nLab flat (infinity,1)-functor



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


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.


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., 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:


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.


  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 itself κ\kappa-flat.

This is HTT, prop.

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.


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.


Section 5.3.2 of

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