Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.

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Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute

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Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

Grothendieck wrote to Serre:

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

Serre replied using the same loose sense of derived functor:

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that

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Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.

Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((fi)i∈I, A), sometimes denoted by (fi,A)I or (Aifi A)I consisting of an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai)i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.

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In Set, a sink (Aifi A)I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪i∈I fi[Ai]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.

Philosophical Isomorphism of Category Theory. Note Quote.

One philosophical reason for categorification is that it refines our concept of ‘sameness’ by allowing us to distinguish between isomorphism and equality. In a set, two elements are either the same or different. In a category, two objects can be ‘the same in a way’ while still being different. In other words, they can be isomorphic but not equal. Even more importantly, two objects can be the same in more than one way, since there can be different isomorphisms between them. This gives rise to the notion of the ‘symmetry group’ of an object: its group of automorphisms.

Consider, for example, the fundamental groupoid Π1(X) of a topological space X: the category with points of X as objects and homotopy classes of paths with fixed endpoints as morphisms. This category captures all the homotopy-theoretic information about X in dimensions ≤ 1. The group of automorphisms of an object x in this category is just the fundamental group π1(X,x). If we decategorify the fundamental groupoid of X, we forget how points in X are connected by paths, remembering only whether they are, and we obtain the set of components of X. This captures only the homotopy 0-type of X.

This example shows how decategorification eliminates ‘higher-dimensional information’ about a situation. Categorification is an attempt to recover this information. This example also suggests that we can keep track of the homotopy 2-type of X if we categorify further and distinguish between paths that are equal and paths that are merely isomorphic (i.e., homotopic). For this we should work with a ‘2-category’ having points of X as objects, paths as morphisms, and certain equivalence classes of homotopies between paths as 2-morphisms. In a marvelous self-referential twist, the definition of ‘2-category’ is simply the categorification of the definition of ‘category’. Like a category, a 2-category has a class of objects, but now for any pair x,y of objects there is no longer a set hom(x,y); instead, there is a category hom(x,y). Objects of hom(x,y) are called morphisms of C, and morphisms between them are called 2-morphisms of C. Composition is no longer a function, but rather a functor:

◦: hom(x, y) × hom(y, z) → hom(x, z)

For any object x there is an identity 1x ∈ hom(x,x). And now we have a choice. On the one hand, we can impose associativity and the left and right unit laws strictly, as equational laws. If we do this, we obtain the definition of ‘strict 2-category’. On the other hand, we can impose them only up to natural isomorphism, with these natural isomorphisms satisfying the coherence. This is clearly more compatible with the spirit of categorification. If we do this, we obtain the definition of ‘weak 2-category’. (Strict 2-categories are traditionally known as ‘2-categories’, while weak 2-categories are known as ‘bicategories’.)

The classic example of a 2-category is Cat, which has categories as objects, functors as morphisms, and natural transformations as 2-morphisms. The presence of 2-morphisms gives Cat much of its distinctive flavor, which we would miss if we treated it as a mere category. Indeed, Mac Lane has said that categories were originally invented, not to study functors, but to study natural transformations! A good example of two functors that are not equal, but only naturally isomorphic, are the identity functor and the ‘double dual’ functor on the category of finite-dimensional vector spaces. Given a topological space X, we can form a 2-category Π>sub>2(X) called the ‘fundamental 2-groupoid’ of X. The objects of this 2-category are the points of X. Given x, y ∈ X, the morphisms from x to y are the paths f: [0,1] → X starting at x and ending at y. Finally, given f, g ∈ hom(x, y), the 2-morphisms from f to g are the homotopy classes of paths in hom(x, y) starting at f and ending at g. Since the associative law for composition of paths holds only up to homotopy, this 2-category is a weak 2-category. If we decategorify the fundamental 2-groupoid of X, we obtain its fundamental groupoid.

From 2-categories it is a short step to dreaming of n-categories and even ω-categories — but it is not so easy to make these dreams into smoothly functioning mathematical tools. Roughly speaking, an n-category should be some sort of algebraic structure having objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, and so on up to n-morphisms. There should be various ways of composing j-morphisms for 1 ≤ j ≤ n, and these should satisfy various laws. As with 2-categories, we can try to impose these laws either strictly or weakly.

