Fosco Loregian

"When you meet your Buddha, kill him"

Name Fosco G. Loregian
Born May 23, 1987
Address Milky Way
Languages Italian (mothertongue),
English (fluently),
French (poorly),
Good places First,

(wannabe) Category Theorist, Bodhisattva and TEXnic.

A kind of CV

Jun 2016


in stable ∞-categories

This is the thesis collecting the three works below.

Jul 2015


in stable ∞-categories

This is the third joint work with D. Fiorenza, about t-structures in stable ∞-categories, which studies recollements. We develop the theory of recollements in a stable ∞-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 D, given t-structures t0, t1 on D0, D1. Such a classical result, well-known in the setting of triangulated categories, acquires a new taste when t-structure are described as suitable (∞-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 ∞-category.

Jan 2015

Hearts and Towers

in stable ∞-categories

This is the second joint work with D. Fiorenza, about t-structures in stable ∞-categories, which shows that in the ∞-categorical setting semiorthogonal decompositions on a stable ∞-category C arise decomposing morphisms in the Postnikov tower induced by a chain of t-structures, regarded (thanks to our previous work) as multiple factorization systems on C. A slightly unexpected result is that t-structures having stable classes, i.e. such that both classes are stable ∞-subcategories of C, are precisely the fixed points for the natural action of Z on the set of t-structures, given by the shift endofunctor.

Dec 2014


This is the (co)end, my only (co)friend

A short note about coend calculus. Co/ends are awesome, once you try to use them, your mathematical life changes forever. I put a considerable effort in making the arguments and constructions rather explicit: even if at some point I decided to come up with an arXiv-ed version, this document must be thought as a never-ending accumulation of examples, constructions and techniques which are better understood by means of co/ends. Feel free to give advices on how to improve the discussion!

Aug 2014

t-structures as factorizations

t-structures are normal torsion theories

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

Jan 2014-Jun 2014

Kan Extension Seminar

under the supervision of E. Riehl

Since January 2014 I am 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.

Jan-Jun 2014

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...).

Aug 2013


This is the first experiment of a (would be) annual meeting I would like to organize with my friends and colleagues (M. Porta, A. Gagna, G. Mossa and many others) in order to get updated (and -simplicially-enriched) about their research and interests. I want to warmly thank M. Porta for its patience in introducing me to the misteries of "higher" language, exposing me little pieces of his thesis and of the collective seminar Autour de DAG.
  As for its philosophical side, GoC-CoG can be defined as an experimental window open to autonomous research, where the word "research" has to be understood in etymological sense: the daily struggle of a bunch of curious minds towards Gnosis, the firm determination to avoid the fragmented, edonistic tendency of a certain modern mathematical practice, which concentrates collective efforts on solving a particular instance of a problem instead of building a theory eroding our questions millennium after millennium.
  (Someday you will also see the videos of our "conferences"...)

Jun 2013

Categorical Groups

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 Grp, internal groups in Cat, crossed modules, strict 2-groupoids witha 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.

  • ▶ 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-Vect carries an astoundingly rich structure, and so does the category of representations Fun(G, 2-Vect).
  • ▶ 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".

Jun 2013

Moerdijk & Ara talks

Notes of two seminars held in Paris 7 on June 17-18, 2013: I. Moerdijk spoke about Dendroidal sets and test categories, and a handwritten copy of the notes is here. D. Ara spoke about Foncteurs lax normalisés entre n-catégories strictes: here you can find a handwritten copy of the notes. Both have been written by F. Genovese, which I warmly thank. Maybe in the future I could merge Francesco's notes with mine and [;\LaTeX;] them.

Apr 2013

Homotopical interpretation of stack theory

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 Gpd(E) are called ''strong stacks'', as they appear as a strengthening of the notion of stack in E (i.e. an internal groupoid object in E subject to a certain condition which involves ''descent data''). The main application is when 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 Gpd(Sets) = Gpd (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).

Mar 2013

Categorification on AQFT

Classical AQFT can be defined as a cosheaf A of C*-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 A(U) and A(V) pairwise commute as subalgebras of 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 B(H) and Von Neumann algebras, leading to the definition of a Von Neumann category as a subcategory of HilbH which equals its double commutant.

Dec 2012

Homotopical Algebra for C*-algebras

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 C*-Alg: all in all, Gel'fand-Naimark's theorem asserts nothing but an (anti-)equivalence of categories C*-AlgLCHaus. Starting from this we shouldn't be surprised by the existence of homotopical methods in C*-algebra theory, and it should be natural to spend a considerable effort to endow C*-Alg with a model structure, maybe exploiting one of the various pre-existing model structures on Top: this is (almost) what [Uuye] proposed in his article.
  The main problem is that the category of C*-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 C*-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.

Jul 2012

My (graduate) thesis

about derived categories as an invariant

Orlov spent lots of years studying the derived category Dbcoh(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. As an example of this, we present in Section 1.3 a theorem which can be found clearly presented in [Cal] (see References) and due to Orlov-Bondal-Kapranov: the category Dbcoh(Pn) can be presented in a "simple" way by means of suitable shifts, mapping cones and cylinders of a finite number of twisting sheaves O(n). Orlov's study culminates into the central theorem of this work: Section 2.4 is entirely devoted to a complete account of the Theorem appeared in Bondal-Orlov's paper [Orl], by means of which one can exhibit an isomorphism of varieties X ~ Y starting from an equivalence of categories Dbcoh(X) ~ Dbcoh(Y), provided the canonical sheaf/bundle of X is ample or antiample. Orlov's idea is to explicitly build a space isomorphic to X by means of suitable objects in Dbcoh(X) which play the role of its points and invertible sheaves: we are allowed to do that mainly thanks to the good behaviour of the canonical sheaf of X, and by means of (a suitable categorification of) Grothendieck-Serre duality.

