nLab essentially surjective and full functor

Idea

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”.

Properties

As 0-connected morphisms

Proposition

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

• $F$ is essentially surjective and full

• $i(F)$ is a 0-connected morphism

Proof

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 $f$, being 0-connected is equivalent to being full.

The homotopy pullback $X \times_Y X$ is given by the groupoid whose objects are triples $(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2))$ and whose morphisms are corresponding tuples of morphisms in $X$ making the evident square in $Y$ commute.

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

First assume that $f$ is full. Then for $(x_1,x_2, \alpha) \in X \times_Y X$ any object, by fullness of $f$ there is a morphism $\hat \alpha : x_1 \to x_2$ in $X$, such that $f(\hat \alpha) = \alpha$.

Accordingly we have a morphism $(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2)$ in $X \times_Y X$

$$\array{ f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{id}{\to}& f(x_2) }$$

to an object in the diagonal.

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

$$\array{ f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2) \\ \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}} \\ f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2) } \,.$$

Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$, and hence $f$ 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 $(n = 0)$ and restricted to 1-truncated objects. More on this is at infinity-image – Of Functors between groupoids.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor