#topoi

[isomorphismes]

There are no theorems in category theory.

Emily Riehl, Category Theory In Context

Mathematicians often tell her this; hence the book.

If I had to summarise her views in one sentence, it would be:

Everything is an adjunction.

I also like the division these mathematicians are making to her: essentially, a theorem is anything that solves Feynman’s challenge: by a series of clear, unsurprising steps, one arrives at an unexpected conclusion.

Examples for me include:

17 possible tessellations

6 ways to foliate a surface

27 lines on a cubic

1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4 ways to link any-dimensional spheres.

the existence of sporadic groups

surprising rep-theory consequences of Young diagrams, Ferrers sequences, and so on (you could say the strangeness of integer partitions is really to blame here…)

59 icosahedra

8 geometric layouts

Books which are bristling with mathematical ideas of this kind include Montesinos on tessellations, Geometry and the Imagination (the original one), and Coxeter’s book on polyhedra (start with Baez on A-D-E if you want to follow my path). Moonshine and anything by Thurston or his students, I’ve found similarly flush with shockng content—quite different to what I thought mathematics would be like. (I had pictured something more like a formal logic book: row by row of symbols. But instead, the deeper I got into mathematics, the fewer the symbols and the more the surnames thanking the person who came up with some good idea.)

Note that a theorem is different here to some geometry — as in The Geometry of Schemes. The word geometry used in that sense, I feel, is to have a comprehensive enough vision of a subject to say how it “looks” — but the word theorem means the result is surprising or unintuitive.

This definition of a theorem, to me, presents a useful challenge to annoying pop-psychology that today lurks under the headings of Bayesianism, cognitive _______, behavioural econ/finance, and so on.

Following Buliga and Thurston to understand the nature of mathematical progress, within mathematics at least (where it’s clearer than elsewhere whether you understand something or not—compare to economic theory for example), there is a clear delination of what’s obvious and what’s not.

What is definitely not the case in mathematics, is that every logical or computable consequence of a set of definitions is computed and known immediately when the definitions are stated! You can look at a (particularly a good) mathematical exposition as walking you through the steps of which shifts in perspective you need to take to understand a conclusion. For example start with some group, then consider it as a topological object with a cohomology to get the centraliser. Or in Fourier analysis: re-present line-elements on a series of widening circles. Use hyperbolic geometry to learn about integers. Use stable commutator length (geometry) to learn about groups. Or read about Teichmüller stuff and mapping class groups because it’s the confluence of three rivers.

Sometimes mathematical explanations require fortitude (Gromov’s "energy") and sometimes a shift in perspective (Gromov’s (neg)"entropy").

This view of theorems should be contrasted to the disease of generalisation in mathematical culture. Citing two real-life grad students and a tenured professor in logic (one philosophical, one mathematical, the professor in computer science):

I like your distinction between hemi-toposes, demi-toposes, and semi-toposes

I care about hyper-reals, sur-reals, para-consistency, and so on

Abstract thought — like mathematicians do — is the best kind of thought.

(twitter.com/replicakill, the author of twitter.com/logicians, ragged on David Lewis by saying “What do mathematicians like?” “What do mathematicians think?” —— And Corey Mohler has done a wonderful job of mocking Platonism, which is how I guess the thirst for over-generalisation reaches non-mathematicians.)

Paul Halmos knew that cool examples beat generalisations for generalisation’s sake, as did V. I. Arnol’d. And it seems that the people a Harvard mathematician spends her time with make reasonable demands of a mathematical idea as well. It shouldn’t just contain previous theories; it should surprise. In Buliga’s Blake/Reynolds dispute, Blake wins hands down.

[kariaminari]

Map of the perfect vacations (actually my last vacations)

[tinyzal]

[ss2017] - EGG by Zalosta

for @okenki , part of the Topoï secret santa !

