arXiv subjects
Publications and source records attributed to Raffael Stenzel.
We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic theorem of Joyal and Street for monoidal categories. The proof presented in this paper is an application of the $\infty$-operadic Additivity Theorem and thereby averts any considerable calculations.
We show that braided, sylleptic and symmetric monoidal bicategories are precisely the $\mathsf{E}_k$-monoids in the cartesian monoidal $(\infty,1)$-category of bicategories for respective integers $k$. To manage the underlying computations, we use the geometry of the little cubes operads to mediate between the 2-dimensional algebra underlying the former and the $\infty$-categorical algebra underlying the latter.
We give a direct proof of the fact that Lurie's Unstraightening functor induces an equivalence between the strict $(\infty,2)$-category of indexed quasi-categories and the strict $(\infty,2)$-category of fibered quasi-categories over any given quasi-categorical base. We conclude that Unstraightening preserves simplicial cotensors up to a (strictly) natural homotopy equivalence, and thus gives rise to an accordingly weakened notion of cosmological biequivalence between the two underlying $\infty$-cosmoses.
We define and study the $(\infty,2)$-category $\mathbf{Cat}_{\infty}(\mathcal{C})$ of $(\infty,1)$-categories internal to a general $(\infty,1)$-category $\mathcal{C}$ via an associated externalization construction. In the first part, we show various formal closure properties of $\mathbf{Cat}_{\infty}(\mathcal{C})$ regarding limits, tensors, cotensors and internal mapping objects under the assumption of various suitable closure properties of $\mathcal{C}$. In particular, we show that $\mathbf{Cat}_{\infty}(\mathcal{C})$ defines a cartesian closed full sub-$\infty$-cosmos of the $\infty$-cosmos $\mathbf{Fun}(\mathcal{C}^{op},\mathbf{Cat}_{\infty})$ of $\mathcal{C}$-indexed $(\infty,1)$-categories under suitable assumptions on $\mathcal{C}$. We furthermore characterize the objects of $\mathbf{Cat}_{\infty}(\mathcal{C})$ by means of a Yoneda lemma that expresses indexed diagrams of internal shape over $\mathcal{C}$ in terms of an $(\infty,1)$-categorical totalization. In the second part, we relate the general theory developed to this point to results in the model categorical literature. We show that every model category $\mathbb{M}$ gives rise to a ``hands-on'' $\infty$-cosmos $\mathbf{Cat}_{\infty}(\mathbb{M})$ directly by restriction of the Reedy model structure on $\mathbb{M}^{\Delta^{op}}$. We then define a corresponding right derived model categorical externalization functor, and use it to show that the $(\infty,1)$-categorical and the model categorical constructions correspond to one another whenever $\mathbb{M}$ is a suitable model category.
We study various characterizations of higher sites over a given $\infty$-category $\mathcal{C}$ which are conceptually in line with their classical ordinary categorical counterparts, and extract some new results about $\infty$-topos theory from them. First, in terms of formal $(\infty,2)$-category theory, we define a notion of higher Lawvere-Tierney operators on $\infty$-toposes which internalizes a parametrized version of the left exact modalities of Anel, Biedermann, Finster and Joyal and the left exact modalities of Rijke, Shulman and Spitters. Second, in the spirit of Lawvere's hyperdoctrines, we describe the $\infty$-toposes embedded in the $\infty$-category $\hat{\mathcal{C}}$ of presheaves over $\mathcal{C}$ as the sheaves of ideals of what we call the logical structure sheaf on $\mathcal{C}$. This naturally induces a notion of ''geometric kernels'' on $\mathcal{C}$ which play the part of higher Grothendieck topologies from the given perspective. Lastly, we study the $\infty$-category of cartesian $(\infty,1)$-sites. We generalize the notion of canonical Grothendieck topologies from Lurie's book appropriately to all geometric kernels and show an according ''Comparison Lemma'' in the best case scenario. However, we show that a corresponding topological version of the lemma in the context of Grothendieck topologies fails.
We prove a correspondence between $κ$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $κ$-compact maps in their underlying quasi-categories for suitably large regular cardinals $κ$. We thus obtain a transition result between weakly universal small fibrations in the (type theoretic) injective Dugger-Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$-toposes in the sense of Lurie.
Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in $(\infty,1)$-categories. Completeness is a property of Segal spaces introduced by Rezk that characterizes those Segal spaces which are $(\infty,1)$-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study its ramifications. The core aspect of this analogy can be understood as a translation between internal and external notions, motivated by model categorical considerations of Joyal and Tierney. As a result, we characterize the internal notion of univalence in logical model categories by the external notion of completeness defined as the right Quillen condition of suitably indexed Set-weighted limit functors. Furthermore, we extend the analogy and show that univalent completion in the sense of van den Berg and Moerdijk translates to Rezk-completion of associated Segal objects as well. Motivated by these correspondences, we exhibit univalence as a homotopical locality condition whenever univalent completion exists.
We introduce the notion of a higher covering diagram in a base $\infty$-category $\mathcal{C}$. The theory of higher covering diagrams in $\mathcal{C}$ will be shown to recover various descent conditions known from the $\infty$-categorical literature in a uniform manner. In fact, higher covering diagrams always assemble to what we refer to as a structured colimit pre-topology on the base $\mathcal{C}$. It hence always defines a sub-canonical sheaf theory over $\mathcal{C}$, and indeed defines the canonical such whenever $\mathcal{C}$ has pullbacks. This ``higher geometric'' sheaf theory will be shown to differ from the usual infinitary-coherent sheaf theory by a cotopological localization whenever $\mathcal{C}$ is infinitary-coherent itself. We prove that this localization is generally non-trivial. For instance, every $\infty$-topos is the theory of higher geometric sheaves over itself, but the according infinitary-coherent sheaf theory over it is generally strictly larger. The higher geometric sheaves are hence characterized by a limit preservation property that is generally not captured by the classical sheaf condition. We define an $\infty$-category of higher geometric $\infty$-categories, and show that the (opposite of the) $\infty$-category of $\infty$-toposes embeds fully faithfully therein. We show that the higher $\kappa$-geometric sheaf theory on a higher $\kappa$-geometric $\infty$-category defines the free $\infty$-topos generated by it, and consequently that it faithfully generalizes Lurie's definition of a ``sheaf'' over an $\infty$-topos.
This paper is a study of Bousfield-Segal spaces, a notion introduced by Julie Bergner drawing on ideas about Eilenberg-Mac Lane objects due to Bousfield. In analogy to Rezk's Segal spaces, they are defined in such a way that Bousfield-Segal spaces naturally come equipped with a homotopy-coherent fraction operation in place of a composition. In this paper we show that Bergner's model structure for Bousfield-Segal spaces in fact can be obtained from the model structure for Segal spaces both as a localization and a colocalization. We thereby prove that Bousfield-Segal spaces really are Segal spaces, and that they characterize exactly those with invertible arrows. We note that the complete Bousfield-Segal spaces are precisely the homotopically constant Segal spaces, and deduce that the associated model structure yields a model for both $\infty$-groupoids and Homotopy Type Theory.
We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing B\'{e}nabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural generalizations of smallness and relative definability in this context, and show for instance that the universal cartesian fibration is small. Furthermore, by building on Johnstone's notion of comprehension schemes for ordinary fibered categories, we characterize and relate numerous higher categorical properties and structures such as left exactness, local cartesian closedness, univalent morphisms and internal $(\infty,1)$-categories in terms of comprehension schemes.