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k-surjective functor

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The notion of kk-surjective functor is the continuation of the sequence of notions

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

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

Generalization to \infty-categories

kk-Surjectivity

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

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

For kk \in \mathbb{N}, k1k \geq 1 the functor is kk-surjective if the universal morphism

C kP k C_k \to P_k

to the pullback P kP_k in

P k D k s×t C k1×C k1 F k1×F k1 D k1×D k1 \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

C k f k D k s×t s×t C k1×C k1 F k1×F k1 D k1×D k1 \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 ff) is an epimorphism.

If you interpret C kC_k and P kP_k as sets and take ‘epimorphism’ in a strict sense (the sense in Set, a surjection), then you have a strictly kk-surjective functor. But if you interpret C kC_k and P kP_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 kk-surjective functor; equivalently, project C kC_k and P kP_k to ω\omega-equivalence-classes before testing surjectivity. A functor is essentially kk-surjective if and only if it is equivalent to some strictly kk-surjective functor, so essential kk-surjectivity is the non-evil notion.

Proposition

For CC and DD categories we have

  1. ff is (essentially) 00-surjective \Leftrightarrow ff is (essentially) surjective on objects;
  2. ff is (essentially) 11-surjective \Leftrightarrow ff is full;
  3. ff is (essentially) 22-surjective \Leftrightarrow ff is faithful;
  4. ff is always 33-surjective.

In terms of lifting diagrams

Proposition

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

G k C f G k D. \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:CDf : C \to D of simplicial sets is an acyclic fibration with respect to the model structure on simplicial sets if it all diagrams

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

have a lift

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

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

Weak equivalences, acyclic fibrations and hypercovers

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

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 XYX \to Y for which all diagrams

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

have a lift

Δ[n] X Δ[n] Y. \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 kk-surjectivity is described around definition 4 of

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

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

For the analogous discussion for simplicial sets see

and references given there.

Revised on June 20, 2012 18:31:43 by Bernhard Stadler? (132.231.1.104)