[asfaltics]

it was under this tree

  here is the place / ilappak oka here is the place / ilappak okato here is the place / ilappakosh       1915   here is the place where he is not       1768 here is the place where he prepared to rob himself       1819 here is the place of understanding       1806   because here is smooth walking, and easier access to each others company, which is agreeable to fowls as well as beasts; and here is the place that they are best acquainted with       1738   Here is the place where the path divides itself into two parts       1828   Here is the place to make it       1903 not unfavorable to roots. Here is the place       1840 for others, here is the place       1831 under which we walk; here is the place of exercise; here are numbers of retiring places       1822   Here is the place; it was under this tree!       1799   and here is the place to observe       1835 here is the place, pointing to a dark mound       1831 here is the place where the slightest       1831 every where, or any where else. Here is the place for you to work       1843   Here is the place, if any where, to bring       1833

—

Here is the Place of Methodizing these Stories into their proper Time and Order       1730   Here is the place, which is, in due order, assigned to figures       1832  

[isomorphismes]

The concept of (Grothendieck) topos was introduced by Alexandre Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the 'exotic' cohomology theories needed in algebraic geometry. Every topological space 𝑿 gives rise to a topos (the category of sheaves of sets on 𝑿) and every topos in Grothendieck’s sense can be considered as a 'generalized space'. At the end of the same decade, William Lawvere and Myles Tierney realized that the concept of Grothendieck topos also yielded an abstract notion of mathematical universe within which one could carry out most familiar set-theoretic constructions, but which also, thanks to the inherent 'flexibility' of the notion of topos, could be profitably exploited to construct ‘new mathematical worlds’ having particular properties. A few years later, the theory of classifying toposes added a further fundamental viewpoint to the above-mentioned ones: a topos can be seen not only as a generalized space or as a mathematical universe, but also as a suitable kind of first-order theory (considered up to a general notion of equivalence of theories). In fact, every first-order... geometric theory) admits a canonical topos-theoretic model lying in the classifying topos of the theory, satisfying the universal property that every topos-theoretic model of the theory can be obtained as a pullback of this canonical model along a unique morphism of toposes. Such canonical models of mathematical theories do not exist in the classical set-theoretic setting, but any classical (i.e., set-based) model of the theory can be seen as a point of its classifying topos. Unfortunately, apart from a few isolated exceptions, the study of Grothendieck toposes in the context of Logic was essentially not pursued afterwards. In fact, a great part of the efforts of the categorical community in the past years had been focused on the study of elementary toposes, a more general (elementarily axiomatizable) class of categories in which one can consider models of arbitrary finitary first-order theories, but in which one cannot do geometry in a Grothendieckian sense due to the fact that elementary toposes do not in general admit sites of definition. Even though the study of these more general concepts has shed important light on the original concept of topos (in fact, it was seen that several concepts in Grothendieck's topos theory admitted an 'elementary' formulation), the concept of elementary topos has not been fruitful as a source of 'concrete' applications in different fields of Mathematics, essentially because of the lack of a non-trivial representation theory 'from below' analogous to that provided by sites (which in the context of Grothendieck toposes play the role of 'carriers of concrete mathematical information').

Olivia Caramello

[wepointeractiontopoi-blog]

Interaction — concerning the range and types of engagement (between people, between people and information) encouraged or allowed by digital designs

James Porter “Recovering Delivery for Digital Rhetoric and Human Computer Interaction”

[affinespaces]

“"[Abstraction] begins with the recognition, through experience and examination of a number of specific situations, that certain phenomena occur repeatedly, that there are a number of common features, that there are formal analogies in the behaviour of different entities. Then comes the actual process of abstraction, wherein these common features are singled out and presented in isolation; an axiomatic description of an "abstract" concept. This is precisely how we obtained our general definition of a category from an inspection of a list of particular categories. It is the same process by which all of the abstract structures that mathematics investigates (group, vector space, topological space etc) were arrived at.

