nLab
epimorphism

Contents

Idea

In category theory, the concept of epimorphism is a generalization or strengthening of the concept of surjective functions between sets (example below).

The formally dual concept is that of monomorphism, similarly related to the concept of injective function.

Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”.

Definition

A morphism f:XYf \colon X \to Y in some category is called an epimorphism (sometimes abbrieviated to epi, or described as being epic), if for every other object ZZ and every pair of parallel morphisms g 1,g 2:YZg_1,g_2 \colon Y \to Z then

(g 1f=g 2f)(g 1=g 2). \left( g_1\circ f \,=\, g_2 \circ f \right) \;\Rightarrow\; \left( g_1 = g_2 \right) \,.

Stated more abstractly, this says that ff is an epimorphism precisely if for every object ZZ the hom-functor Hom(,Z)Hom(-,Z) sends it to an injective function

Hom(Y,Z)f *Hom(X,Z) Hom(Y,Z) \xhookrightarrow{\;\; f^* \;\;} Hom(X,Z)

between hom-sets. Since injective functions are the monomorphisms in Set (example below) this means that ff is an epimorphism precisely if Hom(f,Z)Hom(f,Z) is a monomorphism for all ZZ.

Finally, this means that ff is an epimorphism in a category 𝒞\mathcal{C} precisely if it is a monomorphism in the opposite category 𝒞 op\mathcal{C}^{op}.

Examples

General

Example

Every isomorphism is both an epimorphism and a monomorphism.

Example

(epimorphisms of sets)

The epimorphisms in the category Set of sets are precisely the surjective functions.

Thus the concept of epimorphism may be thought of as a category-theoretic generalization of the concept of surjection.

But beware that in categories of sets with extra structure, epimorphisms need not be surjective (in contrast to monomorphisms, which are usually injective).

Example

(epimorphisms of rings)

In the categories Ring or CRing of (commutative) rings and ring homomorphisms between them, then every surjective ring homomorphisms is an epimorphism,

For more see for instance at Stacks Project, 10.106 Epimorphisms of rings.

Often, though, the surjections correspond to a stronger notion of epimorphism.

But beware that the converse fails:

Examples of monos that are epi but not iso

The following lists some examples of morphisms that are both monomorphisms and epimorphisms, but not necessarily isomorphisms.

Example

In the category of Hausdorff topological spaces, the inclusion AXA \hookrightarrow X of a dense subspace is an epimorphism.

See this Prop. for proof.

Example

In unital Rings, the canonical inclusion i\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q} of the integers into the rational numbers is an epimorphism.

See this Prop. for proof.

Properties

Proposition

The following are equivalent:

  • f:xyf : x \to y is an epimorphism in CC;

  • ff is a monomorphism in the opposite category C opC^{op};

  • precomposition with ff is a monomorphism in Set: that is, for all cCc \in C, f:Hom(y,c)Hom(x,c)- \circ f : Hom(y,c) \to Hom(x,c) is an injection;

  • the commuting diagram

    x f y f Id y Id y \array{ x &\stackrel{f}{\to}& y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Id}} \\ y &\underset{Id}{\to}& y }

    is a pushout diagram.

Proposition

If f:xyf \colon x \to y and g:yzg \colon y \to z are epimorphisms, so is their composite gfg f. If gfg f is an epimorphism, so is gg.

Proposition

Every coequalizer xyx \to y

zxy z \stackrel{\to}{\to} x \to y

is an epimorphism.

The converse of the above proposition fails, and an epimorphism is called a regular epimorphism if it is the coequalizer of some pair of morphisms.

Proposition

Epimorphisms are preserved by pushout: if f:xyf : x \to y is an epimorphism and

x a f g y b \array{ x &\to& a \\ {}^{\mathllap{f}}\downarrow && \downarrow^{g} \\ y &\to& b }

is a pushout diagram, then also gg is an epimorphism.

Proof

Let h 1,h 2:bch_1,h_2 : b \to c be two morphisms such that gh 1=gh 2\stackrel{g}{\to} \stackrel{h_1}{\to} = \stackrel{g}{\to} \stackrel{h_2}{\to} . Then by the commutativity of the diagram also xybh 1cx \to y \to b \stackrel{h_1}{\to} c equals xybh 2cx \to y \to b \stackrel{h_2}{\to} c. Since xyx \to y is assumed to be epi, it follows that ybh 1cy \to b \stackrel{h_1}{\to} c equals ybh 2cy \to b \stackrel{h_2}{\to} c. But this means that h 1h_1 and h 2h_2 define the same cocone. By the universality of the pushout bb there is a unique map of cocones from bb to cc. Hence h 1h_1 must equal h 2h_2. Therefore gg is epi.

