Fosco Loregian


Ma olen matemaatik.

Since October 2023, I am a Researcher at the Compositional Systems and Methods group at TalTech.

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. An updated research statement is here (tl;dr: I do category theory here, category theory there).


The research I have done so far, grouped by topic.

Homotopy theory

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

✅ Triangulated factorization systems and t-structures
  title    = {Triangulated factorization systems and t-structures},
  journal  = {Journal of Algebra},
  volume   = {550},
  pages    = {219-241},
  year     = {2020},
  issn     = {0021-8693},
  doi      = {},
  url      = {},
  author   = {F. Loregian and S. Virili},
✅ t-structures in stable $\infty$-categories
  title    = {t-structures on stable (infinity,1)-categories},
  year     = {2016},
  school   = {SISSA},
  url      = {},
  author   = {F. Loregian}

or (better)

  doi       = {10.48550/ARXIV.2005.14295},
  url       = {},
  author    = {Loregian, F.},
  title     = {t-structures on stable infinity-categories},
  publisher = {arXiv},
  year      = {2020},
  copyright = { perpetual, non-exclusive license}
🟥 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.

✅ Hearts and Towers in stable $\infty$-categories
  doi       = {10.1007/s40062-019-00237-0},
  url       = {},
  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
  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     = {}


🟥 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.

✅ Accessibility and presentability in 2-categories
  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     = {},
  url     = {},
  author  = {I. {D}i Liberti and F. Loregian},

Coend calculus

✅ Profunctor optics, a categorical update
  doi       = {10.48550/ARXIV.2001.07488},
  url       = {},
  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 = { perpetual, non-exclusive license}
✅ Escrows are optics

To appear on thejft, V 1 issue 1.

  doi       = {10.48550/ARXIV.2105.10028},
  url       = {},
  author    = {Genovese, F. and Loregian, F. and Palombi, D.},
  title     = {Escrows are optics},
  publisher = {arXiv},
  year      = {2021},
  copyright = {Creative Commons Attribution 4.0 International}
✅ Coends of higher arity
  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     = {}
✅ A Fubini rule for $\infty$-coends
  title   = {A {F}ubini rule for infinity-coends},
  author  = {Loregian, F.},
  year    = {2018},
  month   = {dec},
  journal = {Preprints of the MPIM},
  number  = {68}
✅ Coend Calculus
  doi       = {10.1017/9781108778657},
  url       = {},
  year      = {2021},
  month     = jun,
  publisher = {Cambridge University Press},
  author    = {F. Loregian},
  title     = {(Co)end Calculus}

Petri nets

✅ A Categorical Semantics for Hierarchical Petri Nets
  doi       = {10.4204/eptcs.350.4},
  url       = {},
  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
  doi       = {10.4204/eptcs.372.5},
  url       = {},
  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
  doi       = {10.1007/978-3-030-78946-6_10},
  url       = {},
  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}

Automata theory

🟥 Automata and coalgebras in categories of species
      title={Automata and coalgebras in categories of species}, 
      author={Loregian, F.},
✅ Bicategories of automata, automata in bicategories

Published in ACT2023

  author    = {Boccali, G. and Laretto, A. and Loregian, F. and Luneia, S.},
  year      = {2023},
  title     = {Bicategories of Automata, Automata in Bicategories},
  editor    = {Staton, Sam and Vasilakopoulou, Christina},
  booktitle = {Proceedings of the Sixth International Conference on
               Applied Category Theory 2023,
               University of Maryland, 31 July - 4 August 2023},
  series    = {Electronic Proceedings in Theoretical Computer Science},
  volume    = {397},
  publisher = {Open Publishing Association},
  pages     = {1-19},
  doi       = {10.4204/EPTCS.397.1},
✅ Completeness for categories of generalized automata

Published in CALCO2023.

