# Fosco Loregian

I am a mathematician. I enjoy stuff like category theory, stable homotopy theory, computations with the Serre spectral sequence, TeXniques to write math notes, lager beers, artificial languages, Japanese philosophy, maps, typography and how to link all these things.

See my CV here. You can email me at fosco.loregian@gmail.com (istitutional emails change fast!).

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

## Stuff

⚠⚠⚠ 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.

⚠⚠⚠ 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-!

The scope of these lectures is to breach in Riehl-Verity’s theory of $\infty$-cosmoi. After a few classical readings

• Street, Ross, and Robert Walters. “Yoneda structures on 2-categories.” Journal of Algebra 50.2 (1978): 350-379.
• Street, Ross. “Fibrations in bicategories.” Cahiers de topologie et géométrie différentielle catégoriques 21.2 (1980): 111-160.
• Street, Ross. “Conspectus of variable categories.” Journal of Pure and Applied Algebra 21.3 (1981): 307-338.
• Street, Ross. “Fibrations and Yoneda’s lemma in a 2-category.” Category seminar. Springer Berlin/Heidelberg, 1974.
• Wood, Richard J. “Abstract pro arrows I.” Cahiers de topologie et géometrie différentielle categoriques 23.3 (1982): 279-290.
• Wood, R. J. “Proarrows II.” Cahiers de topologie et géométrie différentielle catégoriques 26.2 (1985): 135-168.
• Rosebrugh, Robert, and R. J. Wood. “Proarrows and cofibrations.” Journal of Pure and Applied Algebra 53.3 (1988): 271-296.

we embark in the study of RV theory:

• Riehl, Emily, and Dominic Verity. “Infinity category theory from scratch.” arXiv preprint arXiv:1608.05314 (2016). refcards here
• Riehl, Emily, and Dominic Verity. “The 2-category theory of quasi-categories.” Advances in Mathematics 280 (2015): 549-642.
• Riehl, Emily, and Dominic Verity. “Homotopy coherent adjunctions and the formal theory of monads.” Advances in Mathematics 286 (2016): 802-888.
• Riehl, Emily, and Dominic Verity. “Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions.” arXiv preprint arXiv:1401.6247 (2014).
• Riehl, Emily, and Dominic Verity. “Fibrations and Yoneda’s lemma in an $\infty$-cosmos.” Journal of Pure and Applied Algebra 221.3 (2017): 499-564.
• Riehl, Emily, and Dominic Verity. “Kan extensions and the calculus of modules for$\infty$-categories.” Algebraic & Geometric Topology 17.1 (2017): 189-271.
• Riehl, Emily, and Michael Shulman. “A type theory for synthetic $\infty$-categories.” arXiv preprint arXiv:1705.07442 (2017).
• Riehl, Emily, and Dominic Verity. “The comprehension construction.” arXiv preprint arXiv:1706.10023 (2017).
⚠⚠⚠ Homotopy Type Theory

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

REFCARDS - Kan extensions

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

[PDF] A one-page cheatsheet giving equivalent condition for which $F\dashv G$ in formal category theory.

Exercises in Category theory (IT)

[PDF] A few exercises in Category Theory. Last version October 2017.

REFCARDS - Monoidal and enriched derivators

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

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.

Homotopical algebra is not concrete

arXiv:1704.00303 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.

The paper has been accepted for publication on JHRS.

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.

The paper has been published in

Fiorenza, Domenico, and Fosco Loregiàn. “t-structures are normal torsion theories.” Applied Categorical Structures 24.2 (2016): 181-208.

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”…)

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”.
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.

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).

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.

[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

[PDF] 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):

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.
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…

## Contacts

Fosco G. Loregian

Kotlářská 2
611 37 Brno (CZ)
✉: fosco.loregian@gmail.com
✉: loregianf@math.muni.cz

## Friends

P.B.: a smart 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.

I.d.L.: A totem

## Works in progress

2-dimensional aspects of PDer

We want to describe the advantages of developing the theory of derivators initiated by Grothendieck and developed by Groth, Shulman and others in terms of 2-dimensional category theory. The axioms of derivator become natural requests on well-behaved’ objects in the larger 2-topos of prederivators. A few original results stem from this perspective (most notably, an elegant theory of monads).

Analytic functors

Let $\mathbf{Bij}(\omega)$ be the category having objects finite sets and morphisms only bijections. An endofunctor $F\colon \mathbf{Set}\to \mathbf{Set}$ is called analytic if it results as the left Kan extension of $f\colon \mathbf{Bij}(\omega) \to \mathbf{Set}$ (the “generating species” of $F$) along the natural inclusion $\mathbf{Bij}(\omega) \hookrightarrow \mathbf{Set}$. This means that $F$ acts on objects as $X \mapsto \int^{k\in\mathbf{Bij}(\omega)} X^k \times f(k)$ which is only a funny way to write the “formal power series” $\sum_{k\ge 0} \frac{f(k)}{k!}X^k$.

