Topos Theory - Lecture 1: =1=Overview of the course · Topos theory can be regarded as aunifying...

Topos Theory

Olivia Caramello

The “unifyingnotion” of topos

The multifacetednature of toposes

Prerequisites andexam

A bit of history

Toposes asbridges

A new way ofdoingMathematics

Topos TheoryLecture 1: Overview of the course

Olivia Caramello

Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


The “unifying notion” of topos

“It is the topos theme which is this “bed” or “deep river”where come to be married geometry and algebra, topology and arithmetic,mathematical logic and category theory, the world of the “continuous” and

that of “discontinuous” or discrete structures. It is what I have conceived ofmost broad to perceive with finesse, by the same language rich of

geometric resonances, an “essence” which is common to situationsmost distant from each other coming from one region or another

of the vast universe of mathematical things”.

A. Grothendieck

Topos theory can be regarded as a unifying subject in Mathema-tics, with great relevance as a framework for systematically inves-tigating the relationships between different mathematical theoriesand studying them by means of a multiplicity of different points ofview. Its methods are transversal to the various fields and com-plementary to their own specialized techniques. In spite of theirgenerality, the topos-theoretic techniques are liable to generate in-sights which would be hardly attainable otherwise and to establishdeep connections that allow effective transfers of knowledge bet-ween different contexts.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


The multifaceted nature of toposes

The role of toposes as unifying spaces is intimately tied to theirmultifaceted nature; for instance, a topos can be seen as:

• a generalized space• a mathematical universe• a theory (modulo a certain notion of equivalence).

The course will start by presenting the relevant categorical andlogical background and extensively treat these different perspecti-ves on the notion of topos, with the final aim of providing the stu-dent with tools and methods to study mathematical theories froma topos-theoretic perspective, extract new information about cor-respondences, dualities or equivalences, and establish new andfruitful connections between distinct fields.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Prerequisites and exam

Prerequisites:A Bachelor’s degree in Mathematics (or equivalent mathematicalmaturity). Although some familiarity with the language of categorytheory and the basics of first-order logic would be desirable, noprevious knowledge of these subjects is required. Indeed, thecourse will present all the relevant preliminaries as they areneeded.

Exam:The student will be able to choose between two alternative exammodes:

• Solutions (prepared at home) to exercises assigned by thelecturer and oral presentation on a suitable topic (chosen inagreement with the lecturer) not treated in the course.

• Taking two partial written exams, one administered half waythrough the course and the other immediately after the end ofthe course.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


A bit of history

• Toposes were originally introduced by Alexander Grothendieck in theearly 1960s, in order to provide a mathematical underpinning for the‘exotic’ cohomology theories needed in algebraic geometry. Everytopological space gives rise to a topos and every topos inGrothendieck’s sense can be considered as a ‘generalized space’.

• At the end of the same decade, William Lawvere and Myles Tierneyrealized that the concept of Grothendieck topos also yielded anabstract notion of mathematical universe within which one could carryout most familiar set-theoretic constructions, but which also, thanks tothe inherent ‘flexibility’ of the notion of topos, could be profitablyexploited to construct ‘new mathematical worlds’ having particularproperties.

• A few years later, the theory of classifying toposes added a furtherfundamental viewpoint to the above-mentioned ones: a topos can beseen not only as a generalized space or as a mathematical universe,but also as a suitable kind of first-order theory (considered up to ageneral notion of equivalence of theories).

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as generalized spaces• The notion of topos was introduced in the early sixties by A.

Grothendieck with the aim of bringing a topological orgeometric intuition also in areas where actual topologicalspaces do not occur.

• Grothendieck realized that many important properties oftopological spaces X can be naturally formulated as(invariant) properties of the categories Sh(X ) of sheaves ofsets on the spaces.

