Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

Abstract

For a small category D we define fibrations of simplicial presheaves on the category D × ∆, which we call localized D-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized D-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on D, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma.We apply this general framework to study Cartesian fibrations of (∞, n)-categories, for models of (∞, n)-categories that arise via simplicial presheaves, such as n-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of (∞, n)categories. CONTENTS 0. Introduction 1 1. A Deep Dive into Enrichments and Some Fibrations 8 2. From Left Fibrations to D-left Fibrations 29 3. Grothendieck Construction for D-Simplicial Spaces 39 4. Localized D-Left fibrations 57 5. (Segal) Cartesian Fibrations of (∞, n)-Categories 86 6. What needs to be done? 102 References 104

Cocartesian fibrations and straightening internal to an $\infty$-topos

“…In the special case where B is the ∞-topos of spaces, Theorem 6.3.1 recovers Lurie's straightening equivalence. One can therefore regard our proof of Theorem 6.3.1 as another approach to the straightening equivalence, complementing existing proofs such as Lurie's original account in [Lur09] and the more recent approaches by Boavida de Brito [BdB18], Nuiten [Nui21], Hebestreit-Heuts-Ruit [HHR21] and Rasekh [Ras21].…”

“…We have already mentioned above that by now there exist several proofs for the ∞-categorical straightening equivalence [Lur09,BdB18,Nui21,HHR21,Ras21]. In [Sha18], Jay Shah builds upon this result to derive a straightening equivalence for parametrised higher categories.…”

On straightening for Segal spaces

“…This makes Cat(Cat In particular, using the point-set model of Cat d -enriched categories, a copresheaf C −→ Cat d can be modeled by an enriched functor C −→ Cat d . Following [6], we can also describe such copresheaves more homotopy-invariantly in terms of Segal objects (see also [34] for an in-depth discussion): Definition 4.23. Let C be a (d + 1)-categorical algebra.…”

“…Work of Boavida [6] then identifies the category c 1 Fun d+1 (C, Cat d ) with the category of Segal copresheaves, i.e. maps of (d + 1)-fold simplicial spaces X −→ C with the following two conditions (see also [34]):…”

“…One should not think of a Segal copresheaf X −→ C as some sort of cocartesian fibration, since its domain is typically not a (d + 1)-category (unless d = 0); unfortunately, this is in contrast with the terminology employed in [6,34]. Instead, Segal copresheaves can better be viewed as encoding a homotopy coherent action of the (d + 1)-category C on the d-category X(0).…”