[PDF] This is the thesis collecting the three works below.
[PDF] This is the third joint work with D. Fiorenza, about -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” -structure on , given -structures , on , . Such a classical result, well known in the setting of triangulated categories, acquires a new taste when -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 -fold gluings, valid in any stable -category.
This is the second joint work with D. Fiorenza, about -structures in stable -categories, which shows that in the -categorical setting semiorthogonal decompositions on a stable -category arise decomposing morphisms in the Postnikov tower induced by a chain of -structures, regarded (thanks to our previous work) as multiple factorization systems on .
A slightly unexpected result is that -structures having stable classes, i.e. such that both classes are stable -subcategories of , are precisely the fixed points for the natural action of on the set of -structures, given by the shift endofunctor.
[PDF] 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!
[PDF] My first joint work with D. Fiorenza, laying the foundations of the theory of -structures in stable -categories under the unifying notion of a “normal torsion theory”: as you can see in the abstract, we characterize -structures in stable -categories as suitable quasicategorical factorization systems. More precisely we show that a -structure on a stable -category is equivalent to a normal torsion theory on , i.e. to a factorization system where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.
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…).
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] This is the first experiment of a meeting I organized with some friends and colleagues (M. Porta, A. Gagna, G. Mossa and many others) in order to get updated (and “enriched” –pun intended) about their research and interests. M. Porta patiently introduced me to the arcane misteries of bits of “higher” language, exposing me to 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”…)
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 them.
[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 , internal groups in , 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] 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 are called ‘‘strong stacks’’, as they appear as a strengthening of the notion of stack in (i.e. an internal groupoid object in subject to a certain condition which involves ‘‘descent data’’). The main application is when 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 (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] Classical AQFT can be defined as a cosheaf of -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 are spacelike separated regions, then and pairwise commute as subalgebras of .
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 and Von Neumann algebras, leading to the definition of a Von Neumann category as a subcategory of 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 : all in all, Gel’fand-Naimark’s theorem tells that there exists an equivalence
Starting from this we shouldn’t be surprised by the existence of homotopical methods in -algebra theory, and it should be natural to spend a considerable effort to endow with a model structure, maybe exploiting one of the various pre-existing model structures on : this is (almost) what [Uuye] proposed in his article.
The main problem is that the category of -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 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 of coherent sheaves on a variety ; in the spirit of reconstruction theory, lots of algebraic properties of the category itself reflect into geometric properties of the space .
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
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
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] 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…
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.
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?
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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.
D.F.: the man who taught me things.
U.S.: the man who writes. A lot.
Let be a model cosmos (e.g. ); let the bicategory of -profunctors. This is a bicategory where every hom-category is a model category is a (non unique, but) natural way: . This is the paradigmatic example (the unique one at the moment) of a 2-category where every hom-category is a model category. Let be a locally model 2-category.
That’s the starting question: click.
I need it because of my survey for coends.
I’m rather sure that the result is true; also, it should have a rather formal proof using universal properties of adjoints. Obviously, I need it because if the philosophy “coends are integrals” is correct, then you definitely want this result to ensure that you can “integrate a functor by slices”. A different approach to the solution of ths problem is the following: it’s enough to prove the theorem for constant functors instead than for colimits, given the adjointness relation between the two: the left adjoint of a permuted composition is, up to an additional permutation, the composition of the left adjoints to the composands. The problem is that in the finite-set case this additional permutation is the “reversing order” one, sending 1 into , 2 into , etc. Nothing of this sort is available in the case of infinite sets (where should 0 go in such a bijection for ?).