Now, the coend above still makes sense if we replace $\mathbf{Bij}(\omega)$ with $\mathbf{Bij}(\lambda)$ for an ordinale $\lambda > \omega$. This means that we can define a functor $F\colon \mathbf{Set}\to\mathbf{Set}$ to be $\lambda$-analytic if $F$ acts on objects as $% $ which is precisely the left Kan etension of a $\lambda$-generating species along the inclusion $\mathbf{Bij}(\lambda)\hookrightarrow \mathbf{Set}$.

• Is it possible to study the combinatorics of infinite sets using this notion, as Joyal did for the combinatorics of finite sets with his generating species’ theory?

• Is there any hope to describe in a nifty way $\lambda$-analytic functors in a similar way to analytic ones (“$F$ is analytic iff it preserves weak pullbacks” should correspond to “$F$ is $\lambda$-analytic iff it preserves $\lambda$-ary pullbacks”)?

• Is it possible to generalize this construction from $\mathbf{Set}$ to a generic symmetric monoidal (bicomplete and closed?) category $\mathcal V$? This leads to the following question:

Is there a general theory of monoidal categories where the tensor product has arbitrarily large arity?

Examples: every small-complete, cartesian category; every category of modules over a ring $R$.

Weird things happen when one tries to perform transfinite tensor products on the category $\text{Mod}(R)$ (to be more concrete, let’s take $R=$a field): in this case $R\otimes R \otimes R \otimes\cdots$ is infinitely-dimensional, and it is really painful to find a basis for it. Also, it is not clear how to retrieve the expected results “formally” (does the pentagon identity still ensure that “all diagrams commute”?).

Let’s decategorify this property to monoids: we’re looking for a monoid $M$ such that every sequence (every net, to be more precise, but let’s pretend that countable sequences are enough) of elements ${x_i\mid i\in\mathbb{N}}$ has a well-defined product.

Let’s assume that $M$ has a topology: we’re asking that every sequence ${x_i\mid i\in\mathbb{N}}$ is such that $\lim_{k\to \infty}\prod_{i=1}^k x_i$ exists in $M$. I suspect that this entails that $M$ has trivial (=initial) topology, because if you consider $M = {1,e}$ and the jumping sequence $x_{2i}=1, x_{2i+1}=e$ its convergence means that the topology cannot distinguish $e$ from the identity.

This is weird, but happens in rather common situations ($R$-modules, sets…). Are there any other examples?

Coend calculus in $\infty$-category theory

What is co/end calculus in a $\infty$-category, and can one recover all the nifty descriptions that co/end calculus classically allows (unification of elementary constructions, descriptions of operads as monoids for the substitution tensor product, the theory of profunctors…)?

Weighted limits in $\infty$-category theory

What is a weighted co/limit in a $\infty$-category?

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.

1. What is the universal property (in $2\text{-}\mathbf{Cat}$) of the 2-category having the same objects as $\mathbb{A}$ and hom-categories $\text{Ho}(\mathbb{A}(X,Y))$?
2. Is the notion of a locally model 2-category the right notion of a 2-model category?
3. 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 $\mathbf{Prof}(\mathbf{sSet})$?
4. Let $\mathbf{sSet}^Q$ and $\mathbf{sSet}^J$ be the categories of simplicial sets endowed with the Quillen and Joyal model structure, respectively. There is a Bousfield localization of model categories $1\colon \mathbf{sSet}^Q \leftrightarrow \mathbf{sSet}^J \colon 1$. Does this adjunction induces a 2-adjunction $\mathbf{Prof}(\mathbf{sSet}^Q) \leftrightarrows \mathbf{Prof}(\mathbf{sSet}^J)$ which can be thought as “Bousfield localization of a 2-model category”?
5. 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 $\mathbf{Prof}(\mathbb{A},\mathbb{X}) \leftrightarrows \mathbf{Prof}(\mathbb{A},\mathbb{Y})$
6. Is there a way to link the homotopy theory of profunctors to linear logic?
Morse theory via factorization systems

Is Morse theory the theory of some factorization systems on the category of cobordisms?

Let $J : \mathbb{R} \to FS(\mathcal{C})$ be a family of factorization systems indexed by $\mathbb R$. Then, one can define a real number $\lambda$ to be “$J$-critical” if the $\lambda+\varepsilon$ and $\lambda -\varepsilon$ factorizations of $f\colon X\to Y$ are both different from the $\lambda$ factorization.

The idea is: you slice at time $\lambda$, and what you get is different from what you get before, and from what you get after. In other words, “the topology of the slice changes”. In other words, critical values in Morse theory.

How many things true in Morse theory can be rewritten in this language?

Plots

A small category is a monoid with more than one object.