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Other approaches to n-categories use j-morphisms with other shapes, such as simplices, or opetopes. We believe that there is basically a single notion of weak n-category lurking behind these different approaches. If this is true, they will eventually be shown to be equivalent, and choosing among them will be merely a matter of convenience. However, the precise meaning of ‘equivalence’ here is itself rather subtle and n-categorical in flavor.

The first challenge to any theory of n-categories is to give an adequate treatment of coherence laws. Composition in an n-category should satisfy equational laws only at the top level, between n-morphisms. Any law concerning j-morphisms for j < n should hold only ‘up to equivalence’. Here a n-morphism is defined to be an ‘equivalence’ if it is invertible, while for j < n a j-morphism is recursively defined to be an equivalence if it is invertible up to equivalence. Equivalence is generally the correct substitute for the notion of equality in n-categorical mathematics. When laws are formulated as equivalences, these equivalences should in turn satisfy coherence laws of their own, but again only up to equivalence, and so on. This becomes ever more complicated and unmanageable with increasing n unless one takes a systematic approach to coherence laws.

The second challenge to any theory of n-categories is to handle certain key examples. First, for any n, there should be an (n + 1)-category nCat, whose objects are (small) n-categories, whose morphisms are suitably weakened functors between these, whose 2-morphisms are suitably weakened natural transformations, and so on. Here by ‘suitably weakened’ we refer to the fact that all laws should hold only up to equivalence. Second, for any topological space X, there should be an n-category Πn(X) whose objects are points of X, whose morphisms are paths, whose 2-morphisms are paths of paths, and so on, where we take homotopy classes only at the top level. Πn(X) should be an ‘n-groupoid’, meaning that all its j-morphisms are equivalences for 0 ≤ j ≤ n. We call Πn(X) the ‘fundamental n-groupoid of X’. Conversely, any n-groupoid should determine a topological space, its ‘geometric realization’.

In fact, these constructions should render the study of n-groupoids equivalent to that of homotopy n-types. A bit of the richness inherent in the concept of n-category becomes apparent when we make the following observation: an (n + 1)-category with only one object can be regarded as special sort of n-category. Suppose that C is an (n+1)-category with one object x. Then we can form the n-category C ̃ by re-indexing: the objects of C ̃ are the morphisms of C, the morphisms of C ̃ are the 2-morphisms of C, and so on. The n-categories we obtain this way have extra structure. In particular, since the objects of C ̃ are really morphisms in C from x to itself, we can ‘multiply’ (that is, compose) them.

The simplest example is this: if C is a category with a single object x, C ̃ is the set of endomorphisms of x. This set is actually a monoid. Conversely, any monoid can be regarded as the monoid of endomorphisms of x for some category with one object x. We summarize this situation by saying that ‘a one-object category is a monoid’. Similarly, a one-object 2-category is a monoidal category. It is natural to expect this pattern to continue in all higher dimensions; in fact, it is probably easiest to cheat and define a monoidal n-category to be an (n + 1)-category with one object.

Things get even more interesting when we iterate this process. Given an (n + k)-category C with only one object, one morphism, and so on up to one (k − 1)-morphism, we can form an n-category whose j-morphisms are the (j + k)-morphisms of C. In doing so we obtain a particular sort of n-category with extra structure and properties, which we call a ‘k-tuply monoidal’ n-category. Table below shows what we expect these to be like for low values of n and k. For example, the Eckmann-Hilton argument shows that a 2-category with one object and one morphism is a commutative monoid. Categorifying this argument, one can show that a 3-category with one object and one morphism is a braided monoidal category. Similarly, we expect that a 4-category with one object, one morphism and one 2-morphism is a symmetric monoidal category, though this has not been worked out in full detail, because of our poor understanding of 4-categories. The fact that both braided and symmetric monoidal categories appear in this table seems to explain why both are natural concepts.

