# Fosco Loregian

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.

## Stuff

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:

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”.
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.

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):

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…

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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## Contacts

Fosco G. Loregian

1151 Richmond St,
London, ON N6A 3K7

✉: tetrapharmakon@gmail.com
✉: fosco.loregian@gmail.com
✉: floregia@uwo.ca
skype: killing_buddha

## Friends

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.

## Works in progress

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 $X \map­sto \in­t^{\mu < \lamb­da} X^\mu \times f(\mu)$ 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 $\{x_i\}_{i\in\mathbb{N}}$ has a well-de­fined product.

Let’s assume that $M$ has a topol­o­gy: we’re ask­ing that every sequence $\{x_i\}_{i\in\mathbb{N}}$ 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 $\mathbf{sSet}_\text{Quil}$ 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”?
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?

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 $\lambda > \omega$)?

My inter­est in this ques­tion is threefold:

• I’ve always had this ques­tion in my head

• 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.