nLab k-surjective functor

Idea

The notion of $k$-surjective functor is the continuation of the sequence of notions

• essentially surjective functor

• essentially surjective and full functor

• essentially surjective and full and faithful functor

from category theory to an infinite sequence of notions in higher category theory.

Roughly, a functor $F : C \to D$ between ∞-categories $C$ and $D$ is $k$-surjective if for each boundary of a k-morphisms in $C$, each $k$-morphism between the image of that boundary in $D$ is in the image of $F$.

Generalization to $\infty$-categories

$k$-Surjectivity

For the moment, this here describes the notion for globular models of $\infty$-categories. See below for the simplicial reformulation.

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

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

$$C_k \to P_k$$

to the pullback $P_k$ in

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

coming from the commutativity of the square

$$\array{ C_k &\stackrel{f_{k}}{\to}& D_k \\ \downarrow^{s \times t} && \downarrow^{s \times t} \\ C_{k-1} \times C_{k-1} &\stackrel{F_{k-1} \times F_{k-1}}{\to}& D_{k-1} \times D_{k-1} }$$

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

If you interpret $C_k$ and $P_k$ as sets and take ‘epimorphism’ 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 $\infty$-categories or $\infty$-groupoids and take ‘epimorphism’ in a weak sense (the homotopy sense from $\infty$-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 notion that respects the principle of equivalence.

Proposition

For $C$ and $D$ categories we have

1. $f$ is (essentially) $0$-surjective $\Leftrightarrow$ $f$ is (essentially) surjective on objects;
2. $f$ is (essentially) $1$-surjective $\Leftrightarrow$ $f$ is full;
3. $f$ is (essentially) $2$-surjective $\Leftrightarrow$ $f$ 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.

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

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

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

have a lift

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

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: an acyclic fibration in the standard model structure on simplicial sets is a morphism $X \to Y$ for which all diagrams

$$\array{ \partial \Delta[n] &\to& X \\ \downarrow && \downarrow \\ \Delta[n] &\to& Y }$$

have a lift

$$\array{ \partial \Delta[n] &\to& X \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& Y } \,.$$

This is precisely in simplicial language the condition formulated above in globular language.

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.