equivalences in/of $(\infty,1)$-categories
The generalization of the notion of effective epimorphism from category theory to (∞,1)-category theory.
See also at 1-epimorphism. However, beware of the red herring principle: effective epimorphisms in an $(\infty, 1)$-category need not be epimorphisms.
A morphism $f : Y \to X$ in an (∞,1)-category is an effective epimorphism if it has a Cech nerve, of which it is the (∞,1)-colimit; in other words the augmented simplicial diagram
is an colimiting diagram.
This appears below HTT, cor. 6.2.3.5 for $C$ a (∞,1)-semitopos, but seems to be a good definition more generally.
In an (∞,1)-topos the effective epis are the n-epimorphisms for $n = 1$ sitting in the (n-epi, n-mono) factorization system for $n = 1$ with the monomorphism in an (∞,1)-category, factoring every morphism through its 1-image.
In an (∞,1)-semitopos, effective epimorphisms are stable under (∞,1)-pullback.
This appears as (Lurie, prop. 6.2.3.15).
For $C$ an (∞,1)-semitopos we have that $f : X \to Y$ is an effective epimorphism precisely if its (-1)-truncation is a terminal object in the over-(∞,1)-category $C/Y$.
This is HTT, cor. 6.2.3.5.
More generally,
The effective epimorphisms in any (∞,1)-topos are precisely the (-1)-connected morphisms, and form a factorization system together with the monomorphisms (the (-1)-truncated morphisms).
See n-connected/n-truncated factorization system for more on this.
For $C$ an (∞,1)-topos, a morphism $f : X \to Y$ in $C$ is effective epi precisely if the induced morphism on subobjects ((∞,1)-monos, they form actually a small set) by (∞,1)-pullback
is injective.
This appears as (Rezk, lemma 7.9).
Useful is also the following characterization:
A morphism in an (∞,1)-topos is an effective epimorphism precisely if its 0-truncation is an effective epimorphism in the underlying 1-topos.
This is (Lurie, prop. 7.2.1.14).
In words this means that a map is an effective epimorphism if it induces an epimorphism on connected components.
This is true generally in the internal logic of the $(\infty,1)$-topos (i.e. in homotopy type theory, see at 1-epimorphism for more on this), but in ∞Grpd $\simeq L_{whe}$ sSet it is also true externally (prop. 7 below).
In the infinity-topos of infinity-sheaves on an $\infty$-site (i.e. in a topological localization), one has the following characterization of morphisms which become effective epimorphisms after applying the associated sheaf functor.
Let $(C,t)$ be an $\infty$-site and $f : F \to G$ a morphism of presheaves in $P(C)$. The morphism $a_t(f)$ is an effective epimorphism in $Shv_t(C)$ if and only if $f$ is a local epimorphism, i.e.
is $t$-covering, or in other words
is a $t$-covering sieve for all morphisms $h(X) \to G$, where $h$ is the Yoneda embedding. (Here $a_t$ denotes the associated sheaf functor.)
This is clear. See MO/177325/2503 by David Carchedi for the argument.
As a corollary of prop. 5 we have:
(effective epis of $\infty$-groupoids)
In $C =$ ∞Grpd a morphism $f : Y \to X$ is an effective epimorphism precisely if it induces an epimorphism $\pi_0 f : \pi_0 Y \to \pi_0 X$ in Set on connected components.
This appears as HTT, cor. 7.2.1.15.
If $S^1 = \ast \underset{\ast \coprod \ast}{\coprod} \ast$ denotes the homotopy type of the circle, then the unique morphism $S^1 \to \Delta^0$ is an effective epimorphism, by prop. 7, but it not an epimorphism, because the suspension of $S^1$ is the sphere $S^2$, which is not contractible.
effective epimorphism in an $(\infty,1)$-category
Section 7.7 of
Section 6.2.3 of