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, underground brawls, lager beers, artificial languages, Japanese philosophy, maps, typography and how to link all these things. See my CV here. You can email me at I do lots of things, usually failing. You can see a more detailed version of my research interests here


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 PDer\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 D\mathbb{D}, a suit­able class of deriva­tor fac­tor­iza­tion sys­tems (the nor­mal deriva­tor tor­sion the­o­ries) cor­re­spond bijec­tively with t-struc­tures on the under­ly­ing cat­e­gory D(e)\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 &am­p;in­f;-­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 M\mathcal{M}, its homo­topy cat­e­gory ho(M\mathcal{M}) can­not be con­crete with respect to the uni­verse where M\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.

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 tt-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” tt-struc­ture on D\mathcal{D}, given tt-struc­tures t0t_0, t1t_1 on D0\mathcal{D}_0, D1\mathcal{D}_1. Such a clas­si­cal result, well known in the set­ting of tri­an­gu­lated cat­e­gories, acquires a new taste when tt-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 nn-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 tt-struc­tures in sta­ble \infty-cat­e­gories, which shows that in the \infty-cat­e­gor­i­cal set­ting semi­orthog­o­nal decom­po­si­tions on a sta­ble \infty-cat­e­gory C\mathcal{C} arise decom­pos­ing mor­phisms in the Post­nikov tower induced by a chain of tt-struc­tures, regarded (thanks to our pre­vi­ous work) as mul­ti­ple fac­tor­iza­tion sys­tems on C\mathcal{C}.

  A slightly unex­pected result is that tt-struc­tures hav­ing sta­ble class­es, i.e. such that both classes are sta­ble \infty-sub­cat­e­gories of C\mathcal{C}, are pre­cisely the fixed points for the nat­ural action of Z\mathbb Z on the set of tt-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!

tt-structures are normal torsion theories

[PDF] My first joint work with D. Fiorenza, lay­ing the foun­da­tions of the the­ory of tt-struc­tures in sta­ble \infty-cat­e­gories under the uni­fy­ing notion of a “nor­mal tor­sion the­o­ry”: as you can see in the abstract, we char­ac­ter­ize tt-struc­tures in sta­ble \infty-cat­e­gories as suit­able qua­si­cat­e­gor­i­cal fac­tor­iza­tion sys­tems. More pre­cisely we show that a tt-struc­ture on a sta­ble \infty-cat­e­gory C\mathcal{C} is equiv­a­lent to a nor­mal tor­sion the­ory F\mathbb{F} on C\mathcal{C}, i.e. to a fac­tor­iza­tion sys­tem (E,M)(\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 Grp\mathbf{Grp}, inter­nal groups in Cat\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-Vect\mathbf{Vect} carries an astoundingly rich structure, and so does the category of representations Fun(G\mathbf{G}, 2-Vect\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 Gpd(E)\mathbf{Gpd}(\mathcal{E}) are called ‘‘strong stack­s’’, as they appear as a strength­en­ing of the notion of stack in E\mathcal{E} (i.e. an inter­nal groupoid object in E\mathcal{E} sub­ject to a cer­tain con­di­tion which involves ‘‘de­s­cent data’’). The main appli­ca­tion is when E\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 Gpd(Set)\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 A\mathcal{A} of C\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,VU,V are space­like sep­a­rated regions, then A(U)\math­cal A(U) and A(V)\math­cal A(V) pair­wise com­mute as sub­al­ge­bras of A(M)\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 B(H)\math­cal{B}(\­mathbb H) and Von Neu­mann alge­bras, lead­ing to the def­i­n­i­tion of a Von Neu­mann category as a sub­cat­e­gory of HilbH\math­bf{Hilb}_{\­mathbb H} which equals its dou­ble commutant.

Homotopical Algebra for C\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 C-Alg\mathrm{C}^\ast\text{-}\mathbf{Alg}: all in all, Gel’­fand-­Naimark’s the­o­rem tells that there exists an equiv­a­lence C-AlgLCHaus.\math­rm{C}^\ast\­tex­t{-}\­math­bf{Al­g}\­cong \mathbf{LCHaus}.

  Start­ing from this we should­n’t be sur­prised by the exis­tence of homo­topi­cal meth­ods in C\mathrm{C}^\ast-al­ge­bra the­o­ry, and it should be nat­ural to spend a con­sid­er­able effort to endow C-Alg\mathrm{C}^\ast\text{-}\mathbf{Alg} with a model struc­ture, maybe exploit­ing one of the var­i­ous pre-ex­ist­ing model struc­tures on Top\mathbf{Top}: this is (al­most) what [Uuye] pro­posed in his article.

