## Fosco Loregian

Ma olen matemaatik.
I enjoy category theory, stable homotopy theory, computations with the Serre spectral sequence, functional programming, proof assistants, bicategories of profunctors, the deep meaning of the integral sign, lager beers, artificial and natural languages, sharpening knives, Shunga paintings, drawing maps, calligraphy, typography. Most of all, I love linking all these things.

See my CV here. My email is below. My interests grew too far apart to be described by a research statement any more. Let's just say that I do category theory here, category theory there. Here, however, I compiled a short presentation about my past work.

### Publications

Profunctor optics, a categorical update
@misc{arxiv.2001.07488,
doi       = {10.48550/ARXIV.2001.07488},
url       = {https://arxiv.org/abs/2001.07488},
author    = {Clarke, B. and
Elkins, D. and
Gibbons, J. and
Loregian, F. and
Milewski, B. and
Pillmore, E. and
Román, M.},
title     = {Profunctor Optics, a Categorical Update},
publisher = {arXiv},
year      = {2020},
}

Escrows are optics

To appear on thejft, V 1 issue 1.

@misc{arxiv.2105.10028,
doi       = {10.48550/ARXIV.2105.10028},
url       = {https://arxiv.org/abs/2105.10028},
author    = {Genovese, F. and Loregian, F. and Palombi, D.},
title     = {Escrows are optics},
publisher = {arXiv},
year      = {2021},
}

Fibrational linguistics I and II
@misc{arxiv.2201.01136,
doi       = {10.48550/ARXIV.2201.01136},
url       = {https://arxiv.org/abs/2201.01136},
author    = {Genovese, F. and Loregian, F. and Puca, C.},
title     = {Fibrational linguistics: First concepts},
publisher = {arXiv},
year      = {2022},
}


and

@misc{arxiv.2207.06765,
doi       = {10.48550/ARXIV.2207.06765},
url       = {https://arxiv.org/abs/2207.06765},
author    = {Genovese, F. and Loregian, F. and Puca, C.},
title     = {Fibrational linguistics: Language acquisition},
publisher = {arXiv},
year      = {2022},
}

Differential 2-rigs
@misc{arxiv.2103.00938,
doi       = {10.48550/ARXIV.2103.00938},
url       = {https://arxiv.org/abs/2103.00938},
author    = {Loregian, F. and Trimble, T.},
title     = {Differential 2-rigs},
publisher = {arXiv},
year      = {2021},
}


A Categorical Semantics for Hierarchical Petri Nets
@article{Genovese2021,
doi       = {10.4204/eptcs.350.4},
url       = {https://doi.org/10.4204/eptcs.350.4},
year      = {2021},
month     = dec,
publisher = {Open Publishing Association},
volume    = {350},
pages     = {51--68},
author    = {F. Genovese and F. Loregian and D. Palombi},
title     = {A Categorical Semantics for Hierarchical Petri Nets},
journal   = {Electronic Proceedings in Theoretical Computer Science}
}

A Categorical Semantics for Bounded Petri Nets
@article{Genovese2022,
doi       = {10.4204/eptcs.372.5},
url       = {https://doi.org/10.4204/eptcs.372.5},
year      = {2022},
month     = nov,
publisher = {Open Publishing Association},
volume    = {372},
pages     = {59--71},
author    = {F. Genovese and F. Loregian and D. Palombi},
title     = {A Categorical Semantics for Bounded Petri Nets},
journal   = {Electronic Proceedings in Theoretical Computer Science}
}

Nets with Mana - A Framework for Chemical Reaction Modelling
@incollection{Genovese2021,
doi       = {10.1007/978-3-030-78946-6_10},
url       = {https://doi.org/10.1007/978-3-030-78946-6_10},
year      = {2021},
publisher = {Springer International Publishing},
pages     = {185--202},
author    = {F. Genovese and F. Loregian and D. Palombi},
title     = {Nets with Mana: a framework for chemical reaction modelling},
booktitle = {Graph Transformation}
}

