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Department of Mathematics

Photo: UHH/Denstorf

Research Group Higher Structures

Fernando Abellan Garcia

profilbild-abellan-garcia-fernando-180x240

Doctoral Researcher

Address

University of Hamburg
Faculty of Mathematics, Informatics and Natural Sciences
Department of Mathematics
AZ - Bereich: Algebra und Zahlentheorie
Bundesstraße 55
20146 Hamburg

Office

Geomatikum
Room: 340

Contact

Tel: +49 40 42838-5197

I am a PhD student of Prof. Tobias Dyckerhoff, currently at Universität Hamburg.

Higher category theory, categorified homological algebra.

Publications

Publication type
Sorted by:
@article{025d3da1b73842f19216e3eda6cfbab6,
title = "2-Cartesian Fibrations I: A Model for ∞ -Bicategories Fibred in ∞ -Bicategories",
abstract = "In this paper, we provide a notion of ∞-bicategories fibred in ∞-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling ∞-bicategories, which we show is Quillen equivalent to Lurie{\textquoteright}s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an ∞-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of SetΔ+-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.",
keywords = "2-Cartesian fibration, Infinity bicategory, Model structure, Scaled simplicial set",
author = "Garc{\'i}a, {Fernando Abell{\'a}n} and Stern, {Walker H.}",
note = "Comments welcome!",
year = "2022",
month = sep,
day = "28",
doi = "10.1007/s10485-022-09693-x",
language = "English",
volume = "30",
pages = "1341--1392",
journal = "Applied categorical structures",
issn = "0927-2852",
publisher = "Springer Netherlands",
number = "6",

}

@article{107dc144c08843b3b2d4779b13600a81,
title = "Theorem A for marked 2-categories",
abstract = "In this work, we prove a generalization of Quillen's Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on (∞,1)-localizations. When restricted to 1-categories with all morphisms marked, our theorem retrieves the classical Theorem A of Quillen. We additionally state and provide evidence for a new conjecture: the cofinality conjecture, which describes the relation between a conjectural theory of marked (∞,2)-colimits and our generalization of Theorem A.",
keywords = "2-Category, Cofinality, Infinity category, Localization, Marked colimit, Quillen's Theorem A",
author = "Garc{\'i}a, {Fernando Abell{\'a}n} and Stern, {Walker H.}",
note = "36 pages",
year = "2022",
month = sep,
doi = "10.48550/arXiv.2002.12817",
language = "English",
volume = "226",
journal = "Journal of pure and applied algebra",
issn = "0022-4049",
publisher = "Elsevier BV",
number = "9",

}

@techreport{53c21432d5c54e50a92b37d248f231c4,
title = "2-Cartesian fibrations II: Higher cofinality",
abstract = "In this work, we characterize cofinal functors of (∞, 2)-categories via generalizations of the conditions of Quillen's Theorem A. In a special case, our main result recovers Joyal's well-known characterization of cofinal functors of (∞, 1)-categories. As a stepping stone to the proof of this characterization, we use the theory of 2-Cartesian fibrations developed in previous work to provide an (∞, 2)-categorical Grothendieck construction. Given a scaled simplicial set S we construct a 2-categorical version of Lurie's straightening-unstraightening adjunction, thereby furnishing an equivalence between the ∞-bicategory of 2-Cartesian fibrations over S and the ∞-bicategory of contravariant functors Sop →픹icat∞ with values in the ∞-bicategory of ∞-bicategories.",
keywords = "math.AT, math.CT",
author = "Garc{\'i}a, {Fernando Abell{\'a}n} and Stern, {Walker H.}",
note = "Comments welcome! v1: New title and abstract. Improved introduction. Typos fixed",
year = "2022",
month = jan,
day = "24",
doi = "10.48550/arXiv.2201.09589",
language = "English",
type = "WorkingPaper",

