Fosco Loregian


I am a mathematician. I enjoy category theory, stable homotopy theory, computations with the Serre spectral sequence, functional programming (mainly Haskell and Wolfram), bicategories of profunctors seen as universal semantics for 2-dimensional algebraic theories, lager beers, artificial languages (I'm learning Esperanto and Toki Pona; Ithkuil remains wishful thinking), Shunga paintings, drawing maps, typography (I crafted Giordano Bruno's de Magia Mathematica, a small booklet of Rumi poems, old alchemical tokens, a picture from Athanasius Kircher's Cabala Hebraeorum, and a lame edition of Laozi in XeLaTeX). 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.


Fibrational linguistics I and II

2201.01136 • Accepted for ACT2022.

Differential 2-rigs

2103.00938 • ArXiv contains an abridged version accepted for ACT2022.

A Categorical Semantics for Hierarchical Petri Nets

2101.09100 and GCM2021 • We show how a particular flavor of hierarchical nets, where the firing of a transition in the parent net must correspond to an execution in some child net, can be modelled utilizing a functorial semantics from a free category - representing the parent net - to the category of sets and spans between them. This semantics can be internalized via Grothendieck construction, resulting in the category of executions of a Petri net representing the semantics of the overall hierarchical net. We conclude the paper by giving an engineering-oriented overview of how our model of hiearchic nets can be implemented in a transaction-based smart contract environment.

A Categorical Semantics for Bounded Petri Nets

2101.09100 and ACT2021 • We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construction. The individual-token case is harder, as it requires a more explicit reliance on abstract methods.

Nets with Mana - A Framework for Chemical Reaction Modelling

2101.06234 and ICGT2021 • We use categorical methods to define a new flavor of Petri nets which could be useful in modelling chemical reactions.

Coends of higher arity

2011.13881 and doi • We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein (p,q)-ends (resp., (p,q)-coends). While higher arity co/ends are particular instances of “totally symmetrised” (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory.
  The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends.
  The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends.

Functorial semantics for partial theories

[PDF] • We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

A standard theorem on adjunctions in two variables

1902.06074 and REF • We record an explicit proof of the theorem that lifts a two-variable adjunction to the arrow categories of its domains.

A Fubini rule for $\infty$-coends

1902.06074 and REF • We prove a Fubini rule for $\infty$-co/ends of $\infty$-functors $F : \mathcal{C}^\text{op}\times\mathcal{C}\to \mathcal{D}$.

Categorical notions of fibration

1806.06129 and REF • Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^\text{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^\text{op} \to {\bf Set}$. A two-sided discrete variation encodes functors $B^\text{op} \times A \to {\bf Set}$, which are also known as profunctors from $A$ to $B$. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to ${\bf Cat}$, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model ${\mathcal V}$-profunctors internally to ${\mathcal V}$-${\bf Cat}$.

Accessibility and presentability in 2-categories

1804.08710 and JPAA • We outline a definition of accessible and presentable objects in a 2-category K endowed with a “KZ context”, that is to say a pair of lax-idempotent monads interacting in a prescribed way; this perspective suggests a unified treatment of many “Gabriel-Ulmer like” theorems, asserting how presentable objects arise as reflections of generating ones. We outline the notion of “(Gabriel-Ulmer) envelope” for a KZ context, sufficient to concoct Gabriel-Ulmer duality. We end the paper with a roundup of examples, involving classical (set-based and enriched), low dimensional category theory, and a perspective for future work, rooted in higher category theory and homotopy theory.

Factorization systems on (stable) derivators

1705.08565 • We define triangulated factorization systems on a given triangulated category, and prove that a suitable subclass thereof (the normal triangulated torsion theories) corresponds bijectively to t-structures on the same category. This result is then placed in the framework of derivators regarding a triangulated category as the underlying category of a stable derivator. More generally, we define derivator factorization systems in the 2-category $\mathbf{PDer}$, also formally describing them as algebras for a suitable strict 2-monad (this result is of independent interest), and prove that a similar characterization still holds true: for a stable derivator $\mathbb{D}$, a suitable class of derivator factorization systems (the normal derivator torsion theories) correspond bijectively with t-structures on the underlying category $\mathbb{D}(e)$ of the derivator. These two result can be regarded as the triangulated- and derivator- analogues, respectively, of the theorem that says that `t-structures are normal torsion theories’ in the setting of stable ∞-categories, showing how the result remains true whatever the chosen model for stable homotopy theory is.

