nLab
flat (infinity,1)-functor

Contents

Idea

The generalization of the notion of flat functor from category theory to (∞,1)-category theory.

Definition

As for 1-categorical flat functors, there is a general definition of flat functors that restricts, in the case when finite limits exist, to the condition that these are preserved.

Definition

For κ\kappa a regular cardinal, an (∞,1)-functor F:CDF : C \to D is κ\kappa-flat, if, when modeled as a morphism of quasicategories, for any left Kan fibration DDD' \to D with DD' a κ\kappa-cofiltered (∞,1)-category, the pullback C:=C× DDC' := C \times_D D' (in sSet) is also κ\kappa-cofiltered.

If κ=ω\kappa = \omega then we just say FF is flat.

The dual of this is HTT, def. 5.3.2.1, under the name “κ\kappa-right exact”. But in 1-category theory, the terminology “left/right exact” is almost universally reserved for the case when finite limits/colimits do exist, so we continue that tradition in the \infty-case. We do have:

Proposition

If CC has κ\kappa-small limits, then FF is κ\kappa-flat precisely if it preserves these κ\kappa-small limits.

In particular, if CC has all finite limits, then FF is flat precisely if it preserves these.

The dual of this is HTT, prop. 5.3.2.9.

Properties

Proposition
  1. κ\kappa-flat (,1)(\infty,1)-functors are closed under composition.

  2. Every (∞,1)-equivalence is κ\kappa-flat.

  3. An (,1)(\infty,1)-functor equivalent (in the (∞,1)-category of (∞,1)-functors) to a κ\kappa-flat one is itsels κ\kappa-flat.

This is HTT, prop. 5.3.2.4.

Internal flatness

For 1-categories, there are two notions of flat functor: the above corresponds to the “representable” one, while (for functors valued in a topos) there is also a notion of “internal flatness” (and a notion of “covering flatness” that generalizes them both. I do not know whether internally-flat or covering-flat (,1)(\infty,1)-functors have been defined, but the following shows that left exact (,1)(\infty,1)-functors valued in an (,1)(\infty,1)-topos, at least, satisfy a condition that ought to characterize internally-flat ones.

Proposition

If CC is an (,1)(\infty,1)-category with finite limits, DD is an (,1)(\infty,1)-topos, and F:CDF:C\to D preserves finite limits, then its Yoneda extension 𝒫(C)D\mathcal{P}(C) \to D also preserves finite limits.

This is HTT, prop. 6.1.5.2.

References

Section 5.3.2 of

Revised on April 4, 2017 04:39:09 by Mike Shulman (76.167.222.204)