}

@phdthesis{5c03fd2521924e128dbdeb44a2d0bb2d,
title = "On cofinal functors of ∞-bicategories",
abstract = "In this thesis we investigate the notion of cofinal functor of ∞-bicategories and establish foundational results in the theory of ∞-bicategories along the way. We start with an introductory section where we present and motivate the main results achieved to later move into the main body of the thesis which is structured as follows: • In Chapter 1 we review the relevant (∞,1)-categorical theory that will be later generalized to the (∞,2)-categorical realm. •In Chapter 2 we construct a model structure on the category of marked biscaled simplicial sets over a scaled simplicial set S which models outer 2-Cartesian fibrations: An (∞,2)-categorical upgrade of the notion of Cartesian fibration. •In Chapter 3 we prove an ∞-bicategorical Grothendieck construction relating outer 2-Cartesian fibrations and contravariant functors with values in ∞-bicategories. •In Chapter 4 we characterize cofinal functors of ∞-bicategories via generalizations of the conditions of Quillen's Theorem A. •In Chapter 5 we provide applications of our cofinality criterion as well pointing out the next steps in the research programme of the author.",
author = "{Abellan Garcia}, Fernando",
year = "2022",
language = "English",
school = "University of Hamburg",

}

@article{ddf4ae3e02b04bcc9a88c611587c7066,
title = "Marked colimits and higher cofinality",
abstract = "Given a marked ∞-category D† (i.e. an ∞-category equipped with a specified collection of morphisms) and a functor F: D→ B with values in an ∞-bicategory, we define [InlineEquation not available: see fulltext.], the marked colimit of F. We provide a definition of weighted colimits in ∞-bicategories when the indexing diagram is an ∞-category and show that they can be computed in terms of marked colimits. In the maximally marked case D♯, our construction retrieves the ∞-categorical colimit of F in the underlying ∞-category B⊆ B. In the specific case when [InlineEquation not available: see fulltext.], the ∞-bicategory of ∞-categories and D♭ is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable ∞-localization of the associated coCartesian fibration Un D(F) computes [InlineEquation not available: see fulltext.]. Our main theorem is a characterization of those functors of marked ∞-categories f: C†→ D† which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits.",
keywords = "Cofinality, Infinity bicategories, Weighted colimits, localization",
author = "Garc{\'i}a, {Fernando Abell{\'a}n}",
note = "17 pages. Minor revisions. Submitted for publication. Comments welcome",
year = "2021",
month = dec,
day = "16",
doi = "10.48550/arXiv.2006.12416",
language = "English",
volume = "17",
pages = "1--22",
journal = "Journal of Homotopy and Related Structures",
issn = "1512-2891",
publisher = "Springer Science + Business Media",
number = "1",

}

@article{aa3824a2a35d44d8b682d27035bd057f,
title = "A relative 2-nerve",
abstract = "In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.",
author = "Garc{\'i}a, {Fernando Abell{\'a}n} and Tobias Dyckerhoff and Stern, {Walker H.}",
year = "2020",
month = dec,
day = "8",
doi = "10.2140/agt.2020.20.3147",
language = "English",
volume = "20",
pages = "3147–3182",
journal = "Algebraic & geometric topology",
issn = "1472-2747",
publisher = "Mathematical Sciences Publishers",

}

@techreport{06acec6dc9d44e12babc3452d8c9360a,
title = "Enhanced twisted arrow categories",
abstract = " Given an $\infty$-bicategory $\mathbb{D}$ with underlying $\infty$-category $\mathcal{D}$, we construct a Cartesian fibration $\operatorname{Tw}(\mathbb{D})\to \mathcal{D} \times \mathcal{D}^{\operatorname{op}}$, which we call the enhanced twisted arrow $\infty$-category, classifying the restricted mapping category functor $\operatorname{Map}_{\mathbb{D}}:\mathcal{D}^{\operatorname{op}}\times \mathcal{D} \to \mathbb{D}^{\operatorname{op}} \times \mathbb{D} \to \operatorname{Cat}_{\infty}$. With the aid of this new construction, we provide a description of the $\infty$-category of natural transformations $\operatorname{Nat}(F,G)$ as an end for any functors $F$ and $G$ from an $\infty$-category to an $\infty$-bicategory. As an application of our results, we demonstrate that the definition of weighted colimits presented in arXiv:1501.02161 satisfies the expected 2-dimensional universal property. ",
keywords = "math.CT",
author = "Garc{\'i}a, {Fernando Abell{\'a}n} and Stern, {Walker H.}",
note = "35 pages",
year = "2020",
month = sep,
day = "24",
language = "English",
type = "WorkingPaper",

}

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