Coends of higher arity
@article{Loregian2022,
author  = {Loregian, F. and de Oliveira Santos, E.},
title   = {Coends of Higher Arity},
journal = {Applied Categorical Structures},
year    = {2022},
month   = {Feb},
day     = {01},
volume  = {30},
number  = {1},
pages   = {173-221},
issn    = {1572-9095},
doi     = {10.1007/s10485-021-09653-x},
url     = {https://doi.org/10.1007/s10485-021-09653-x}
}

Functorial semantics for partial theories
@article{10.1145/3434338,
author     = {Di Liberti, I. and
Loregian, F. and
Nester, C. and
Soboci\'{n}ski, P.},
title      = {Functorial Semantics for Partial Theories},
year       = {2021},
issue_date = {January 2021},
publisher  = {Association for Computing Machinery},
address    = {New York, NY, USA},
volume     = {5},
number     = {POPL},
url        = {https://doi.org/10.1145/3434338},
doi        = {10.1145/3434338},
journal    = {Proc. ACM Program. Lang.},
month      = {jan},
articleno  = {57},
numpages   = {28},
}

A standard theorem on adjunctions in two variables
@article{stadj_MPIM767,
title   = {A standard theorem on adjunctions in two variables},
author  = {Loregian, F.},
year    = {2018},
month   = {dec},
journal = {Preprints of the MPIM},
number  = {67}
}

A Fubini rule for $\infty$-coends
@article{infub_MPIM768,
title   = {A {F}ubini rule for infinity-coends},
author  = {Loregian, F.},
year    = {2018},
month   = {dec},
journal = {Preprints of the MPIM},
number  = {68}
}

Categorical notions of fibration
@article{riehl2020496,
title    = {Categorical notions of fibration},
journal  = {Expositiones Mathematicae},
volume   = {38},
number   = {4},
pages    = {496-514},
year     = {2020},
issn     = {0723-0869},
doi      = {https://doi.org/10.1016/j.exmath.2019.02.004},
url      = {https://www.sciencedirect.com/science/article/pii/S0723086918300872},
author   = {F. Loregian and E. Riehl},
}

Accessibility and presentability in 2-categories
@article{accpres2023107155,
title   = {Accessibility and presentability in 2-categories},
journal = {Journal of Pure and Applied Algebra},
volume  = {227},
number  = {1},
pages   = {107155},
year    = {2023},
issn    = {0022-4049},
doi     = {https://doi.org/10.1016/j.jpaa.2022.107155},
url     = {https://www.sciencedirect.com/science/article/pii/S0022404922001517},
author  = {I. {D}i Liberti and F. Loregian},
}

Triangulated factorization systems and t-structures
@article{loregian_virili_2020219,
title    = {Triangulated factorization systems and t-structures},
journal  = {Journal of Algebra},
volume   = {550},
pages    = {219-241},
year     = {2020},
issn     = {0021-8693},
doi      = {https://doi.org/10.1016/j.jalgebra.2019.12.021},
url      = {https://www.sciencedirect.com/science/article/pii/S0021869320300296},
author   = {F. Loregian and S. Virili},
}

Homotopical algebra is not concrete
@article{concreteness2018,
author  = {Di Liberti, I. and Loregian, F.},
title   = {Homotopical algebra is not concrete},
journal = {Journal of Homotopy and Related Structures},
year    = {2018},
month   = {Sep},
day     = {01},
volume  = {13},
number  = {3},
pages   = {673-687},
issn    = {1512-2891},
doi     = {10.1007/s40062-018-0197-3},
url     = {https://doi.org/10.1007/s40062-018-0197-3}
}

t-structures in stable $\infty$-categories
@phdthesis{loregian2016tstructures,
title    = {t-structures on stable (infinity,1)-categories},
year     = {2016},
school   = {SISSA},
url      = {http://urania.sissa.it/xmlui/handle/1963/35202},
author   = {F. Loregian}
}


or (better)

@misc{10.48550/arxiv.2005.14295,
doi       = {10.48550/ARXIV.2005.14295},
url       = {https://arxiv.org/abs/2005.14295},
author    = {Loregian, F.},
title     = {t-structures on stable infinity-categories},
publisher = {arXiv},
year      = {2020},
}

