nLab
k-surjective functor

Essentially surjective, faithful, and full functors

The standard stuff:

• http://math.ucr.edu/home/baez/qg-spring2004/discussion.html

• stuff, structure, property

Generalization to $\mathrm{\infty }$-categories

$k$-surjectivity

An $\omega$-functor $f:C\to D$ between $\mathrm{\infty }$-categories is 0-surjective if $f_0:C_0\to D_0$ is an epimorphism.

For $k\in \mathbb{N}$, $k\ge 1$ the functor is $k$-surjective if the universal morphism

$$C_k\to P_k$$C_k \to P_k

to the pullback $P_k$ in

$$\begin{array}{ccc}{P}_k& \to & D_k\\ \downarrow & & {\downarrow }^{s\times t}\\ C_{k-1}\times C_{k-1}& \stackrel{F_{k-1}\times F_{k-1}}{\to }& D_{k-1}\times D_{k-1}\end{array}$$\array{ P_k &\to& D_k \\ \downarrow && \downarrow^{s \times t} \\ C_{k-1} \times C_{k-1} &\stackrel{F_{k-1} \times F_{k-1}}{\to}& D_{k-1} \times D_{k-1} }

coming from the commutativity of the square

$$\begin{array}{ccc}{C}_k& \stackrel{f_k}{\to }& D_k\\ {\downarrow }^{s\times t}& & {\downarrow }^{s\times t}\\ C_{k-1}\times C_{k-1}& \stackrel{F_{k-1}\times F_{k-1}}{\to }& D_{k-1}\times D_{k-1}\end{array}$$\array{ C_k &\stackrel{f_{k}}{\to}& D_k \\ \downarrow^{s \times t} && \downarrow^{s \times t} \\ C_{k-1} \times C_{k-1} &\stackrel{F_{k-1} \times F_{k-1}}{\to}& D_{k-1} \times D_{k-1} }

(which commutes due to the functoriality axioms of $f$) is an epimorphism.

If you interpret $C_k$ and $P_k$ as sets and take ‘epimorphim’ in a strict sense (the sense in Set, a surjection), then you have a strictly $k$-surjective functor. But if you interpret $C_k$ and $P_k$ as $\mathrm{\infty }$-categories or $\mathrm{\infty }$-groupoids and take ‘epimorphism’ in a weak sense (the homotopy sense from $\mathrm{\infty }$-Grpd), then you have an essentially $k$-surjective functor; equivalently, project $C_k$ and $P_k$ to $\omega$-equivalence-classes before testing surjectivity. A functor is essentially $k$-surjective if and only if it is equivalent to some strictly $k$-surjective functor, so essential $k$-surjectivity is the non-evil notion.

Proposition

For $C$ and $D$ categories we have

1. $f$ is (essentially) $0$-surjective $\iff$ $f$ is (essentially) surjective on objects;
2. $f$ is (essentially) $1$-surjective $\iff$ $f$ is full;
3. $f$ is (essentially) $2$-surjective $\iff$ $f$ is faithful;
4. $f$ is always $3$-surjective.

In terms of lifting diagrams

Proposition

An $\omega$-functor $f:C\to D$ is $k$-surjective for $k\in \mathbb{N}$ precisely if it has the right lifting property with respect to the inclusion $\partial G_k\to G_k$ of the boundary of the $k$-globe into the $k$-globe.

$$\begin{array}{ccc}\partial G_k& \to & C\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^f\\ G_k& \to & D\end{array}.$$\array{ \partial G_k &\to& C \\ \downarrow &{}^{\exists}\nearrow& \downarrow^f \\ G_k &\to& D } \,.

One recognizes the similarity to situation for geometric definition of higher category. A morphism $f:C\to D$ of simplicial sets is an acyclic fibration with respect to the model structure on simplicial sets if it all diagrams

$$\begin{array}{ccc}\partial \Delta [k]& \to & C\\ \downarrow & & {\downarrow }^f\\ \Delta [k]& \to & D\end{array}$$\array{ \partial \Delta[k] &\to& C \\ \downarrow && \downarrow^f \\ \Delta[k] &\to& D }

have a lift

$$\begin{array}{ccc}\partial \Delta [k]& \to & C\\ \downarrow & \nearrow & {\downarrow }^f\\ \Delta [k]& \to & D\end{array}$$\array{ \partial \Delta[k] &\to& C \\ \downarrow &\nearrow& \downarrow^f \\ \Delta[k] &\to& D }

for all $k$, where now $\Delta [k]$ is the $k$-simplex.

Weak equivalences, acyclic fibrations and hypercovers

With respect to the folk model structure on $\omega$-categories an $\omega$-functor is

• an acyclic fibration? if it is $k$-surjective for all $k\in \mathbb{N}$;

• a weak equivalence if it is essentially $k$-surjective for all $k\in \mathbb{N}$. See also equivalence of categories.

Remarks

All this has close analogs in other models of higher structures, in particular in the context of simplicial sets. Simplicial maps which are $k$-surjective for all $k$ are called hypercovers.

Literature

The general idea of $k$-surjectivity is described around definition 4 of

• Baez-Shulman, Lectures on n-Categories and Cohomology (arXiv)

The concrete discussion in the context of strict omega-categories is in

• Yves Lafont, Francois Métayer, Krzysztof Worytkiewicz, A folk model structure on $\omega$-cat (arXiv).

For the analogous discussion for simplicial sets see

• model structure on simplicial sets

and references given there.