Flat functors in the context of fibration categories

While "general" models for (∞, 1)-categories ought to replicate the concepts of category theory in a homotopy-enabled context, they do not necessarily provide strict notions of composition and identity morphisms. As such, objects such as quasicategories do not usually come with a 1-categorical structure. It is however convenient for many purposes to work present (∞, 1)categories in term of 1-categories with additional structure, such as model categories or fibration categories. When working with specific categories of (∞, 1)-categories, the process of replacing a quasicategory with a strict presentation thereof need not be restrictive. Specifically, there exist DKequivalences:

CMC → PrL from the relative category CMC of combinatorial model categories, left Quillen functors and left Quillen equivalence to the relative category PrL of presentable quasicategories, left adjoint ∞-functors and equivalences (this is established in [Pav25, Theorem 1.1]).

FibCat → QCat lex from the relative category of fibration categories, exact functors and DK-equivalences to the relative category of finitely complete quasicategories, left exact ∞-functors and equivalences (as proved in [Szu14, Theorem 4.9]).

In this document, our goal is to provide a new outlook on the correspondence between fibration categories and finitely complete quasicategories by means of the theory of flat functors and its (∞, 1)-categorical counterpart.

A functor between finitely complete categories is flat if and only if it preserves finite limits. While the definition of flat functors makes sense in a more general context (see Section 6.3 of [Bor94]), we will only be considering flat functors between finitely complete categories, so that the notion coincides with that of finite limit-preserving functors. As a matter of fact, we will consider flat ∞-functors between (∞, 1)-categories presented by fibration categories. A subtle point is that fibration categories need not have all finite limits as 1-categories, but the (∞, 1)-categories they present are indeed finitely complete. Likewise, exact functors present finite limit-preserving ∞functors, namely, the flat functors we consider, but they need not be limitpreserving as 1-functors.

Additionally, the framework of fibration categories provides a tool to compute finite limits in the corresponding (∞, 1)-category, especially pullbacks, by means of (special) 1-categorical limits (e.g, pullbacks along fibrations).

As such, the morphisms between fibration categories, that is, the exact functors, are not just presenting flat ∞-functors: they also preserve these special 1-categorical limits, which encode the ∞-categorical ones.

One cannot expect exact functors to be flat in the 1-categorical sense, as the preservation property they satisfy, by definition, only deals with particular pullbacks (pullbacks along fibrations). It is reasonable, nonetheless, to expect the theory of such functors to be richer, in some sense, than the theory of flat ∞-functors in general. Precisely, we are interested in the connection between flat functors and left Kan extensions. It is known that a functor F : C → E, valued in some Grothendieck topos E, is (internally) flat precisely when its left Kan extension Lan y : Set C op → E along the Yoneda embedding preserves finite limits (see [MM12], Corollary 3 in VII.9.1). In this section, we will study the left Kan extensions of exact functors. Explicitly, we will show in Proposition 4.4 that it is possible to define such an extension, or rather an approximation of it, as a span of exact functor; hence reconciling the correct homotopy behavior with the rigidify of a 1-categorical presentation that relies on fibrations and weak equivalences.

Recall the definition of a fibration category: Definition 1.1. A fibration category is a category F equipped with two classes of morphisms W (the weak equivalences) and F (the fibrations) that are stable under composition, and such that:

• F admits a terminal object * , and the unique map x → * is a fibration for every object x.

• F admits pullbacks along fibrations, and the base change of a fibration is a fibration.

• Trivial fibrations (fibrations that are also weak equivalences) are stable under pullback.

• The class of weak equivalence W satisfies the 2-out-of-3 property.

• For every object x, there is a factorization of the diagonal

where x → px is a weak equivalence and px → x × x is a fibration.

A functor F : F → F ′ between fibration categories is exact when it preserves the corresponding structure: it maps fibrations (resp. weak equivalences) to fibrations (resp. weak equivalences), and preserves the terminal object as well as pullbacks along fibrations. Definition 1.2. We define Sp w to be the “homotopical span” category, that is, the following category

where both maps are weak equivalences. Sp w admits a Reedy category structure (which is an inverse one): the apex has degree 1, and the two others objects have degree 0.

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