Coend Calculus
@book{Loregian2021,
doi       = {10.1017/9781108778657},
url       = {https://doi.org/10.1017/9781108778657},
year      = {2021},
month     = jun,
publisher = {Cambridge University Press},
author    = {F. Loregian},
title     = {(Co)end Calculus}
}

Hearts and Towers in stable $\infty$-categories
@article{Fiorenza2019,
doi       = {10.1007/s40062-019-00237-0},
url       = {https://doi.org/10.1007/s40062-019-00237-0},
year      = {2019},
month     = {May},
publisher = {Springer Science and Business Media {LLC}},
volume    = {14},
number    = {4},
pages     = {993--1042},
author    = {D. Fiorenza and F. Loregian and G.L. Marchetti},
title     = {Hearts and towers in stable $\infty$-categories},
journal   = {Journal of Homotopy and Related Structures}
}

$t$-structures are normal torsion theories
@Article{Fiorenza2016,
author  = {Fiorenza, D. and Loregian, F.},
title   = {t-Structures are Normal Torsion Theories},
journal = {Applied Categorical Structures},
year    = {2016},
month   = {Apr},
day     = {01},
volume  = {24},
number  = {2},
pages   = {181-208},
issn    = {1572-9095},
doi     = {10.1007/s10485-015-9393-z},
url     = {https://doi.org/10.1007/s10485-015-9393-z}
}


### Preprints

Completeness for categories of generalized automata
@misc{arxiv.2303.03867,
doi       = {10.48550/ARXIV.2303.03867},
url       = {https://arxiv.org/abs/2303.03867},
author    = {Boccali, G. and
Laretto, A. and
Loregian, F. and
Luneia, S.},
title     = {Completeness for categories of generalized automata},
publisher = {arXiv},
year      = {2023},
}

Bicategories of automata, automata in bicategories
@misc{arxiv.2303.03865,
doi       = {10.48550/ARXIV.2303.03865},
url       = {https://arxiv.org/abs/2303.03865},
author    = {Boccali, G. and
Laretto, A. and
Loregian, F. and
Luneia, S.},
title     = {Bicategories of automata, automata in bicategories},
publisher = {arXiv},
year      = {2023},
}

Functorial Erkennen

[PDF] • The second chapter of a series devoted to join category theory and ontology.

Categorical Ontology I - Existence

[PDF] • The first chapter of a series devoted to join category theory and ontology.

On the unicity of formal category theories

1901.01594 • We prove an equivalence between cocomplete Yoneda structures and certain proarrow equipments on a 2-category K. In order to do this, we recognize the presheaf construction of a cocomplete Yoneda structure as a relative, lax idempotent monad sending each admissible 1-cell to an adjunction. Each cocomplete Yoneda structure on K arises in this way from a relative lax idempotent monad “with enough adjoint 1-cells”, whose domain generates the ideal of admissibles, and the Kleisli category of such a monad equips its domain with proarrows. We call these structures “yosegi”. Quite often, the presheaf construction associated to a yosegi generates an ambidextrous Yoneda structure; in such a setting there exists a fully formal version of Isbell duality.

Localization theory for derivators

1802.08193 • We outline the theory of reflections for prederivators, derivators and stable derivators. In order to parallel the classical theory valid for categories, we outline how reflections can be equivalently described as categories of fractions, reflective factorization systems, and categories of algebras for idempotent monads. This is a further development of the theory of monads and factorization systems for derivators.

Recollements in stable $\infty$-categories

1507.03913 • This is the third joint work with D. Fiorenza, about $t$-structures in stable $\infty$-categories, which studies recollements. We develop the theory of recollements in a stable $\infty$-categorical setting. In the axiomatization of Beilinson, Bernstein and Deligne, recollement situations provide a generalization of Grothendieck’s “six functors” between derived categories. The adjointness relations between functors in a recollement induce a “recollee” $t$-structure on $\mathcal{D}$, given $t$-structures $t_0$, $t_1$ on $\mathcal{D}_0$, $\mathcal{D}_1$. Such a classical result, well known in the setting of triangulated categories, acquires a new taste when $t$-structure are described as suitable ($\infty$-categorical) factorization systems: the corresponding factorization system enjoys a number of interesting formal properties and unexpected autodualities. In the geometric case of a stratified space, various recollements arise, which “interact well” with the combinatorics of the intersections of strata to give a well-defined, associative operation. From this we deduce a generalized associative property for $n$-fold gluings, valid in any stable $\infty$-category.

