逢佛殺佛

I am a mathematician. I enjoy stuff like category theory, underground brawls, lager beers, artificial languages, Japanese philosophy, apple juice, maps and typography. Drop me a line at **fosco.loregian@gmail.com**.

- t-structures in stable $\infty$-categories
[PDF] This is the thesis collecting the three works below.

- Recollements in stable $\infty$-categories
[PDF] 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
This is the second joint work with D. Fiorenza, about $t$-structures in stable $\infty$-categories, which shows that in the $\infty$-categorical setting semiorthogonal decompositions on a stable $\infty$-category $\mathcal{C}$ 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 $\mathcal{C}$.A slightly unexpected result is that $t$-structures having stable classes, i.e. such that both classes are stable $\infty$-subcategories of $\mathcal{C}$, are precisely the

*fixed points*for the natural action of $\mathbb Z$ on the set of $t$-structures, given by the shift endofunctor.- This is the (co)end, my only (co)friend
[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!

- $t$-structures are normal torsion theories
[PDF] 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.

- 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…).- 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.

- COG-GOC 2013
[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”…*)- 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 them.- 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:- arXiv: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-$\mathbf{Vect}$).
- arXiv: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”.

- 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 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 $\mathbf{Gpd}(\mathbf{Set})$ (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).- 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.- 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.

- My (graduate) 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 $X$.

- Functorial topology
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- Galois theory notes
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- 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):

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

- Hamiltonian Mech
[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…- 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.- 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?

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✉: fosco.loregian@gmail.com

✉: floregia@uwo.ca

skype: killing_buddha

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.

- Locally model bicategories
Let $\mathcal{V}$ be a model cosmos (e.g. $\mathbf{sSet}$); let $\mathbf{Prof}(\mathcal{V})$ the bicategory of $\mathcal{V}$-profunctors. This is a bicategory where every hom-category is a model category is a (non unique, but) natural way: $\mathbf{Prof}(\mathcal{V})(C,D) = \text{Fun}(C^\text{op}\times D, \mathbf{sSet})$. This is the paradigmatic example (the unique one at the moment) of a 2-category where every hom-category is a model category. Let $\mathbb{A}$ be a locally model 2-category.

- What is the universal property (in $2\text{-}\mathbf{Cat}$) of the 2-category having the same objects as $\mathbb{A}$ and hom-categories $\text{Ho}(\mathbb{A}(X,Y))$?
- Is the notion of a locally model 2-category the right notion of a 2-model category?
- Starting from the hom-wise model structure on each hom-category, is there a way to unravel a class of 1-cells in $\mathbb{A}$ which can be inverted (à la Pronk-Tommasini)? What’s the answer in the case of $\mathbf{Prof}(\mathcal{V})$, or more specifically $\mathbf{Prof}(\mathbf{sSet})$?
- Let $\mathbf{sSet}_\text{Quil}$ and $\mathbf{sSet}_\text{Joy}$ be the categories of simplicial sets endowed with the Quillen and Joyal model structure, respectively. There is a Bousfield localization of model categories $1\colon \mathbf{sSet}_\text{Quil} \leftrightarrow \mathbf{sSet}_\text{Joy}\colon 1$. Does this adjunction induces a 2-adjunction $\mathbf{Prof}(\mathbf{sSet}_{\text{Quil}}) \leftrightarrows \mathbf{Prof}(\mathbf{sSet}_\text{Joy})$ which can be thought as “Bousfield localization of a 2-model category”?
- Is it true that the functors $\text{Lan}_\varphi \dashv \varphi^*\dashv \text{Ran}_\varphi$ and $\text{Lift}_\varphi \dashv \varphi_*\dashv \text{Rift}_\varphi$ form triples of Quillen functors $\mathbf{Prof}(\mathbb{A},\mathbb{X}) \leftrightarrows \mathbf{Prof}(\mathbb{A},\mathbb{Y})$
- Is there a way to link the homotopy theory of profunctors to linear logic?

- A transfinite Fubini theorem for co/ends
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 $n$, 2 into $n-1$, etc. Nothing of this sort is available in the case of infinite sets (where should 0 go in such a bijection for $\omega$?).