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.
Accepted for ACT 2023.
@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},
copyright = {Creative Commons Attribution 4.0 International}
}
Accepted for CALCO2023.
@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},
copyright = {Creative Commons Attribution 4.0 International}
}
@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},
copyright = {arXiv.org perpetual, non-exclusive license}
}
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},
copyright = {Creative Commons Attribution 4.0 International}
}
@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},
copyright = {Creative Commons Attribution 4.0 International}
}
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},
copyright = {Creative Commons Attribution 4.0 International}
}
@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},
copyright = {Creative Commons Attribution 4.0 International}
}
@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}
}
@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}
}
@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}
}
@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}
}
@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},
}
@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}
}
@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}
}
@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},
}
@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},
}
@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},
}
@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}
}
@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},
copyright = {arXiv.org perpetual, non-exclusive license}
}
@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}
}
@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}
}
@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}
}
@misc{boccali2023semibicategory,
title={The semibicategory of Moore automata},
author={Boccali, G. and
Femić, B. and
Laretto, A. and
Loregian, F. and
Luneia, S.},
year={2023},
eprint={2305.00272},
archivePrefix={arXiv},
primaryClass={math.CT}
}
[PDF] • The second chapter of a series devoted to join category theory and ontology.
[PDF] • The first chapter of a series devoted to join category theory and ontology.
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.
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.
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.
A course on 2-category theory (hopefully) held in Padua; there will be notes (hopefully), and beer afterwards.
[pdf] A talk at the Turin-Udine Logic Seminar.
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…).
A course on category theory held in Taltech.
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-!
A reading seminar at Masaryk University; each other Wednesday (first lecture October 11) from 2pm to 4pm. There will be cookies!
A running seminar on functorial semantics, with an eye on profunctors as an equivalent definition of algebraic theories.
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):
[PDF] A list of exercises in Category Theory. Last version October 2017. No particular order, nor particular care in their structure.
[html] A running seminar on category theory, at CMUC.
A running seminar on the Chu construction; videos and slides of talks; my talk at Coimbra pointfree topology seminar: slides and video here.
[pdf] Slides of my talk at ItaCa 2022 in Pisa. Also, a video recording.
[pdf] A talk for “New Directions in Group Theory & Triangulated Categories” online seminar; check their website!
[pdf] A talk @ItaCa about arXiv:2011.13881.
[html] The ItaCa website.
[html] The ItaCa project continues online, every month.
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.
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 at BTM2017
Speaker at STA2017
Speaker at the categorical day in Turin
Attended the categorical day in Turin
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.
[PDF] A presentation by T. Massacrier on his internship project at TalTech, about 2-rigs and differential equations therein.
[pdf] Slides for a talk on Fibrational Linguistics.
[pdf] Slides for my talk at ItaCa 2019.
[pdf] A talk at Sapy about axiomatic cohesion. The slides are better appreciated with acroread. Sorry.
[html] A fast talk at unipd about category theory and functional programming
[pdf] A talk at ULB about an almost finished preprint with I. Di Liberti.
[html] A brief slideshow about what I’m doing as a guest at MPI. The file was written using emacs and org-reveal
[PDF] A brief cheatsheet on Kan extensions, written for TiCT.
[PDF] A brief cheatsheet on enriched derivators and the {0,1,2}-Grothendieck construction.
[html] A few exercises from Jacobs’ book.
[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’).
[html] I didn’t care about the object in itself, but it’s been an interesting typographical proof of concept.
[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.
[PDF] A booklet collecting a few poems by Jalāl ad-Dīn Moḥammad Rūmī
[pdf] from A. Kircher’s Oedipus Aegyptiacus.
[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.
[PDF] A copy of G. Bruno’s De Magia Mathematica. I liked TeXing this!
[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 spaces 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).
[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:
[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.
[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.
[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 $X$.
Akadeemia tee 21B,
12618 Tallinn, Estonia 🇪🇪
mail: fosco.loregian@gmail.com
skype: fosco.loregian
github: tetrapharmakon