  The main prob­lem is that the cat­e­gory of C\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 C-Alg\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 Dcohb(X)\mathbf{D}^b_\text{coh}(X) of coher­ent sheaves on a vari­ety XX; in the spirit of recon­struc­tion theory, lots of alge­braic prop­er­ties of the cat­e­gory itself reflect into geo­met­ric prop­er­ties of the space XX.

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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Fosco G. Loregian

Kotlářská 2
611 37 Brno


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.

I.d.L.: A totem

Works in progress

Analytic functors

Let Bij(ω)\mathbf{Bij}(\omega) be the cat­e­gory hav­ing objects finite sets and mor­phisms only bijec­tions. An end­o­func­tor F:SetSetF\colon \math­bf{Set}\to \mathbf{Set} is called analytic if it results as the left Kan exten­sion of f:Bij(ω)Setf\colon \math­bf{Bi­j}(\omega) \to \mathbf{Set} (the “gen­er­at­ing species” of FF) along the nat­ural inclu­sion Bij(ω)Set\math­bf{Bi­j}(\omega) \hookrightar­row \mathbf{Set}. This means that FF acts on objects as XkBij(ω)Xk×f(k)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” k0f(k)k!Xk\sum_{k\ge 0} \frac{f(k)}{k!}X^k.

Now, the coend above still makes sense if we replace Bij(ω)\mathbf{Bij}(\omega) with Bij(λ)\mathbf{Bij}(\lambda) for an ordi­nale λ>ω\lambda > \omega. This means that we can define a func­tor F:SetSetF\colon \mathbf{Set}\to\mathbf{Set} to be λ\lambda-analytic if FF acts on objects as Xμ<λXμ×f(μ)X \map­sto \in­t^{\mu <\lamb­da} X^\mu \times f(\mu) which is pre­cisely the left Kan eten­sion of a λ\lambda-gen­er­at­ing species along the inclu­sion Bij(λ)Set\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 λ\lambda-an­a­lytic func­tors in a sim­i­lar way to ana­lytic ones (“FF is ana­lytic iff it pre­serves weak pull­backs” should cor­re­spond to “FF is λ\lambda-an­a­lytic iff it pre­serves λ\lambda-ary pullbacks”)?

  • Is it pos­si­ble to gen­er­al­ize this con­struc­tion from Set\mathbf{Set} to a generic sym­met­ric monoidal (bi­com­plete and closed?) cat­e­gory V\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 RR.

Weird things hap­pen when one tries to per­form trans­fi­nite ten­sor prod­ucts on the cat­e­gory Mod(R)\text{Mod}(R) (to be more con­crete, let’s take R=R=a field): in this case RRRR\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 MM such that every sequence (ev­ery net, to be more pre­cise, but let’s pre­tend that count­able sequences are enough) of ele­ments {xi}iN\{x_i\}_{i\in\mathbb{N}} has a well-de­fined product.

Let’s assume that MM has a topol­o­gy: we’re ask­ing that every sequence {xi}iN\{x_i\}_{i\in\mathbb{N}} is such that limki=1kxi\lim_{k\to \in­fty}\prod_{i=1}^k x_i exists in MM. I sus­pect that this entails that MM has triv­ial (=ini­tial) topol­o­gy, because if you con­sider M={1,e}M = \{1,e\} and the jump­ing sequence x2i=1,x2i+1=ex_{2i}=1, x_{2i+1}=e its con­ver­gence means that the topol­ogy can­not dis­tin­guish ee from the identity.

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

Homotopical algebra is not concrete

There’s this old result by Freyd that says that “ho­mo­topy is not con­crete”. In 2017, we write this result say­ing that there is a homo­topi­cal category (Top,W)(\math­bf{­Top}, \mathcal{W}), such that the localization Top[W1]\mathbf{Top}\!\!\left[\mathcal{W}^{-1}\right] is not (Set\mathbf{Set}-)concrete. This the­o­rem is as old as Quil­len’s def­i­n­i­tion of a model cat­e­go­ry, so I doubt that Freyd ignored that you can ask the fol­low­ing ques­tion: how often, and why, the local­iza­tion of a cat­e­gory at a class of weak equiv­a­lences is not concrete?

