Name  Fosco G. Loregian 
Born  May 23, 1987 
Address  Milky Way 
Languages  Italian (mothertongue),
English (fluently), French (poorly), 
Good places  First,
Second, (∞,n)^{th} 
tetrapharmakon@gmail.com floregi@sissa.it 
(wannabe) Category Theorist, Bodhisattva and T_{E}Xnic.
Jul 2015 
in stable ∞categories This is the third joint work with D. Fiorenza, about tstructures 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" tstructure on D, given tstructures t_{0}, t_{1} on D_{0}, D_{1}. Such a classical result, wellknown in the setting of triangulated categories, acquires a new taste when tstructure 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 welldefined, associative operation. From this we deduce a generalized associative property for nfold gluings, valid in any stable ∞category. 
Jan 2015 
in stable ∞categories This is the second joint work with D. Fiorenza, about tstructures 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 tstructures, regarded (thanks to our previous work) as multiple factorization systems on C. A slightly unexpected result is that tstructures 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 tstructures, 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 arXived version, this document must be thought as a neverending 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 
tstructures as factorizations tstructures are normal torsion theories My first joint work with D. Fiorenza, laying the foundations of the theory of tstructures in stable ∞categories under the unifying notion of a "normal torsion theory": as you can see in the abstract, we characterize tstructures in stable ∞categories as suitable quasicategorical factorization systems. More precisely we show that a tstructure 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 3for2 cancellation property, and a certain compatibility with pullbacks/pushouts. 
Jan 2014Jun 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. 
JanJun 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 gungho must meet at least once in a lifetime (bits of enriched category theory, toposes, spectral sequences, homotopy theory, weighted limits, coendjuggling, 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 simpliciallyenriched) 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. 
Jun 2013 
Categorical groups (or "strict 2groups") 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 2groupoids witha single object...

Jun 2013 
Moerdijk & Ara talks Notes of two seminars held in Paris 7 on June 1718, 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 ncatégories 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 spaces of simplicial groups.

Mar 2013 
Classical AQFT can be defined as a cosheaf A of C^{*}algebras
on the manifold of spacetime (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). 
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
stepbystep 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'fandNaimark's
theorem asserts nothing but an (anti)equivalence of categories C*Alg ∼ LCHaus.
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 preexisting model structures on Top: this is
(almost) what [Uuye] proposed in his article. 
Jul 2012 
about derived categories as an invariant Orlov spent lots of years studying the derived category D^{b}_{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. 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 OrlovBondalKapranov: the category D^{b}_{coh}(P^{n}) 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 BondalOrlov's paper [Orl], by means of which one can exhibit an isomorphism of varieties X ~ Y starting from an equivalence of categories D^{b}_{coh}(X) ~ D^{b}_{coh}(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 D^{b}_{coh}(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) GrothendieckSerre 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 shortterm 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 
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 topset E and a bottomset F, we can turn to study Aut(EF), the group of automorphisms of the topset, fixing pointwise the bottomset. 
JanJul 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).

Jun 2011 
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 onedimensional manifold: asking the transition functions between charts to be (bi)holomorphisms between domains of the complex line obstructs the general (even smooth) twodimensional 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 nonflat 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 nonmathematicians 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. 
OctJan 2012 
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. 
2009 
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. 
Fosco G. Loregian
Trst, Italy; via Bonomea, 265  34136
Email: tetrapharmakon[at]gmail[dot]com
Email: floregi[at]sissa[dot]it
skype: killing_buddha