Jun 2012

Tentative complements on functorial topology

and A short introduction to triangulated categories

The first reason I chose to study Mathematics is Algebraic Topology. Despite the intrinsic complexity of the topic, I can't abandon the idea that this is the most elegant (=abstract) way to look at Geometry, so with the passing of time I cared to refine my understanding about homotopy theory, homological algebra and suchlike, accepting that the main reason Category Theory was invented is to turn Algebraic Topologist's deliria into rigorous statements. The "tentative complements" arose with two short-term goals, but rapidly fell off to become the draft of a draft: 1) explicitly solve some exercises nobody publicly solves (they're often left to the conscious reader, but mathematicians are often lazy people) and 2) give a categorical flavour even to basic statements on both General and basic Algebraic Topology. The "short intro" arose to extend and publicly propose one of the cornerstones in advanced Homological Algebra: triangulated categories.

Mar 2012

Galois Theory

One of the most beautiful pieces of Abstract Algebra discovered by mankind. It is indeed one of the subtlest incarnation of the mathematical notion of duality between two entities. Whenever we are interested in studying the (partially ordered) set of intermediate structure between a top-set E and a bottom-set F, we can turn to study Aut(E|F), the group of automorphisms of the top-set, fixing pointwise the bottom-set.

Jan-Jul 2011

The Jacobian Mathematicians

A dissident simple group

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. Here an (italian) manifesto explaining our intent: students talking to other students bring their own researches on the scene. Feel free to mail me if you want to reach us; meetings "illegally" took place in Padua @math department (63, via Trieste: Google maps puts it here).
  I gave seven lectures until now (but three more people talked about Game Theory, Fourier analysis, and analytical solutions to PDEs):

  • ONE Fibrations between spheres and Hopf theorem
  • TWO The importance of being abstract aka A gentle introduction to the categorical point of view to reality;
  • THREE low dimensional Topological Quantum Field Theories;
  • FOUR Chatting about complex geometry (from symplectic to Kahler manifolds);
  • FIVE Connections and Fiber Bundles, with a glance to the geometry of Classical Field Theory;
  • SIX: A short lecure about Computational Homological Algebra, my first piece of (!) applied Mathematics.
  • SEVEN [indeed, yet to come]: Monoidal Categories for the working physicist, a tentative introduction to Categorical approach to Quantum Mechanics.

Jun 2011

Hamiltonian Mechanics

My first love is Mathematical Physics, I cannot hide it. In writing these poor and chaotic pages I wanted to give myself some sort of glossa about basic mathematical methods used in Physics; in fact there's neither something original, nor something new in them, and I should have hidden them to your eyes if I had wanted to avoid a bad impression. But I definitely fell in love with Wheeler's idea that "Physics is [a part of] Geometry", and I'm fascinated by the ill genius of A. Fomenko, so I can't quit my quixotic quest for a rigorous foundation of Mathematical Physics...

Jun 2010

Lecture notes about Riemann surfaces

A Riemann surface is a complex one-dimensional manifold: asking the transition functions between charts to be (bi)holomorphisms between domains of the complex line obstructs the general (even smooth) two-dimensional manifold to be a RS. Algebraic, analytical and geometrical methods work in sinergy to give a beautiful and (at least in the case of compact spaces) complete theory.

Jan 2010

Notes on elementary Differential Geometry

The study of the Geometry of curves and surfaces culminates with Gauss' masterpiece Disquisitiones generales circa superficies curvas, where he defines the concept of intrinsical geometric property. Can a small ant lying on a sphere notice it is walking on a globally non-flat surface? And what if it was on a cylinder? And what if it was on a torus?

Mar 2013

A compact History of Geometry

The slides of a talk I gave @Caffè dei libri, Bassano del Grappa, chatting about the foundations of Modern Geometry. I started with a precise idea: making a public of non-mathematicians understand that the leading idea behind Algebraic Topology ("Thou shalt look at algebraic invariants of objects") is blatantly simple. You can judge if I did it: the person who invited me put on YT the video of the lecture, and also the further discussion.

Oct-Jan 2012

History of Mathematics

A collaborative, compact but hopefully pervasive acoount. It's neither a masterpiece, nor it contains something new, but the spirit we used to write it should be preserved. Beware! The pdf file is ~17MB.


Two Poems

As a mathematician, I'm interested in the whole human culture, seeking the more general and comprehensive view to explain reality; this said, it is natural that I'm particularly linked to Dante's Commedia, one of the greatest masterpieces of the whole human history.
  I tried (indeed, twice) to grasp the meaning of Dante's work composing a couple of "clownish" imitations, inspired by the political and social facts (or at least of those which are linked to my student's life) of the period. Both started trying to describe the "cut-throat savings" promoted by the Italian government in the years 2009-2011, which affected (and will destroy during the next years) the quality of teaching. They are obviously written in Italian: enjoy them.

Fosco G. Loregian

Trst, Italy; via Bonomea, 265 - 34136

E-mail: tetrapharmakon[at]gmail[dot]com

E-mail: floregi[at]sissa[dot]it

skype: killing_buddha

People I know
  • M. F.: a smart guy.
  • P.B.: an even smarter guy.
  • F. G.: a good mathematician.
  • M. G.: a group-theorist in love with the world.
  • S. T.: "I have lived through much, and now I think I have found what is needed for happiness."
  • D. T.: L'homme le plus bon du monde.