### Teaching

SEMINARS - A course on 2-categories

A course on 2-category theory (hopefully) held in Padua; there will be notes (hopefully), and beer afterwards.

SEMINAR - Functorial semantics

[pdf] A talk at the Turin-Udine Logic Seminar.

SEMINARS - Categorical Tools

I started another project (similar to the Jacobians mathematicians) called Categorical Tools, where I tried to propose a bit of categorical language to the “heathens”, and in order to introduce the youngsters here in math@unipd to the “classical” constructions any functorial gung-ho must meet at least once in a lifetime (bits of enriched category theory, toposes, spectral sequences, homotopy theory, weighted limits, coend-juggling, higher category theory…).

COURSE - Introduction to category theory @taltech

A course on category theory held in Taltech.

SEMINARS - Formal category theory

A reading seminar at Masaryk University; each other Wednesday (but the first lecture is on Monday, October 9) from 2pm to 4pm. There will be cookies -and notes-!

SEMINARS - Homotopy Type Theory

A reading seminar at Masaryk University; each other Wednesday (first lecture October 11) from 2pm to 4pm. There will be cookies!

SEMINARS - Profunctorial semantics

A running seminar on functorial semantics, with an eye on profunctors as an equivalent definition of algebraic theories.

SEMINARS - Jacobian Mathematicians

Our name is a pun between Jacobian and Jacobins; it is intended to be some kind of open window towards the scientific attitude to knowledge. We talk about Maths, also developing its interconnection with culture and Philosophy.

I gave seven lectures until now (but three more people talked about Game Theory, Fourier analysis, and analytical solutions to PDEs):

1. [PDF] Fibrations between spheres and Hopf theorem
2. [PDF] The importance of being abstract aka A gentle introduction to the categorical point of view to reality;
3. [PDF] low dimensional Topological Quantum Field Theories;
4. [PDF] Chatting about complex geometry (from symplectic to Kahler manifolds);
5. [PDF] Connections and Fiber Bundles, with a glance to the geometry of Classical Field Theory;
6. [PDF]: A short lecure about Computational Homological Algebra, my first piece of (!) applied Mathematics.
7. [PDF] [indeed, yet to come]: Monoidal Categories for the working physicist, a tentative introduction to Categorical approach to Quantum Mechanics.
Exercises in Category theory (IT)

[PDF] A list of exercises in Category Theory. Last version October 2017. No particular order, nor particular care in their structure.

SEMINARS - ¢øsmøs

[html] A running seminar on category theory, at CMUC.

SEMINARS - Chu construction

A running seminar on the Chu construction; videos and slides of talks; my talk at Coimbra pointfree topology seminar: slides and video here.

### Conferences

SPEAKER - A few remarks on the fibration of algebras

[pdf] Slides of my talk at ItaCa 2022 in Pisa. Also, a video recording.

SPEAKER - A formal category theory of derivators

[pdf] A talk for “New Directions in Group Theory & Triangulated Categories” online seminar; check their website!

SPEAKER - ItaCa Fest I

[pdf] A talk @ItaCa about arXiv:2011.13881.

SPEAKER - ItaCa 2019

[html] The ItaCa website.

ORGANISER - ItaCa Fest I

[html] The ItaCa project continues online, every month.

PARTICIPANT - Applied CT 2019

Member of ACT2019, in the Profunctor Optics group. We study the description of Haskell’s optics as certain endo-profunctors $p : \mathcal{A} \to \mathcal{A}$ with a compatible action of their domain.

ORGANISER - PSSL 103

Masaryk University will host the 103rd edition of PSSL. We look forward to see you in Brno! all the relevant informations are on the conference website.

