nLab essentially surjective and full functor




An essentially surjective and full functor is a functor which is both essentially surjective and full.

Sometimes this condition is abbreviated eso and full, where “eso” is short for “essentially surjective on objects”.


As 0-connected morphisms


Let f:XYf \colon X \longrightarrow Y be a functor between small categories that happen to be groupoids, and write i:GrpdGrpdi \;\colon\; Grpd \hookrightarrow \infty Grpd for the full inclusion of groupoids into the (∞,1)-topos ∞Grpd of ∞-groupoids. Then the following are equivalent:


As discussed there, an effective epimorphism in ∞Grpd between 1-groupoids is precisely an essentially surjective functor.

So it remains to check that for an essentially surjective ff, being 0-connected is equivalent to being full.

The homotopy pullback X× YXX \times_Y X is given by the groupoid whose objects are triples (x 1,x 2X,α:f(x 1)f(x 2))(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2)) and whose morphisms are corresponding tuples of morphisms in XX making the evident square in YY commute.

By prop. it is sufficient to check that the diagonal functor XX× YXX \to X \times_Y X is (-1)-connected, hence, as before, essentially surjective, precisely if ff is full.

First assume that ff is full. Then for (x 1,x 2,α)X× YX(x_1,x_2, \alpha) \in X \times_Y X any object, by fullness of ff there is a morphism α^:x 1x 2\hat \alpha : x_1 \to x_2 in XX, such that f(α^)=αf(\hat \alpha) = \alpha.

Accordingly we have a morphism (α^,id):(x 1,x 2)(x 2,x 2)(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2) in X× YXX \times_Y X

f(x 1) f(α^) f(x 2) α id f(x 2) id f(x 2) \array{ f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{id}{\to}& f(x_2) }

to an object in the diagonal.

Conversely, assume that the diagonal is essentially surjective. Then for every pair of objects x 1,x 2Xx_1, x_2 \in X such that there is a morphism α:f(x 1)f(x 2)\alpha : f(x_1) \to f(x_2) we are guaranteed morphisms h 1:x 1x 2h_1 : x_1 \to x_2 and h 2:x 2x 2h_2 : x_2 \to x_2 such that

f(x 1) f(h 1) f(x 2) α id f(x 2) f(h 2) f(x 2). \array{ f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2) } \,.

Therefore h 2 1h 1h_2^{-1}\circ h_1 is a preimage of α\alpha under ff, and hence ff is full.

Eso+full/faithful factorization system

In the 2-topos Cat, the pair of classes of morphisms consisting of

  • left class: essentially surjective and full functors

  • right class: faithful functors

form a factorization system in a 2-category – the (eso and full, faithful) factorization system.

When restricted to the (2,1)-topos Grpd and in view of Prop. , this is the special case of the n-connected/n-truncated factorization system in the (∞,1)-topos ∞Grpd for the case that (n=0)(n = 0) and restricted to 1-truncated objects. More on this is at infinity-image – Of Functors between groupoids.

basic properties of…

Created on May 27, 2020 at 16:09:47. See the history of this page for a list of all contributions to it.