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In any reasonable approach to n-categories there should be an n-category nCatk whose objects are k-tuply monoidal weak n-categories. One should also be able to treat nCatk as a full sub-(n + k)-category of (n + k)Cat, though even for low n, k this is perhaps not as well known as it should be. Consider for example n = 0, k = 1. The objects of 0Cat1 are one-object categories, or monoids. The morphisms of 0Cat1 are functors between one-object categories, or monoid homomorphisms. But 0Cat1 also has 2-morphisms corresponding to natural transformations.

• Decategorification: (n, k) → (n − 1, k). Let C be a k-tuply monoidal n-category C. Then there should be a k-tuply monoidal (n − 1)-category DecatC whose j-morphisms are the same as those of C for j < n − 1, but whose (n − 1)-morphisms are isomorphism classes of (n − 1)-morphisms of C.

• Discrete categorification: (n, k) → (n + 1, k). There should be a ‘discrete’ k-tuply monoidal (n + 1)-category DiscC having the j-morphisms of C as its j-morphisms for j ≤ n, and only identity (n + 1)-morphisms. The decategorification of DiscC should be C.

• Delooping: (n, k) → (n + 1, k − 1). There should be a (k − 1)-tuply monoidal (n + 1)-category BC with one object obtained by reindexing, the j-morphisms of BC being the (j + 1)-morphisms of C. We use the notation ‘B’ and call BC the ‘delooping’ of C because of its relation to the classifying space construction in topology.

• Looping: (n, k) → (n − 1, k + 1). Given objects x, y in an n-category, there should be an (n − 1)-category hom(x, y). If x = y this should be a monoidal (n−1)-category, and we denote it as end(x). For k > 0, if 1 denotes the unit object of the k-tuply monoidal n-category C, end(1) should be a (k + 1)-tuply monoidal (n − 1)-category. We call this process ‘looping’, and denote the result as ΩC, because of its relation to loop space construction in topology. For k > 0, looping should extend to an (n + k)-functor Ω: nCatk → (n − 1)Catk+1. The case k = 0 is a bit different: we should be able to loop a ‘pointed’ n-category, one having a distinguished object x, by letting ΩC = end(x). In either case, the j-morphisms of ΩC correspond to certain (j − 1)-morphisms of C.

• Forgetting monoidal structure: (n, k) → (n, k−1). By forgetting the kth level of monoidal structure, we should be able to think of C as a (k−1)-tuply monoidal n-category FC. This should extend to an n-functor F: nCatk → nCatk−1.

• Stabilization: (n, k) → (n, k + 1). Though adjoint n-functors are still poorly understood, there should be a left adjoint to forgetting monoidal structure, which is called ‘stabilization’ and denoted by S: nCatk → nCatk+1.

• Forming the generalized center: (n,k) → (n,k+1). Thinking of C as an object of the (n+k)-category nCatk, there should be a (k+1)-tuply monoidal n-category ZC, the ‘generalized center’ of C, given by Ωk(end(C)). In other words, ZC is the largest sub-(n + k + 1)-category of (n + k)Cat having C as its only object, 1C as its only morphism, 11C as its only 2-morphism, and so on up to dimension k. This construction gets its name from the case n = 0, k = 1, where ZC is the usual center of the monoid C. Categorifying leads to the case n = 1, k = 1, which gives a very important construction of braided monoidal categories from monoidal categories. In particular, when C is the monoidal category of representations of a Hopf algebra H, ZC is the braided monoidal category of representations of the quantum double D(H).

Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

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As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Idα : SetsCop → Shvs(Cop,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves SetsTop. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define YX as the set of functions X → Y. Then in the category of presheaves,

SetsCopYX(U) ≅ Hom(hU,YX) ≅ Hom(hU × X,Y)

, where hU is the representable sheaf hU(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf YX is actually a sheaf. Finally, for the existence of the subobject-classifier ΩSetsCop, it can be defined as

ΩSetsCop(U) ≅ Hom(hU,Ω) ≅ {sub-presheaves of hU} ≅ {sieves on U}, or alternatively, for the category of proper sheaves Shvs(C,J), as

ΩShvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

frs(S) ∈ J(r) ⇒ frs ∈ S

, where frs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1s(S) = fss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφp(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φp(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

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Badiou’s Diagrammatic Claim of Democratic Materialism Cuts Grothendieck’s Topos. Note Quote.

