## 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), lager beers, artificial languages (I'm learning Esperanto and Toki Pona), oriental art, drawing maps, typography (I like crafting old Mathematics books in XeLaTeX), and how to link all these things.

See my CV here. My email is below.

I do lots of things. You can see a more detailed version of my research interests here.

### Papers

On the unicity of formal category theories

arXiv: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.

t-structures in stable $\infty$-categories

[PDF] • This is my PhD thesis.

Factorization systems on (stable) derivators

arXiv: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.

A standard theorem on adjunctions in two variables

arXiv: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.

$t$-structures are normal torsion theories

arXiv: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.

Recollements in stable $\infty$-categories

arXiv: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.

Localization theory for derivators

arXiv: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.

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.

A Fubini rule for $\infty$-coends

arXiv: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

arXiv: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 ${\mathcal V}$-${\bf Cat}$.

Homotopical algebra is not concrete

arXiv: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.

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!

Accessibility and presentability in 2-categories

arXiv:1804.08710 • We outline a definition of accessible and presentable objects in a 2-category endowed with a Yoneda structure, this perspective suggests a unified treatment of many “Gabriel-Ulmer like” theorems (like the classical Gabriel-Ulmer representation for locally presentable categories, Giraud theorem, Gabriel-Popescu theorem, etc.), asserting how presentable objects arise as reflections of “generating” ones. In a Yoneda structure, whose underlying “presheaf construction” is $\bf P$, two non-equivalent definitions of presentability for $A\in\mathcal{K}$ can be given: in the weakest, it is generally false that all presheaf objects are presentable, this leads to the definition of a Gabriel-Ulmer structure, i.e. a Yoneda structure rich enough to concoct Gabriel-Ulmer duality and to make this asimmetry disappear. We end the paper with a roundup of examples, involving classical (set-based and enriched) and low- as well as higher-dimensional category theory.

### Teaching

A course on 2-categories

A course on 2-category theory (hopefully) held in Padua; there will be notes (hopefully), and beer afterwards.

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…).

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-!

Homotopy Type Theory

A reading seminar at Masaryk University; each other Wednesday (first lecture October 11) from 2pm to 4pm. There will be cookies!

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 few exercises in Category Theory. Last version October 2017.

¢øsmøs

[html] A running seminar on category theory, at CMUC.

### Conferences

Some trends in Algebra

Speaker at STA2017

Maltsiniotis

Attended the Journées en l’honneur de Georges Maltsiniotis held in Paris VII.

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.

ItaCa 2019

[html] The ItaCa website.

Category Theory 2014

Attended the 2014 edition of the 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”…)

catday in Turin

Speaker at the categorical day in Turin

catday in Turin

Attended the categorical day in Turin

British Topology Meeting

Speaker at BTM2017

Applied Category Theory

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.

PSSL 103

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.

### Reference Cards

REFCARDS - Yosegi@ULB

[pdf] A talk at ULB about an almost finished preprint with I. Di Liberti.

CARDS - Rumi / poesie (IT)

[PDF] A booklet collecting a few poems by Jalāl ad-Dīn Moḥammad Rūmī

REFCARDS - Monoidal and enriched derivators

[PDF] A brief cheatsheet on enriched derivators and the {0,1,2}-Grothendieck construction.

REFCARDS - Kan extensions

[PDF] A brief cheatsheet on Kan extensions, written for TiCT.

REFCARDS - Cohesion@Roma1

[pdf] A talk at Sapy about axiomatic cohesion. The slides are better appreciated with acroread. Sorry.

REFCARDS - Categories@unipd

[html] A fast talk at unipd about category theory and functional programming

CARDS - De magia Mathematica (Latin)

[PDF] A copy of G. Bruno’s De Magia Mathematica. I liked TeXing this!

REFCARDS - Formal category theory

[html] A brief slideshow about what I’m doing as a guest at MPI. The file was written using emacs and org-reveal

### Miscellaneous

Upwork

I am a part-time freelancer TeXnic on Upwork.

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).

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.

Hamiltonian Mech (IT)

[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…

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 $X$.

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. 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-$\mathbf{Vect}$).
2. 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”.
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.