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 V\mathcal{V} be a model cos­mos (e.g. sSet\mathbf{sSet}); let Prof(V)\mathbf{Prof}(\mathcal{V}) the bicat­e­gory of V\mathcal{V}-pro­func­tors. This is a bicat­e­gory where every hom-­cat­e­gory is a model cat­e­gory is a (non unique, but) nat­ural way: Prof(V)(C,D)=Fun(Cop×D,sSet)\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 A\mathbb{A} be a locally model 2-category.

  1. What is the universal property (in 2-Cat2\text{-}\mathbf{Cat}) of the 2-category having the same objects as A\mathbb{A} and hom-categories Ho(A(X,Y))\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 A\mathbb{A} which can be inverted (à la Pronk-Tommasini)? What’s the answer in the case of Prof(V)\mathbf{Prof}(\mathcal{V}), or more specifically Prof(sSet)\mathbf{Prof}(\mathbf{sSet})?
  4. Let sSetQuil\mathbf{sSet}_\text{Quil} and sSetJoy\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:sSetQuilsSetJoy:11\colon \mathbf{sSet}_\text{Quil} \leftrightarrow \mathbf{sSet}_\text{Joy}\colon 1. Does this adjunction induces a 2-adjunction Prof(sSetQuil)Prof(sSetJoy)\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 LanφφRanφ\text{Lan}_\varphi \dashv \varphi^*\dashv \text{Ran}_\varphi and LiftφφRiftφ\text{Lift}_\varphi \dashv \varphi_*\dashv \text{Rift}_\varphi form triples of Quillen functors Prof(A,X)Prof(A,Y) \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:RFS(C)J : \math­b­b{R} \to FS(\mathcal{C}) be a fam­ily of fac­tor­iza­tion sys­tems indexed by R\mathbb R. Then, one can define a real num­ber λ\lambda to be “JJ-crit­i­cal” if the λ+ε\lambda+\varepsilon and λε\lambda -\varepsilon fac­tor­iza­tions of f:XYf\colon X\to Y are both dif­fer­ent from the λ\lambda factorization.

The idea is: you slice at time λ\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?


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!) Plt\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 nn-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 Plt\mathbf{Plt} as a full sub­cat­e­gory of a cat­e­gory of presheaves, in the same way Cat[Δop,Sets]\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 X,Y\math­bf{X}, \mathbf{Y} be two toposes of sheaves on topo­log­i­cal spaces X,YX,Y. Let XY\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 CatNsSetLTopos\math­bf{­Cat} \to_N \math­bf{s­Set} \to_L \mathbf{Topos} where NN is the nerve func­tor, Topos\mathbf{Topos} is a cat­e­gory of suf­fi­ciently small topos­es, and LL is the func­tor which arises as the left Kan exten­sion of J:ΔToposJ\colon \Delta \to \mathbf{Topos} (which sends [n][n] to the cat­e­gory of sheaves on the sphere Δ[n]\partial\Delta[n]) along the Yoneda embed­ding. Define XY:=L(N(XY))\math­bf{X} \odot \math­bf{Y} := L(N(\math­bf{X}\star \math­bf{Y}))

  • Is \odot a monoidal structure on Topos\mathbf{Topos}?
  • Under which conditions on X,YX,Y is XY\mathbf{X} \odot \mathbf{Y} equivalent to a topos of sheaves on a topological space XYX \odot Y?
  • What are the properties of the bifunctor (X,Y)XY(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 λ\lambda-sets (=presheaves on the cat­e­gory of ordi­nals smaller than a fixed λω\lambda \ge \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 C\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 gJg\in\mathcal{J} be an ele­ment of a cel­lu­lar class of edges J\math­cal J in a \infty-cat­e­go­ry. Then, gg has the RLP against trans­fi­nite horn inclu­sions ΛμκΔκ\Lamb­da_\­mu^\kappa \sub­set \Delta^\kappa.

  • It is related to my ques­tion about ana­lytic func­tors: a main prob­lem han­dling the def­i­n­i­tion of [Δλ,Set][\Delta_{\le \lamb­da}, \mathbf{Set}] is that one would like to have “gen­er­al­ized faces and degen­era­cies” to present Δλ\Delta_{\le \lambda} as a quo­tient of a free category.

    The prob­lem is that in Δλ\Delta_{\le \lambda} there are no “faces” in the sense we hope. The idea is that we have a “sim­ple” com­bi­na­to­r­ial descrip­tion for Δ\Delta because we have a “sim­ple” com­bi­na­torics for finite sets. Infi­nite sets com­pletely lose this simplicity.