Proposition

Epimorphisms are preserved by any left adjoint functor, or more generally any functor that preserves pushouts: if F:CDF : C \to D is a functor that preserves pushouts and fMor(C)f \in Mor(C) an epimorphism then F(f)Mor(D)F(f) \in Mor(D) is an epimorphism.

Proof

If F:CDF : C \to D is a left adjoint we can argue this way: by the adjunction natural isomorphism we have for all dObj(D)d \in Obj(D)

Hom D(L(f),d)Hom C(f,R(d)). Hom_D(L(f),d) \simeq Hom_C(f,R(d)) \,.

The right hand is a monomorphism by assumption, hence so is the left hand, hence L(f)L(f) is epi.

More generally, if FF preserves pushouts we can use the fact that ff is epic iff

x f y f Id y Id y \array{ x &\stackrel{f}{\to}& y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Id}} \\ y &\underset{Id}{\to}& y }

is a pushout diagram.

Proposition

Epimorphisms are reflected by faithful functors.

Proof

Let F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} be a faithful functor. Consider f:xyf \colon x \longrightarrow y a morphism in 𝒞\mathcal{C} such that F(f):F(x)F(y)F(f) \colon F(x)\longrightarrow F(y) is an epimorphism in 𝒟\mathcal{D}. We need to show that then ff itself is an epimorphism.

So consider morphisms g,h:yzg,h \colon y \longrightarrow z such that gf=hfg \circ f = h \circ f. We need to show that this implies that already g=hg = h (injectivity of Hom(f,z)Hom(f,z)). But functoriality implies that F(g)F(f)=F(h)F(f)F(g)\circ F(f) = F(h) \circ F(f), and since F(f)F(f) is epic this implies that F(g)=F(h)F(g) = F(h). Now the statement follows with the assumption that FF is faithful, hence injective on morphisms.

Epimorphisms get along with colimits in a number of ways, some of which we have seen above. Here is another:

Proposition
  1. Any morphism to an initial object is an epimorphism.

  2. The coproduct of epimorphisms is an epimorphism.

Proof

For the first suppose 0C0 \in C is initial and f:x0f : x \to 0. Given morphisms g,h:0yg,h: 0 \to y with gf=hfg \circ f = h \circ f we have g=hg = h simply because 00 is initial.

For the second suppose f 1:x 1y 1f_1 : x_1 \to y_1 and f 2:x 2y 2f_2 : x_2 \to y_2 are epimorphisms; we wish to show that f 1+f 2:x 1+x 2y 1+y 2f_1 + f_2 : x_1+x_2 \to y_1 + y_2 is an epimorphism. Suppose we have morphisms g,h:y 1+y 2zg, h: y_1+y_2 \to z with g(f 1+f 2)=h(f 1+f 2)g \circ (f_1+f_2) = h \circ (f_1 + f_2). Then gi 1f 1=hi 1f 1g \circ i_1 \circ f_1 = h \circ i_1 \circ f_1 where i 1:x 1x 1+x 2i_1 : x_1 \to x_1 + x_2 is the canonical map into the coproduct. Since f 1f_1 is epic we conclude gi 1=hi 1g \circ i_1 = h \circ i_1. Similarly we have gi 2=hi 2g \circ i_2 = h \circ i_2. If follows that g=hg = h.

Epimorphisms do not get along quite as well with limits. For example, the projections from a Cartesian product onto its factors, e.g. p 1:x 1×x 2x 1p_1 \colon x_1 \times x_2 \to x_1, are not always epimorphisms (even in SetSet: take x 2x_2 to be empty).

Variations

There are a sequence of variations on the concept of epimorphism, which conveniently arrange themselves in a total order. In order from strongest to weakest, we have:

In the category of sets, every epimorphism is effective descent (and even split if you believe the axiom of choice). Thus, it can be hard to know, when generalising concepts from Set to other categories, what kind of epimorphism to use. The following discussion may be helpful in this regard.

First we note:

Moreover, if the category has finite limits, then the picture becomes much simpler:

Also worth noting are:

Thus, in general, the two serious distinctions come

Moreover, even in non-regular categories, there seems to be a strong tendency for strong/extremal epimorphisms to coincide with regular/strict ones. For example, this is the case in Top, where both are the class of quotient maps. (The plain epimorphisms are the surjective continuous functions.)

However, the distinction is real. For instance, in the category generated by the following graph:

C f A hk B g D \array{ &&&& C\\ &&& ^f\nearrow\\ A& \underoverset{h}{k}{\rightrightarrows} & B \\ &&& _g\searrow\\ &&&& D}

subject to the equations fh=fkf h = f k and gh=gkg h = g k, both ff and gg are strong, but not strict, epimorphisms.