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Let us focus on the more abstract, elementary definition of a topos and discuss materiality in the categorical context. The materiality of being can, indeed, be defined in a way that makes no material reference to the category of Sets itself.

The stakes between being and materiality are thus reverted. From this point of view, a Grothendieck-topos is not one of sheaves over sets but, instead, it is a topos which is not defined based on a specific geometric morphism E → Sets – a materialization – but rather a one for which such a materialization exists only when the topos itself is already intervened by an explicitly given topos similar to Sets. Therefore, there is no need to start with set-theoretic structures like sieves or Badiou’s ‘generic’ filters.

Strong Postulate, Categorical Version: For a given materialization the situation E is faithful to the atomic situation of truth (Setsγ∗(Ω)op) if the materialization morphism itself is bounded and thus logical.

In particular, this alternative definition suggests that materiality itself is not inevitably a logical question. Therefore, for this definition to make sense, let us look at the question of materiality from a more abstract point of view: what are topoi or ‘places’ of reason that are not necessarily material or where the question of materiality differs from that defined against the ‘Platonic’ world of Sets? Can we deploy the question of materiality without making any reference – direct or sheaf-theoretic – to the question of what the objects ‘consist of’, that is, can we think about materiality without crossing Kant’s categorical limit of the object? Elementary theory suggests that we can.

Elementary Topos:  An elementary topos E is a category which

  1. has finite limits, or equivalently E has so called pull-backs and a terminal object 1,
  2. is Cartesian closed, which means that for each object X there is an exponential functor (−)X : E → E which is right adjoint to the functor (−) × X, and finally
  3. axiom of truth E retains an object called the subobject classifier Ω, which is equipped with an arrow 1 →true Ω such that for each monomorphism σ : Y ֒→ X in E, there is a unique classifying map φσ : X → Ω making σ : Y ֒→ X a pull-back of φσ along the arrow true.

Grothendieck-topos: In respect to this categorical definition, a Grothendieck-topos is a topos with the following conditions satisfies:

(1) E has all set-indexed coproducts, and they are disjoint and universal,

(2) equivalence relations in E have universal co-equalisers,

(3) every equivalence relation in E is effective, and every epimorphism in E is a coequaliser,

(4) E has ‘small hom-sets’, i.e. for any two objects X, Y , the morphisms of E from X to Y are parametrized by a set, and finally

(5) E has a set of generators (not necessarily monic in respect to 1 as in the case of locales).

Together the five conditions can be taken as an alternative definition of a Grothendieck-topos. We should still demonstrate that Badiou’s world of T-sets is actually the category of sheaves Shvs (T, J) and that it will, consequentially, hold up to those conditions of a topos listed above. To shift to the categorical setting, one first needs to define a relation between objects. These relations, the so called ‘natural transformations’ we encountered in relation Yoneda lemma, should satisfy conditions Badiou regards as ‘complex arrangements’.

Relation: A relation from the object (A, Idα) to the object (B,Idβ) is a map ρ : A → B such that

Eβ ρ(a) = Eα a and ρ(a / p) = ρ(a) / p.

It is a rather easy consequence of these two pre-suppositions that it respects the order relation ≤ one retains Idα (a, b) ≤ Idβ (ρ(a), ρ(b)) and that if a‡b are two compatible elements, then also ρ(a)‡ρ(b). Thus such a relation itself is compatible with the underlying T-structures.

Given these definitions, regardless of Badiou’s confusion about the structure of the ‘power-object’, it is safe to assume that Badiou has demonstrated that there is at least a category of T-Sets if not yet a topos. Its objects are defined as T-sets situated in the ‘world m’ together with their respective equalization functions Idα. It is obviously Badiou’s ‘diagrammatic’ aim to demonstrate that this category is a topos and, ultimately, to reduce any ‘diagrammatic’ claim of ‘democratic materialism’ to the constituted, non-diagrammatic objects such as T-sets. That is, by showing that the particular set of objects is a categorical makes him assume that every category should take a similar form: a classical mistake of reasoning referred to as affirming the consequent.