Homotopical algebra is not concrete

1704.00303 and REF • We generalize Freyd’s well-known result that “homotopy is not concrete” offering a general method to show that under certain assumptions on a model category $\mathcal{M}$, its homotopy category ho($\mathcal{M}$) cannot be concrete with respect to the universe where $\mathcal{M}$ is assumed to be locally small. This result is part of an attempt to understand more deeply the relation between (some parts of) set theory and (some parts of) abstract homotopy theory.

t-structures in stable $\infty$-categories

[PDF] • This is my PhD thesis.

Coend Calculus

[PDF] & [url] • The book on coend calculus.

Hearts and Towers in stable $\infty$-categories

[PDF] and REF • We exploit the equivalence between $t$-structures and normal torsion theories on a stable $\infty$-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded $t$-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a $J$-slicing of a stable $\infty$-category $\mathcal{C}$, where $J$ is a totally ordered set equipped with a monotone $\mathbb{Z}$-action.

$t$-structures are normal torsion theories

1408.7003 and REF • My first joint work with D. Fiorenza, laying the foundations of the theory of $t$-structures in stable $\infty$-categories under the unifying notion of a “normal torsion theory”: as you can see in the abstract, we characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure on a stable $\infty$-category $\mathcal{C}$ is equivalent to a normal torsion theory $\mathbb{F}$ on $\mathcal{C}$, i.e. to a factorization system $(\mathcal{E}, \mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.


Escrows are optics

2105.10028 • We provide a categorical interpretation for escrows, i.e. trading protocols in trustless environment, where the exchange between two agents is mediated by a third party where the buyer locks the money until they receive the goods they want from the seller. A simplified escrow system can be modeled as a certain kind of optic in a monoidal category $\mathcal{M}$ (e.g., the category of sets with cartesian product); escrows can be regarded as morphisms of a category $\mathcal{E}(\mathcal{M})$, with the same objects of $\mathcal{M}$, and where the hom-objects are $\langle X , Y \rangle = \mathsf{Opt}_{\mathcal{M}}( \binom Y X , \binom X Y )$. When $X$ is a comonoid and $Y$ is a monoid in $\mathcal{M}$, $\mathcal{E}(\mathcal{M})(X,Y)$ is a monoid in $\mathsf{Set}$ (or in the base of enrichment chosen to model one’s specific problem), acting on the set of optics $ \binom B B \to \binom X Y $. Moreover, we define a map

\[\textstyle\lhd : \langle Y , X \rangle \times \mathsf{Opt}( \binom Y X , \binom B B ) \to \mathsf{Opt}( \binom Y X , \binom{X\otimes B} {Y\otimes B} )\]

having action-like properties. This has the following interpretation: the object $B$ acts as an intermediary in a transaction between $X$ and $Y$, modeled by an escrow in $\langle Y , X \rangle$.

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.

Profunctor optics, a categorical update

2001.07488 • Profunctor optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. They are modular, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have been studied only using Sets as the enriching category and in the non-mixed case. However, functional programming languages are arguably better described by enriched categories and we have found that some structures in the literature are actually mixed optics. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with $\mathcal{V}$-category structure. We provide some original families of optics and derivations, including an elementary one for traversals that solves an open problem posed by Milewski. Finally, we discuss a Haskell implementation.

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.


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.


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.


[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.


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.


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 $X$.


Fosco G. Loregian

Akadeemia tee 21B,
12618 Tallinn, Estonia 🇪🇪

skype: fosco.loregian
github: tetrapharmakon