  author = {Boccali, G. and 
            Laretto, A. and 
            Loregian, F. and 
            Luneia, S.},
  title =	{Completeness for categories of generalized automata},
  booktitle =	{10th Conference on Algebra and Coalgebra 
                 in Computer Science (CALCO 2023)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in 
             Informatics (LIPIcs)},
  ISBN =	{978-3-95977-287-7},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{270},
  editor =	{Baldan, Paolo and de Paiva, Valeria},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  doi =		{10.4230/LIPIcs.CALCO.2023.20},
🟥 The semibicategory of Moore automata
      title={The semibicategory of Moore automata},
      author={Boccali, G. and
              Femić, B. and
              Laretto, A. and
              Loregian, F. and
              Luneia, S.},

Miscellaneous works

✅ Fibrational linguistics I and II
  doi       = {10.48550/ARXIV.2201.01136},
  url       = {},
  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}


  doi       = {10.48550/ARXIV.2207.06765},
  url       = {},
  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}
✅ Differential 2-rigs
  title = {Differential 2-rigs},
  volume = {380},
  ISSN = {2075-2180},
  url = {},
  DOI = {10.4204/eptcs.380.10},
  journal = {Electronic Proceedings in Theoretical Computer Science},
  publisher = {Open Publishing Association},
  author = {Loregian,  Fosco and Trimble,  Todd},
  year = {2023},
  month = aug,
  pages = {159–182}
🟥 Functorial Erkennen

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

✅ Functorial semantics for partial theories
  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        = {},
  doi        = {10.1145/3434338},
  journal    = {Proc. ACM Program. Lang.},
  month      = {jan},
  articleno  = {57},
  numpages   = {28},
🟥 Categorical Ontology I - Existence

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

✅ Categorical notions of fibration
title    = {Categorical notions of fibration},
journal  = {Expositiones Mathematicae},
volume   = {38},
number   = {4},
pages    = {496-514},
year     = {2020},
issn     = {0723-0869},
doi      = {},
url      = {},
author   = {F. Loregian and E. Riehl},
✅ Homotopical algebra is not concrete
  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     = {}


My teaching experience so far.
🎓 A course on 2-categories

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

🌱 Functorial semantics

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

🌱 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…).

🎓 Introduction to category theory ITI9200

See here


FELINYTY: Formal Education in Literature, Informatics, and NaturallY-Themed Yarns is a reading course on Formal Category Theory.

More info available here

🎓 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.

🌱 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.


As speaker

🎤 Welcome to the machine

[pdf] Slides of my talk at ItaCa 2023 in Torino.

🎤 A few remarks on the fibration of algebras

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

🎤 A formal category theory of derivators

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

🎤 ItaCa Fest I

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

🎤 fiblang

[pdf] Slides for a talk on Fibrational Linguistics.

🎤 ItaCa 2019

[pdf] Slides for my talk at ItaCa 2019.

📣 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.

🎤 Cohesion@Roma1

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

🎤 Categories@unipd

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

🎤 British Topology Meeting

Speaker at BTM2017

🎤 Yosegi@ULB

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

🎤 Some trends in Algebra

Speaker at STA2017

🎤 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

🎤 Categorical day in Turin

Speaker at the categorical day in Turin

📣 Categorical day in Turin

Attended the categorical day in Turin

🛠️ 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.


🎤 ItaCa 2023

[html] The 4th ItaCa in-person meeting, in Turin.

🎤 ItaCa 2019

[html] The first ItaCa in-person meeting, in Milan.

🛠️ ItaCa Fest I,II,III (and counting)

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

🛠️ 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.

Reference Cards

The idea of having "reference cards" comes from here.
✍ Street-Walters-Yoneda structures

[PDF] A presentation on Street-Walters’ “Yoneda structures on 2-categories” for the Bob Walters’ fest in Tallinn.

✍ Differential 2-rigs

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

✍ 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).

✍ 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”.
✍ 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.

✍ Kan extensions

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

✍ 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.

✍ Monoidal and enriched derivators

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


Fosco G. Loregian

Akadeemia tee 21B,
12618 Tallinn, Estonia 🇪🇪

skype: fosco.loregian
github: tetrapharmakon