other words, it is equivalent to a choice function for the family of sets $f^{-1}(y)$ , and consequently the statement that every epimorphism in $\text{Set}$ 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 $C$ if every epimorphism is split.

On the other hand, every monomorphism in $\text{Set}$ (edit: with non-empty domain!) is split with no set-theoretic assumptions: if $f : X \to Y$ is injective, then we can define a left inverse $g : Y \to X$ by defining it to be equal to $f^{-1}(y)$ whenever $y \in \text{im}(f)$ and equal to some fixed element $x \in X$ otherwise.

Example. Let $f : X \to Y$ be a split epimorphism in $\text{Top}$ with right inverse $g : Y \to X$ . Then in particular $f$ is surjective and $g$ is injective. Furthermore, the statement that $f \circ g : Y \to Y$ is the identity implies that $g$ is an embedding and that $f$ is a quotient map; in other words, $Y$ is simultaneously a quotient space and a subspace of $X$ .

Note that a split epimorphism in $\text{Top}$ must in particular be a quotient map, so a generic epimorphism in $\text{Top}$ cannot be split. In other words, the axiom of choice is false in $\text{Top}$ ! 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 $\text{Top}$ cannot be split.

A large class of examples of split epimorphisms $f : X \to Y$ come from deformation retracts. This is the origin of the term “retraction” for $f$ and “retract” for $Y$ .

Another large class of examples is the vector bundles, thought of via their projection maps $\pi : E \to B$ . If $E$ is, for example, the tangent bundle $T(M)$ of a smooth manifold $M$ , then a section of the projection map $\pi : T(M) \to M$ is precisely a vector field on $M$ . This is the origin of the term “section” for $g$ . Any vector bundle has a distinguished section, the zero section, given by sending each point $m \in M$ to the zero vector in the vector space $\pi^{-1}(m)$ .

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 $\text{U}(1)$ -bundles on a topological space $X$ are in natural correspondence with classes in $H^2(X, \mathbb{Z})$ upon taking Chern classes, with trivial bundles corresponding to $0 \in H^2(X, \mathbb{Z})$ , so if the axiom of choice were true in $\text{Top}$ 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 $\text{Top}$ . For example, proving the Brouwer fixed point theorem amounts to proving the nonexistence of a retraction of the ball $B^n$ onto its boundary, the sphere $S^{n-1}$ , and this can be shown using the fact that such a retraction induces a retraction of the homology $H_{n-1}(B^n)$ onto the homology $H_{n-1}(S^{n-1})$ . 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 $f : G \to H$ be a split epimorphism in $\text{Grp}$ with right inverse $g : H \to G$ . Then in particular $f$ is surjective and $g$ is injective, so this data determines a short exact sequence

$\displaystyle 0 \to K \to G \xrightarrow{f} H \to 0$

where $K = \text{Ker}(f)$ is the kernel of $f$ . Because $f$ has a section $g : H \to G$ , we know furthermore that $H$ is both a quotient group and a subgroup of $G$ , and in fact the existence of $g$ is equivalent to $G$ being a semidirect product $K \rtimes H$ .

Many epimorphisms of groups, even abelian groups, are not split; the simplest example is the quotient map $\mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ . Thus the axiom of choice is also false in $\text{Grp}$ .

Similarly, many monomorphisms of groups, even abelian groups, are not split; the simplest example is the inclusion $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z}$ . (This example can be thought of as obtained from the above example by Pontrjagin duality.)

Isomorphisms

In familiar algebraic categories like $\text{Grp}$ and $\text{Ring}$ , 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 $f : a \to b$ in a poset $P$ is a monomorphism and an epimorphism in $P$ which is not an isomorphism if $a \neq b$ .

Example. Any localization $D \mapsto S^{-1} D$ of an integral domain is a monomorphism and an epimorphism in $\text{CRing}$ which is not an isomorphism if $S$ is nontrivial.

Example. Any inclusion $X \subsetneq Y$ 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 $f : a \to b$ are equivalent.

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

Proof. $1 \Rightarrow 2$ : clear.

$2 \Leftrightarrow 3$ : clear.

$3 \Rightarrow 1$ : let $f : a \to b$ be a monomorphism with a right inverse $g : b \to a$ . Then $f \circ g = \text{id}_b$ , hence $f \circ g \circ f = f = f \circ \text{id}_a$ , hence (by left cancellability) $g \circ f = \text{id}_a$ . $\Box$

Dually, we have the following.

Proposition: The following conditions on a morphism $f : a \to b$ are equivalent.

$f$ is an isomorphism.
$f$ is bijective on generalized copoints.
$f$ 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 $\text{Grp}$ 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 $\text{Grp}$ is split.

Posted in math.CT | Tagged universal properties | 21 Comments

21 Responses

on August 27, 2017 at 10:26 am | Reply adi

Sorry, I want to clarify the proof of (2) => (3) in the definition-proposition of split-epi. In general epi not necessarily surjective right? meanwhile in the proof we need the surjectivity of F(f) for the functor F=Hom(-,c). Do I miss something?
- on August 27, 2017 at 11:11 am | Reply Qiaochu Yuan
  
  In $\text{Set}$ , the epimorphisms are precisely the surjective maps, and $\text{Hom}(-, c)$ takes values in $\text{Set}$ .
on December 26, 2015 at 7:09 pm | Reply Harrison Smith

Is 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 | Reply Qiaochu Yuan
  
  Yes, as long as the domain is nonempty. I gave a construction in the post. Your SDG example is irrelevant to what happens in $\text{Set}$ ; that takes place in a different category.
on March 28, 2015 at 5:11 pm | Reply 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 November 29, 2014 at 3:18 pm | Reply 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 April 1, 2013 at 4:03 pm | Reply 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 March 27, 2013 at 6:17 am | Reply Fred

Thanks 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 | Reply Qiaochu Yuan
  
  You’re welcome! I think it’s correct as written.
on November 3, 2012 at 9:13 pm | Reply 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 November 1, 2012 at 10:23 pm | Reply PTB

This 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 | Reply Qiaochu Yuan
  
  Thank 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 October 15, 2012 at 3:44 pm | Reply Aaron Mazel-Gee

Nice 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 | Reply Qiaochu Yuan
  
  I’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 | Reply Aaron Mazel-Gee
    
    Gotcha, 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 October 1, 2012 at 7:31 pm | Reply Owen Biesel

Just 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 | Reply Qiaochu Yuan
  
  Thanks for the correction!
on October 1, 2012 at 12:58 pm | Reply Zhen Lin

There 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 | Reply Qiaochu Yuan
  
  Yes, 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 | Reply Zhen Lin
    
    Oh, 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 | Reply Qiaochu Yuan
      
      I haven’t decided what to say about factorization systems yet…

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