SPEAKER - British Topology Meeting

Speaker at BTM2017

SPEAKER - Some trends in Algebra

Speaker at STA2017

SPEAKER - Categorical day in Turin

Speaker at the categorical day in Turin

PARTICIPANT - Categorical day in Turin

Attended the categorical day in Turin

PARTICIPANT - Kan Extension Seminar

From January to June 2014 I’ve been a proud member of the Kan extension seminar. I wrote about Freyd and Kelly’s paper “Categories of continuous functors, I”, a copy of which you can find here. This experience culminated with the participation to an informal series of short seminars at the Winstanley Lecture Theatre in Trinity College, right before the beginning of the 2014 International Category Theory Conference.

### Reference Cards

REFCARDS - Differential 2-rigs

[PDF] A presentation by T. Massacrier on his internship project at TalTech, about 2-rigs and differential equations therein.

TALK - fiblang

[pdf] Slides for a talk on Fibrational Linguistics.

TALK - ItaCa 2019

[pdf] Slides for my talk at ItaCa 2019.

TALK - Cohesion@Roma1

[pdf] A talk at Sapy about axiomatic cohesion. The slides are better appreciated with acroread. Sorry.

TALK - Categories@unipd

[html] A fast talk at unipd about category theory and functional programming

TALK - Yosegi@ULB

[pdf] A talk at ULB about an almost finished preprint with I. Di Liberti.

REFCARDS - oberseminar FCT

[html] A brief slideshow about what I’m doing as a guest at MPI. The file was written using emacs and org-reveal

TALK - Kan extensions

[PDF] A brief cheatsheet on Kan extensions, written for TiCT.

REFCARDS - Monoidal and enriched derivators

[PDF] A brief cheatsheet on enriched derivators and the {0,1,2}-Grothendieck construction.

### Miscellaneous

LAGDA - a few exercises in agda

[html] A few exercises from Jacobs’ book.

CARDS - Bede the Venerable

[pdf] De natura rerum is a treatise by the Anglo-Saxon monk Bede, composed in 703 as a companion-piece to his De temporibus (‘On Times’).

CARDS - A bullet journal template

[html] I didn’t care about the object in itself, but it’s been an interesting typographical proof of concept.

CARDS - Aurea Catena Homeri

[pdf] Aurea Catena Homeri. Written in German by Dr. A. J. Kirchweger, was first printed in 1723, though it was distributed in a handwritten format prior to that time. It is said to be one of the most important books ever created giving insight into alchemy-the idea that all creation, no matter what its nature, is closely interconnected, that a deeply secret connection pervades all of nature, that one thing relates to the next and things depend upon each other.

CARDS - Rumi / poesie (IT)

[PDF] A booklet collecting a few poems by Jalāl ad-Dīn Moḥammad Rūmī

CARDS - Ovum Zoroastræum

[pdf] from A. Kircher’s Oedipus Aegyptiacus.

CARDS - Dao De Jing

[pdf] The Tao Te Ching (pinyin: Dàodé Jīng) also known as Lao Tzu or Laozi, is a Chinese classic text traditionally credited to the 6th-century BC sage Laozi.

CARDS - De magia Mathematica (Latin)

[PDF] A copy of G. Bruno’s De Magia Mathematica. I liked TeXing this!

NOTES - Homotopical interpretation of stack theory

[PDF] In their paper “Strong stacks and classifying spaces” A. Joyal and M. Tierney provide an ​internal characterization of the classical (or ‘‘folk’’) model structure on the category of groupoids in a Grothendieck topos E. The fibrant objects in the classical model structure on $\mathbf{Gpd}(\mathcal{E})$ are called ‘‘strong stacks’’, as they appear as a strengthening of the notion of stack in $\mathcal{E}$ (i.e. an internal groupoid object in $\mathcal{E}$ subject to a certain condition which involves ‘‘descent data’’). The main application is when $\mathcal{E}$ is the topos of simplicial sheaves on a space or on a site: in that case then strong stacks are intimately connected with classifying space​s of simplicial groups.

Adapting the presentation to the audience needed a ‘‘gentle introduction’’ to Topos Theory and the internalization philosophy of Category Theory, and a more neat presentation of the folk model structure on $\mathbf{Gpd}(\mathbf{Set})$ (not to mention the original article by Joyal and Tierney was utterly hard-to-read, so I tried to fill some holes and unraveled some prerequisites).