    Having obtained our abstract concept we then develop its general theory, and seek further instances of it. These instances are called examples of the concept or models of the axioms that define the concept. Any statement that belongs to the general theory of the concept (i.e. is derivable from the axioms) will hold true in all models. The search for new models is a process of specialisation, the reverse of abstraction. Progress in understanding comes as much from the recognition that a particular new structure is an instance of a more general phenomenon, as from the recognition that several different structures have a common core. Our knowledge of mathematical reality advances through the interplay of these two processes, through movement from the particular to the general then back again.

    Roughly speaking, the stronger the abstraction, i.e. the more we put into the abstract concept, the fewer will be the possible examples. The extreme case is where there is only one model. A classic example of this is the axiomatically presented concept of a complete ordered field. There is in fact only one such field, viz the real number system. The category axioms represent a very weak abstraction."

Robert Goldblatt, Topoi: The Categorical Analysis of Logic

[ao-boshi]

Inutile far finta di nulla, in Italia abbiamo un problema enorme: gli stranieri. Per prima cosa ci rubano i lavori migliori. Infatti chiunque vorrebbe vendere accendini e occhiali da sole sul parcheggio della Coop, specialmente ora d'agosto, che c'è quella brezzolina che ti rimette al mondo. Invece quegli impieghi lì sono già tutti occupati dagli africani mentre i culi merdosi delle vecchie se li accaparrano subito le moldave. Uno farebbe carte false per occuparsene e invece nulla! Una vergogna. Che poi voi pensate davvero che questi arrivino coi barconi? Questi vengono con la Costa Crociere, altrimenti non si spiegherebbe quanto spendono per il viaggio, e poi una volta vicini alle acque italiane li trasferiscono sulle zattere per non pagare l'ormeggio, me l'ha detto un mio amico che conosce una che è venuta dall'Islam e s'è sposata per avere il visto, perché poi sono furbe eh, non ti credere. Ma se dipendesse da me, io la casa non gliela darei mica, prima gli italiani da almeno sette generazioni! Oddio, magari sette sono un po' troppe visto che l'Italia esiste da soli 150 anni ma il sangue è sangue e infatti Salvini lo dona a tutti, con il rischio che possa finire addirittura ai negri, poi ditene male! A proposito di sangue c'è da dire che questi immigrati sono tutti violenti, specie con le donne. Pensate che il 94% degli abusi avviene ad opera di familiari o conoscenti, ma gli altri? Eh? Lo sapete che in quel 6% di estranei ci potrebbero essere anche degli extracomunitari? No, non parlavo dei preti. Sì, ho capito, lo so che il Vaticano è uno Stato extracomunitario ma che c'entra? Loro mica ti trombano la moglie, al limite i figlioli. E poi le tasse? Vogliamo parlare delle tasse? Mi vorrete mica far credere che in quei 150 miliardi all'anno di evasione non ci siano anche quelli che vendono i parei sulle spiagge? No perché son cifre grosse eh! Io un giorno ho controllato, in un pomeriggio a Fetovaia un marocchino n'avrà venduti perlomeno tre. Te fai due conti! No via, bisogna fare qualcosa, e anche alla svelta. Se non altro per le malattie, perché questi qui vai a sapere cosa ci portano! Mica son come gli italiani che quando vanno a trombare le minorenni in Thailandia partono già bell'e vaccinati. Inutile dire, non è questione di essere razzisti, è proprio che noi abbiamo una cultura differente, non è mica un caso se siamo conosciuti nel mondo per essere persone oneste, rispettose delle leggi e incorruttibili! Invece lì, in Mussulmània, sono dei selvaggi: ammazzano le donne, picchiano gli omosessuali! Ve l'immaginate? Come se qui ci fosse ancora bisogno di fare una legge sull'omofobia! Nel 2015! Ma che scherziamo davvero? Via, ora vado sennò mi chiudono i cinesi e devo comprare il profumo per la macchina che stasera vò a troie brasiliane e se mi sgama la filippina poi lo dice alla mi' moglie. Di dov'è la mi' moglie? Rumena, perché?

Ettore Ferrini