nLab flat (infinity,1)-functor

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

If $\kappa = \omega$ then we just say $F$ is flat.

The dual of this is HTT, def. 5.3.2.1, under the name “$\kappa$-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 $C$ has $\kappa$-small limits, then $F$ is $\kappa$-flat precisely if it preserves these $\kappa$-small limits.

In particular, if $C$ has all finite limits, then $F$ is flat precisely if it preserves these.

The dual of this is HTT, prop. 5.3.2.9.

Properties

Proposition

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

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

3. An $(\infty,1)$-functor equivalent (in the (∞,1)-category of (∞,1)-functors) to a $\kappa$-flat one is itself $\kappa$-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 $(\infty,1)$-functors have been defined, but the following shows that left exact $(\infty,1)$-functors valued in an $(\infty,1)$-topos, at least, satisfy a condition that ought to characterize internally-flat ones.

Proposition

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

This is HTT, prop. 6.1.5.2.

Related concepts

• flat functor
• excisive (∞,1)-functor

References

Section 5.3.2 of

• Jacob Lurie, Higher Topos Theory