nLab
effective epimorphism

Definition

An effective epimorphism is a morphism $f:c\to d$ (in a given category) that has a kernel pair $c{\times }_{d}c$ and is the quotient object of this kernel pair in that

$$c{\times }_{d}c\overrightarrow{\to }c\stackrel{f}{\to }d$$c \times_d c \stackrel{\to}{\to} c \stackrel{f}{\to} d

is a colimit diagram (a coequalizer). This is equivalent to saying that $f$ is a regular epimorphism which has a kernel pair. The reader should be aware, however, that some writers use “effective epimorphism” to mean what is here called a strict epimorphism.

In other words this says that $f:c\to d$ is effective if $d$ is the coimage of $f$.

Sometimes we say that such morphism $f$ is an effective quotient.

Remarks

The dual concept is that of effective monomorphism?.

An effective epimorphism is necessarily a regular epimorphism. Conversely, any regular epimorphism that has a kernel pair is an effective epimorphism.

In the category of sets, every epimorphism is effective. Thus, it can be hard to know, when generalising concepts from $Set$ to other categories, what kind of epimorphism to use. In particular, one may define a projective object (and hence the axiom of choice) using effective epimorphisms.

Definition for $(\mathrm{\infty }\mathrm{,1})$-categories

Notice that we may equivalently reinterpret the condition that

• a morphism $\pi :U\to X$ is the quotient object of its kernel pair $U{\times }_{X}U\overrightarrow{\to }U$

by saying that $X$ is the colimit over the entire Čech nerve

$$\cdots U{\times }_{X}U{\times }_{X}U\overrightarrow{\overrightarrow{\to }}U{\times }_{X}U\overrightarrow{\to }U$$\cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\to} U

of $\pi$.

This is the formulation that generalizes to higher categorical context.

In an $(\mathrm{\infty }\mathrm{,1})$-category $C$, an effective epimorphism $f:U\to X$ is a morphism such that its Čech nerve $U_{•}$ is a simplicial resolution of $X$.

This is defined in HTT only for an $(\mathrm{\infty }\mathrm{,1})$-semitopos?, but seems to be correct in general.

Conjecture

In any $(\mathrm{\infty }\mathrm{,1})$-category, an effective epimorphism is precisely a regular epimorphism that has a Čech nerve.

No serious attempt has been made to check this; it may be obvious … or obviously false.

Presumably, we get the general definition in an arbitary $(\mathrm{\infty }\mathrm{,1})$-category by simply including the requirement that $f$ have a Čech nerve. In an $(\mathrm{\infty }\mathrm{,1})$-topos, is there any distinction to be drawn between an effective epimorphism and a regular epimorphism? (There isn't in a topos.)

Urs Schreiber: my understanding is that the effective epimorphisms are equivalently to be thought of as the regulkar epimorphisms in an $(\mathrm{\infty }\mathrm{,1})$-topos, indeed.

In HTT the notion of regular epimorphism is defined more generally in the context of a “semi $(\mathrm{\infty }\mathrm{,1})$-topos”.

The notion of groupoid object in an (infinity,1)-category is the categorification of equivalence relation. With that in mind the relation between regular/effective epimorphisms and equivalence relations/groupoid objects is the same.

Toby: I would like to define both ‘effective epimorphism’ and ‘regular epimorphism’ in the general context of an $(\mathrm{\infty }\mathrm{,1})$-category, rather than in two contexts that only partially overlap (a category, and an $(\mathrm{\infty }\mathrm{,1})$-topos or $(\mathrm{\infty }\mathrm{,1})$-semitopos). Since Lurie doesn't do that, do we at least get a theorem (in an $(\mathrm{\infty }\mathrm{,1})$-topos) that a regular epimorphism (as defined in a semitopos) is the same as an effective one (not just ‘to be thought of’ as that)? If so, …

I would like to say that a regular epimorphism in an $(\mathrm{\infty }\mathrm{,1})$-topos is a morphism that is a simplicial resolution, and that an effective epimorphism is a morphism that has a Čech nerve and is a simplicial resolution of that Čech nerve. I would like this to agree with Lurie's definition in the cases that he gives, and I would like Lurie's proof of the hypothetical theorem above to generalise to a theorem that an effective epimorphism is precisely a regular epimorphism that has a Čech nerve. Does that work?

Urs Schreiber: yes, I agree, that’s how it should be (regular = is simplicial resolution, effective=that simplicial resolution is given by the Cech nerve)

As far as I can see Lurie doesn’t define the term “regular epimorphism” at all.

It’s very late here and I shouldn’t be working anymore, but apart from that I’d say you’d be safe with doing what you propose to do. I am not sure which hypothetical theorem you mean. Seems like just a definition is needed for that particular statement.

Toby: Ah, I see, it's ‘effective epimorphism’ that Lurie defines for an $(\mathrm{\infty }\mathrm{,1})$-semitops, while ‘regular epimorphism’ is never defined.