Badiou Contra Grothendieck Functorally. Note Quote.

What makes categories historically remarkable and, in particular, what demonstrates that the categorical change is genuine? On the one hand, Badiou fails to show that category theory is not genuine. But, on the other, it is another thing to say that mathematics itself does change, and that the ‘Platonic’ a priori in Badiou’s endeavour is insufficient, which could be demonstrated empirically.

Yet the empirical does not need to stand only in a way opposed to mathematics. Rather, it relates to results that stemmed from and would have been impossible to comprehend without the use of categories. It is only through experience that we are taught the meaning and use of categories. An experience obviously absent from Badiou’s habituation in mathematics.

To contrast, Grothendieck opened up a new regime of algebraic geometry by generalising the notion of a space first scheme-theoretically (with sheaves) and then in terms of groupoids and higher categories. Topos theory became synonymous to the study of categories that would satisfy the so called Giraud’s axioms based on Grothendieck’s geometric machinery. By utilising such tools, Pierre Deligne was able to prove the so called Weil conjectures, mod-p analogues of the famous Riemann hypothesis.

These conjectures – anticipated already by Gauss – concern the so called local ζ-functions that derive from counting the number of points of an algebraic variety over a finite field, an algebraic structure similar to that of for example rational Q or real numbers R but with only a finite number of elements. By representing algebraic varieties in polynomial terms, it is possible to analyse geometric structures analogous to Riemann hypothesis but over finite fields Z/pZ (the whole numbers modulo p). Such ‘discrete’ varieties had previously been excluded from topological and geometric inquiry, while it now occurred that geometry was no longer overshadowed by a need to decide between ‘discrete’ and ‘continuous’ modalities of the subject (that Badiou still separates).

Along with the continuous ones, also discrete variates could then be studied based on Betti numbers, and similarly as what Cohen’s argument made manifest in set-theory, there seemed to occur ‘deeper’, topological precursors that had remained invisible under the classical formalism. In particular, the so called étale-cohomology allowed topological concepts (e.g., neighbourhood) to be studied in the context of algebraic geometry whose classical, Zariski-description was too rigid to allow a meaningful interpretation. Introducing such concepts on the basis of Jean-Pierre Serre’s suggestion, Alexander Grothendieck did revolutionarize the field of geometry, and Pierre Deligne’s proof of the Weil-conjenctures, not to mention Wiles’ work on Fermat’s last theorem that subsequentely followed.

Grothendieck’s crucial insight drew on his observation that if morphisms of varieties were considered by their ‘adjoint’ field of functions, it was possible to consider geometric morphisms as equivalent to algebraic ones. The algebraic category was restrictive, however, because field-morphisms are always monomorphisms which makes geometric morphisms: to generalize the notion of a neighbourhood to algebraic category he needed to embed algebraic fields into a larger category of rings. While a traditional Kuratowski covering space is locally ‘split’ – as mathematicians call it – the same was not true for the dual category of fields. In other words, the category of fields did not have an operator analogous to pull-backs (fibre products) unless considered as being embedded within rings from which pull-backs have a co-dual expressed by the tensor operator ⊗. Grothendieck thus realized he could replace ‘incorporeal’ or contained neighborhoods U ֒→ X by a more relational description: as maps U → X that are not necessarily monic, but which correspond to ring-morphisms instead.

Topos theory applies similar insight but not in the context of only specific varieties but for the entire theory of sets instead. Ultimately, Lawvere and Tierney realized the importance of these ideas to the concept of classification and truth in general. Classification of elements between two sets comes down to a question: does this element belong to a given set or not? In category of S ets this question calls for a binary answer: true or false. But not in a general topos in which the composition of the subobject-classifier is more geometric.