Now, if we do Abstract Algebra à la Bourbaki, monoids appear as rather highly-structured objects: they are unital, associative, total magmas (total, opposed to partial, means the operation is everywhere defined).

What happens when you try to understand what is a “partial magma with more than one object”?

In a few words, you get these things called plots studied in Tringali’s “Plots and Their Applications - Part I: Foundations” where your composition is not defined for each pair of consecutive arrows, and even if it’s defined it is possibly non-associative. Finally, you don’t have identities everywhere. Salvatore is a good friend of mine, and I’ve been pleased to slightly collaborate to his visionary project. I definitely fell in love with it.

Even if it’s difficult to believe so, even in such a poorly behaved setting one is able to find interesting results, and the theory established so far is everything but an empty academic exercise.

In (what we would like to call) plot theory you can define isomorphisms (yes, without having identities), and notice that “being an isomorphism” and “admitting an inverse” are different notions in the world of plots: the two notions collapse in category theory (a category is an associative plot, where the composition is defined and every object has a unit, in the same vein a monoid is an “extremely smooth partial magma”). You can then define isoids, i.e. plots where every arrow is an isomorphism.

Salvatore is even able to define morphisms of plots (punctors), natural transformations (trimmings, if I remember well the name I helped him to chose to continue the sartorial inspiration), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category!) $\mathbf{Plt}$ of plots to the category of associative plots, semi-categories [the obvious forgetful has two adjoints, left and right, giving two different completions], and categories.

Salvo to-do list:

• What’s a $n$-dimensional version of these guys?

• How can one define classical category theoretic notions (Kan extensions, adjunctions between punctors and their monads [do we obtain…magmads?], representables…are they still the same or not?) in the setting of plots?

• What is the localization of a plot with respect to a family of arrows, given that we lack associativity and unitality?

• How about a “simplicial” pov? Is there a shape giving $\mathbf{Plt}$ as a full subcategory of a category of presheaves, in the same way $\mathbf{Cat}\subset [\Delta^\text{op}, \mathbf{Sets}]$ and semi-categories arise as suitable semisimplicial sets (“degeneracies give identities”)?

• What about enrichment (whatever this means, in this context)? Can we do “formal” plot theory?

Functional analysis and symplectic geometry provide “natural factories” (read Examples 10-14 of Salvatore’s paper) of examples for such structures. As a toy example, one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.

Join of toposes

Let $\mathbf{X}, \mathbf{Y}$ be two toposes of sheaves on topological spaces $X,Y$. Let $\mathbf{X}\star \mathbf{Y}$ be the join of the two toposes seen as categories. This is not a topos anymore, but there is a canonical procedure to turn it into a topos: just compose the functors $\mathbf{Cat} \to_N \mathbf{sSet} \to_L \mathbf{Topos}$ where $N$ is the nerve functor, $\mathbf{Topos}$ is a category of sufficiently small toposes, and $L$ is the functor which arises as the left Kan extension of $J\colon \Delta \to \mathbf{Topos}$ (which sends $[n]$ to the category of sheaves on the sphere $\partial\Delta[n]$) along the Yoneda embedding. Define $\mathbf{X} \odot \mathbf{Y} := L(N(\mathbf{X}\star \mathbf{Y}))$

• Is $\odot$ a monoidal structure on $\mathbf{Topos}$?
• Under which conditions on $X,Y$ is $\mathbf{X} \odot \mathbf{Y}$ equivalent to a topos of sheaves on a topological space $X \odot Y$?
• What are the properties of the bifunctor $(X,Y)\mapsto X \odot Y$? How much does this operation resemble the join of topological spaces?
Transfinite simplicial homotopy

Is there a theory (and an homotopy theory) for simplicial $\lambda$-sets (=presheaves on the category of ordinals smaller than a fixed $\lambda \ge \omega$)?

My interest in this question is threefold:

• I need it because I want to describe in a homotopy meaningful way the cellularity property of a class of arrows in a $\infty$-category $\mathcal C$: morally, composition of edges corresponds to the choice of a lifting against a horn-inclusion. But if it is so, then “being closed under transfinite composition” must be equivalent to being in the class of maps having the RLP against “transfinite horn inclusions”. More precisely, let $g\in\mathcal{J}$ be an element of a cellular class of edges $\mathcal J$ in a $\infty$-category. Then, $g$ has the RLP against transfinite horn inclusions $\Lambda_\mu^\kappa \subset \Delta^\kappa$.
• It is related to my question about analytic functors: a main problem handling the definition of $[\Delta_{\le \lambda}, \mathbf{Set}]$ is that one would like to have “generalized faces and degeneracies” to present $\Delta_{\le \lambda}$ as a quotient of a free category.
The problem is that in $\Delta_{\le \lambda}$ there are no “faces” in the sense we hope. The idea is that we have a “simple” combinatorial description for $\Delta$ because we have a “simple” combinatorics for finite sets. Infinite sets completely lose this simplicity.