arXiv subjects

Fernando Abellán

Publications and source records attributed to Fernando Abellán.

8 recordsLinked to original sources

arXiv2026

Free bifibrations of $(\infty,2)$-categories, 2-simplicial objects and the walking adjunction

In this work, we develop a fibrational approach to freely adjoining adjoints in an $(\infty,2)$-category. We construct the universal bifibration obtained from a cocartesian fibration of $(\infty,2)$-categories by adjoining cartesian lifts over a chosen class of $1$-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping $(\infty,1)$-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl--Verity, and establish the universal characterization of the simplex $2$-category conjectured by Dyckerhoff--Kapranov--Schechtman--Soibelman.

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arXiv2026

Sphericalization and the Universal Spherical Adjunction

For every adjunction of stable $\infty$-categories -- or more generally, in any locally stable $(\infty,2)$-category -- we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence, we obtain an explicit construction for a left and right adjoint to the inclusion of the $(\infty,2)$-category of spherical adjunctions of stable $\infty$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\infty,2)$-category which classifies spherical adjunctions, and to provide a synthetic proof of the fact that every spherical functor admits infinitely many left and right adjoints.

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arXiv2026

Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases $(\infty,2)$-categories of functors and partially (op)lax transformations preserve fibrations. We also describe free fibrations of $(\infty,2)$-categories, including in the case where we only ask for (co)cartesian lifts of specified 1- and 2-morphisms in the base, and describe the right adjoint to pullback from fibrations to such partial fibrations along an arbitrary functor. In the second part of the paper we apply these results to study colimits and Kan extensions of $(\infty,2)$-categories. Most notably, we give a fibrational description of both partially (op)lax and weighted (co)limits of $(\infty,2)$-categories and construct partially lax Kan extensions. Among other results, we also include a model-independent version of cofinality for $(\infty,2)$-categories and briefly consider presentable $(\infty,2)$-categories, characterizing them as accessible localizations of presheaves of $\infty$-categories.

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arXiv2024

$(\infty,2)$-Topoi and descent

We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as the exponentiability of these fibrations. As our main result, we prove a 2-dimensional version of Giraud's theorem which characterizes $(\infty,2)$-topoi as those $(\infty, 2)$-categories that appear as localizations of $\mathfrak{C}\!\operatorname{at}$-valued presheaves in which the localization functor preserves certain partially lax finite limits which we call oriented pullbacks. We develop the basics of a theory of partially lax Kan extensions internal to an $(\infty,2)$-topos, and we show that every $(\infty,2)$-topos admits an internal version of the Yoneda embedding. Our general formalism recovers the theory of categories internal to a $(\infty,1)$-topos (as develop by the second author and Sebastian Wolf) as a full sub-$(\infty,2)$-category of the $(\infty,2)$-category of $(\infty,2)$-topoi. As a technical ingredient, we prove general results on the theory of presentable $(\infty,2)$-categories, including lax cocompletions and 2-dimensional versions of the adjoint functor theorem, which might be of independent interest.

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arXiv2024

Straightening for lax transformations and adjunctions of $(\infty,2)$-categories

We prove an unstraightening result for lax transformations between functors from an arbitrary $(\infty,2)$-category to that of $(\infty,2)$-categories. We apply this to study partially (op)lax and weighted (co)limits, giving fibrational descriptions of such (co)limits for diagrams valued in $(\infty,2)$-categories, to characterize adjoints in $(\infty,2)$-categories of functors and (op)lax transformations, and to prove a mate correspondence between lax transformations that are componentwise right adjoints and oplax transformations that are componentwise left adjoints, for such transformations among functors between arbitrary $(\infty,2)$-categories.

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arXiv2023

Comparing lax functors of $(\infty,2)$-categories

In this work, we study oplax normalised functors of $(\infty,2)$-categories. Our main theorem is a comparison between the notion of oplax normalised functor of scaled simplicial sets due to Gagna-Harpaz-Lanari and the corresponding notion in the setting of complete Segal objects in $(\infty,1)$-categories studied by Gaitsgory and Rozenblyum. As a corollary, we derive that the Gray tensor product of $(\infty,2)$-categories as defined by Gaitsgory-Rozenblyum is equivalent to that of Gagna-Harpaz-Lanari. Moreover, we construct an $(\infty,2)$-categorical variant of the quintet functor of Ehresmann, from the $(\infty,2)$-category of $(\infty,2)$-categories to the $(\infty,2)$-category of double $(\infty,1)$-categories and show that it is fully faithful. As a key technical ingredient, given $(\mathbb{C},E)$ an $(\infty,2)$-category equipped with a collection of morphisms and a functor of $(\infty,2)$-categories $f:\mathbb{C}\to \mathbb{D}$, we construct a right adjoint to the restriction functor $f^*$ from the $(\infty,2)$-category of functors $\mathbb{D} \to \mathbb{C}\!\operatorname{at}_{(\infty,2)}$ and natural transformations to the $(\infty,2)$-category of functors $\mathbb{C} \to \mathbb{C}\!\operatorname{at}_{(\infty,2)}$ and partially lax (according to $E$) natural transformations. We apply this new technology of partially lax Kan extensions to the study of complete Segal objects in $(\infty,1)$-categories and double $(\infty,1)$-categories which allows us to define the notion of an enhanced Segal object (resp. enhanced double $(\infty,1)$-category), the former yielding yet another model for the theory of $(\infty,2)$-categories.

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arXiv2023

On cofinal functors of $\infty$-bicategories

In this work, we study the notion of cofinal functor of $\infty$-bicategories with respect to the theory of partially lax colimits. The main result of this paper is a characterization of cofinal functors of $\infty$-bicategories via generalizations of the conditions of Quillen's Theorem A. As a key ingredient for the proof of our main theorem we produce for every functor of $\infty$-bicategories $f:\mathbb{C} \to \mathbb{D}$ an outer 2-Cartesian fibration $\mathbb{F}(\mathbb{C})\to \mathbb{D}$ which we identify it as the free fibration on the functor $f$.

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arXiv2023

2-Cartesian fibrations II: A Grothendieck construction for $\infty$-bicategories

In this work, we conclude our study of fibred $\infty$-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set $S$ (which need not be fibrant) we construct a 2-categorical version of Lurie's straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty$-bicategory of 2-Cartesian fibrations over $S$ and the $\infty$-bicategory of contravariant functors $S^{\operatorname{op}} \to \mathbb{B}\mathbf{\!}\operatorname{icat}_\infty$ with values in the $\infty$-bicategory of $\infty$-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

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