NOTES - Categorical Groups

[PDF] Categorical groups (or “strict 2-groups”) arise, like many other notions, as a categorification. They appear in a number of forms: as “fully dualizable” strict monoidal categories, internal categories in $\mathbf{Grp}$, internal groups in $\mathbf{Cat}$, crossed modules, strict 2-groupoids with a single object…
This variety of incarnations gives a very rich theory which can be built by the power of analogy with the set-theoretic case: my exposition will concentrate mostly on two sides of the story:

1. 0407463 As set-theoretic groups can be linearly represented on vectors spaces, so 2-groups can be 2-linearly represented on 2-vector spaces, thanks to a construction by Voevodsky and Kapranov; the category 2-$\mathbf{Vect}$ carries an astoundingly rich structure, and so does the category of representations Fun($\mathbf{G}$, 2-$\mathbf{Vect}$).
2. 0801.3843 As (suitably tame) topological groups give Cech theory of principal G-bundles, so 2-groups give Cech theory of principal 2-bundles; Cech cocycles can be characterized, thanks to an idea by G. Segal, as suitable functors, allowing to recover a categorified Cech theory of “2-bundles”.
NOTES - Categorification on AQFT

[PDF] Classical AQFT can be defined as a cosheaf $\mathcal{A}$ of $\mathrm{C}^\ast$-algebras on the manifold of space-time (or more generally, on a suitable lorentzian manifold playing such rôle) M, satisfying two axioms: locality, ensuring that observables in an open region are a fortiori observables in any superset of that region, and causality, ensuring that If $U,V$ are spacelike separated regions, then $\mathcal A(U)$ and $\mathcal A(V)$ pairwise commute as subalgebras of $\mathcal A(M)$.

Now what if we want to suitably categorify this notion, extending it to the realm of tensor categories (that is, categories equipped with a tensor functor subject to suitable axioms)? Causality has to be replaced by a higher-categorical analogue of the concept of commutators of a subalgebra of $\mathcal{B}(\mathbb H)$ and Von Neumann algebras, leading to the definition of a Von Neumann category as a subcategory of $\mathbf{Hilb}_{\mathbb H}$ which equals its double commutant.

NOTES - Homotopical Algebra for $\mathrm{C}^\ast$-algebras

[PDF] Homotopical Algebra showed to be extremely fruitful in studying categories of “things that resemble spaces” and structured spaces, keeping track of their structure in the step-by-step construction of abstract homotopy invariants; so in a certain sense it is natural to apply this complicated machinery to the category $\mathrm{C}^\ast\text{-}\mathbf{Alg}$: all in all, Gel’fand-Naimark’s theorem tells that there exists an equivalence $$\mathrm{C}^\ast\text{-}\mathbf{Alg}\cong \mathbf{LCHaus}.$$

Starting from this we shouldn’t be surprised by the existence of homotopical methods in $\mathrm{C}^\ast$-algebra theory, and it should be natural to spend a considerable effort to endow $\mathrm{C}^\ast\text{-}\mathbf{Alg}$ with a model structure, maybe exploiting one of the various pre-existing model structures on $\mathbf{Top}$: this is (almost) what [Uuye] proposed in his article.

The main problem is that the category of $\mathrm{C}^\ast$-algebras admits a homotopical calculus which can’t be extended to a full model structure in the sense of [Quillen]. This is precisely Theorem 5.2, which we take from [Uuye], who repeats an unpublished argument by Andersen and Grodal; the plan to overcome this difficulty is to seek for a weaker form of Homotopical Calculus, still fitting our needs. To this end, the main reference is [Brown]’s thesis, which laid the foundations of this weaker abstract Homotopy Theory, based on the notion of “category with fibrant objects”. Instead of looking for a full model structure on $\mathrm{C}^\ast\text{-}\mathbf{Alg}$ we seek for a fibrant one, exploiting the track drawn by [Uuye]’s paper, which is the main reference of the talk together with [Brown]’s thesis.

THESIS - My MSc thesis

[PDF] Orlov spent lots of years studying the derived category $\mathbf{D}^b_\text{coh}(X)$ of coherent sheaves on a variety $X$; in the spirit of reconstruction theory, lots of algebraic properties of the category itself reflect into geometric properties of the space $X$.