Indeed, Lawvere and Tierney then considered this characteristc map ‘either/or’ as a categorical relationship instead without referring to its ‘contents’. It was the structural form of this morphism (which they called ‘true’) and as contrasted with other relationships that marked the beginning of geometric logic. They thus rephrased the binary complete Heyting algebra of classical truth with the categorical version Ω defined as an object, which satisfies a specific pull-back condition. The crux of topos theory was then the so called Freyd–Mitchell embedding theorem which effectively guaranteed the explicit set of elementary axioms so as to formalize topos theory. The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring R and that the embedding is an exact functor. It is easy to see that not every abelian category is equivalent to RMod for some ring R. The reason is that RMod has all small limits and colimits. But for instance the category of finitely generated R-modules is an abelian category but lacks these properties.

But to understand its significance as a link between geometry and language, it is useful to see how the characteristic map (either/or) behaves in set theory. In particular, by expressing truth in this way, it became possible to reduce Axiom of Comprehension, which states that any suitable formal condition λ gives rise to a peculiar set {x ∈ λ}, to a rather elementary statement regarding adjoint functors.

At the same time, many mathematical structures became expressible not only as general topoi but in terms of a more specific class of Grothendieck-topoi. There, too, the ‘way of doing mathematics’ is different in the sense that the object-classifier is categorically defined and there is no empty set (initial object) but mathematics starts from the terminal object 1 instead. However, there is a material way to express the ‘difference’ such topoi make in terms of set theory: for every such a topos there is a sheaf-form enabling it to be expressed as a category of sheaves S etsC for a category C with a specific Grothendieck-topology.

Geach and Relative Identity

Peter-Geachsi_2790822a

The Theory of Relative Identity is a logical innovation due to Peter Thomas Geach  (P.T. Geach-Logic Matters) motivated by the same sort of mathematical examples as Frege’s definition by abstraction. Like Frege Geach seeks to give a logical sense to mathematical talk “up to” a given equivalence E through replacing E by identity but unlike Frege he purports, in doing so, to avoid the introduction of new abstract objects (which in his view causes unnecessary ontological inflation). The price for the ontological parsimony is Geach’s repudiation of Frege’s principle of a unique and absolute identity for the objects in the domain over which quantified variables range. According to Geach things can be same in one way while differing in others. For example two printed letters aa are same as a type but different as tokens. In Geach’s view this distinction does not commit us to a-tokens and a-types as entities but presents two different ways of describing the same reality. The unspecified (or “absolute” in Geach’s terminology) notion of identity so important for Frege is in Geach’s view is incoherent.

Geach’s proposal appears to account better for the way the notion of identity is employed in mathematics since it does not invoke “directions” or other mathematically redundant concepts. It captures particularly well the way the notion of identity is understood in Category theory. According to Baez & Dolan

In a category, two objects can be “the same in a way” while still being different.

So in Category theory the notion of identity is relative in exactly Geach’s sense. But from the logical point of view the notion of relative identity remains highly controversial. Let x,y be identical in one way but not in another, or in symbols: Id(x,y) & ¬Id'(x,y). The intended interpretation assumes that x in the left part of the formula and x in the right part have the same referent, where this last same apparently expresses absolute not relative identity. So talk of relative identity arguably smuggles in the usual absolute notion of identity anyway. If so, there seems good reason to take a standard line and reserve the term “identity” for absolute identity.

We see that Plato, Frege and Geach propose three different views of identity in mathematics. Plato notes that the sense of “the same” as applied to mathematical objects and to the ideas is different: properly speaking, sameness (identity) applies only to ideas while in mathematics sameness means equality or some other equivalence relation. Although Plato certainly recognizes essential links between mathematical objects and Ideas (recall the “ideal numbers”) he keeps the two domains apart. Unlike Plato Frege supposes that identity is a purely logical and domain-independent notion, which mathematicians must rely upon in order to talk about the sameness or difference of mathematical objects, or any other kind at all. Geach’s proposal has the opposite aim: to provide a logical justification for the way of thinking about the (relativized) notions of sameness and difference which he takes to be usual in mathematical contexts and then extend it to contexts outside mathematics (As Geach says):

Any equivalence relation … can be used to specify a criterion of relative identity. The procedure is common enough in mathematics: e.g. there is a certain equivalence relation between ordered pairs of integers by virtue of which we may say that x and y though distinct ordered pairs, are one and the same rational number. The absolute identity theorist regards this procedure as unrigorous but on a relative identity view it is fully rigorous.