• He then defined toposes as more general categories ofsheaves of sets, by replacing the topological space X by apair (C ,J) consisting of a (small) category C and a‘generalized notion of covering’ J on it, and taking sheaves(in a generalized sense) over the pair:

X //

��

Sh(X )

��(C ,J) // Sh(C ,J)

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Topos-theoretic invariants

DefinitionBy a topos-theoretic invariant we mean a property of (or aconstruction involving) toposes which is stable under categoricalequivalence.

• The notion of a geometric morphism of toposes has notablyallowed to build general comology theories starting from thecategories of internal abelian groups or modules in toposes.The cohomological invariants have had a tremendous impacton the development of modern Algebraic Geometry andbeyond.

• On the other hand, homotopy-theoretic invariants such as thefundamental group and the higher homotopy groups can bedefined as invarants of toposes.

• Still, these are by no means the only invariants that one canconsider on toposes: indeed, there are infinitely manyinvariants of toposes (of algebraic, logical, geometric orwhatever nature), the notion of identity for toposes beingsimply categorical equivalence.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as mathematical universes

A decade later, W. Lawvere and M. Tierney discovered that atopos could not only be seen as a generalized space, but also asa mathematical universe in which one can do mathematicssimilarly to how one does it in the classical context of sets (withthe only important exception that one must argue constructively).

Amongst other things, this discovery made it possible to:

• Exploit the inherent ‘flexibility’ of the notion of topos toconstruct ‘new mathematical worlds’ having particularproperties.

• Consider models of any kind of (first-order) mathematicaltheory not just in the classical set-theoretic setting, but insideevery topos, and hence ‘relativise’ Mathematics.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Classifying toposes

• It was realized in the seventies (thanks to the work of severalpeople, notably including W. Lawvere, A. Joyal, G. Reyes andM. Makkai) that to any mathematical theory T (of a generalspecified form) one can canonically associate a topos ET,called the classifying topos of the theory, which represents its‘semantical core’, a sort of ‘DNA’ of the theory.

• Conversely, every Grothendieck topos is the classifying toposof some theory.

• Hence, a Grothendieck topos can be seen as a canonicalrepresentative of equivalence classes of geometric theoriesmodulo Morita-equivalence.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Classifying toposes

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


The idea of bridge

• We can think of a bridge object connecting two objects a andb as an object u which can be ‘built’ from any of the twoobjects and admits two different representations f (a) andg(b) related by some kind of equivalence ', the former beingin terms of the object a and the latter in terms of the object b:

f (a)' u ' g(b)

a b

• The transfer of information arises from the process of‘translating’ invariant (with respect to ') properties of (resp.constructions on) the ‘bridge object’ u into properties of (resp.constructions on) the two objects a and b by using the twodifferent representations f (a) and g(b) of the bridge object.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as bridges I

• Toposes can effectively act as ‘bridge objects’ acrossMorita-equivalent theories (or, more generally, betweentheories to which one can attach equivalent toposes).

• The notion of Morita-equivalence is ubiquitous inMathematics; indeed, it formalizes in many situations thefeeling of ‘looking at the same thing in different ways’, or‘constructing a mathematical object through differentmethods’.

• In fact, many important dualities and equivalences inMathematics can be naturally interpreted in terms ofMorita-equivalences.

• Any two theories which are bi-interpretable in each other areMorita-equivalent but, very importantly, the converse doesnot hold.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as bridges II

• The fact that different mathematical theories have equivalentclassifying toposes translates, at the technical level, into theexistence of different representations of one topos.

• So Topos Theory itself is a primary source ofMorita-equivalences. Indeed, different representations of thesame topos can be interpreted as Morita-equivalencesbetween different mathematical theories.

• We can think of a topos as embodying the ‘common features’of mathematical theories which are Morita-equivalent to eachother.

• The underlying intuition is that a given mathematical propertycan manifest itself in different forms in the context ofmathematical theories which have a common ‘semanticalcore’ but a different linguistic presentation.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as bridges III

• The essential features of Morita-equivalences are all ‘hidden’inside toposes, and can be revealed by using their differentrepresentations.

• For example, imagine starting with a property, saygeometrical, of a certain mathematical object, and being ableto find a topos and a property of it which is (logically)equivalent to the given property of our object; then one canuse e.g. a logical representation for the topos to convert thisproperty of the topos into a logical statement of a certainkind; as a result, one obtains the equivalence of our initialgeometrical property with a logical one.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as bridges IV

• If a property of a mathematical object is formulated as atopos-theoretic invariant on some topos then the expressionof it in terms of the different theories classified by that toposis determined to a great extent by the technical relationshipbetween the topos and the different representations of it.

• Topos-theoretic invariants can thus be used to transferinformation from one theory to another:

ET ' ET′

��T

11

T′

• The transfer of information takes place by expressing a giveninvariant in terms of the different representations of the topos.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


Toposes as bridges V

• The level of generality represented by topos-theoreticinvariants is ideal to capture several important features ofmathematical theories.

• The fact that topos-theoretic invariants specialize toimportant properties or constructions of natural mathematicalinterest is a clear indication of the centrality of theseconcepts in Mathematics. In fact, whatever happens at thelevel of toposes has ‘uniform’ ramifications into Mathematicsas a whole.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


The duality between ‘real’ and ‘imaginary’• The passage from a site (or a theory) to the associated topos

can be regarded as a sort of ‘completion’ by the addition of‘imaginaries’ (in the model-theoretic sense), which materializesthe potential contained in the site (or theory).

• The duality between the (relatively) unstructured world ofpresentations of theories and the maximally strucured world oftoposes is of great relevance as, on the one hand, the‘simplicity’ and concreteness of theories or sites makes it easyto manipulate them, while, on the other hand, computations aremuch easier in the ‘imaginary’ world of toposes thanks to theirvery rich internal structure and the fact that invariants live atthis level.

toposMorita equivalence

starting point= concrete fact

(often quiteelementary)

generationof other

concrete resultsno

directdeduction

REAL

IMAGINARY

liftingchoice of invariants

for computation

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


A mathematical morphogenesis

• The essential ambiguity given by the fact that any topos isassociated in general with an infinite number of theories ordifferent sites allows to study the relations between differenttheories, and hence the theories themselves, by usingtoposes as ’bridges’ between these different presentations.

• Every topos-theoretic invariant generates a veritablemathematical morphogenesis resuting from its expression interms of different representations of toposes, which gives risein general to connections between properties or notions thatare completely different and apparently unrelated from eachother

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


A new way of doing Mathematics I

• Taking these methodologies seriously entails a deepparadigmatic shift as they suggest a new way of doingMathematics which is ‘upside-down’ in that the mathematicalexploration is guided by Morita-equivalences andtopos-theoretic invariants rather than by ‘concrete’considerations.

• These are the objects at the ‘center of the stage’, and it isfrom them that one proceeds to extract concrete informationon the theories that one wishes to study.

• The ‘working mathematician’ would greatly profit attemptingto formulate his or her properties of interest in terms oftopos-theoretic invariants, and derive equivalent versions ofthem by using alternative representations of the relevanttoposes.

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Topos Theory

Olivia Caramello




A bit of history

Toposes asbridges


A new way of doing Mathematics II

• There is an strong element of automatism in thesetechniques; by means of them, one can generate a greatnumber of new mathematical results without really makingany creative effort.

• The results generated in this way are in general non-trivial; insome cases they can be rather ‘weird’ according to the usualmathematical standards (although they might still be quitedeep) but, with a careful choice of Morita-equivalences andinvariants, one can obtain interesting and naturalmathematical results.

• In fact, a lot of information that is not visible with the usual‘glasses’ is revealed by the application of this machinery.

• On the other hand, the range of applicability of thesemethods is extremely broad within Mathematics, by the verygenerality of the notion of topos.

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Topos Theory - Lecture 1: =1=Overview of the course · Topos theory can be regarded as aunifying...

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