Categories of Space & Quantity

0:00 So, let me open the second session this morning. I'm very pleased. Professor Dress will not, if you have not heard the announcement, there will only be one more talk this morning, and after the talk we'll have some time for questions when you, if you want to ask a question, please announce your name before doing so. So I'm very pleased to introduce Professor William Lovaire of Buffalo. Professor Lovaire is known as having made many important contributions to the theory of categories, which is a very general concept which has many applications in theory of structures and mathematics. Originally introduced by Professors Eilenberg and McLean and considerably developed in the last 30 years or so. So I turn it over to Professor Levaire. Thank you. This morning I want to speak about certain ideas which have grown out of mathematics in the past 50 years. Actually, I'm here beginning what might be considered a very ambitious program of reviving the outlook approach of Hermann Grassmann, which is to apply philosophy to mathematics and apply mathematics to philosophy. As I say, they grew out of specific, concrete mathematical problems. They were not invented for the sake of abstraction in its own right. They are not mere analogies or mere side comments on the growth of mathematics. Rather, it was necessary in order to be able to complete certain calculations.

2:30 There are many ways in which these concepts can be used to make explicit certain general ideas, or in some cases it was necessary to complete the calculations, in other cases necessary to be able to explain the calculations in a reasonable manner. But once having been made explicit, these general ideas turned out to have their application in mathematics as a whole. In all the fields of science where mathematics plays a central role, thus the hope is that more explicit attention to ideas of the philosophical character within mathematics will lead to the possibility to assist the process of the learning, development, and use of mathematics, and in particular The complex of these ideas has, I think, reached a point where it is possible to begin to give an explicit account of what is mathematics as a whole. Help me recall what some of these ideas were. It was first the discovery of Eilenberg and MacLean, the concept of category, which, I think above all, in its philosophical content, It is a crucial idea that in order to specify a category of objects of thought, it is necessary to also specify the maps between these objects, that is, the modes in which these objects transform into one another, is as much a part of the specification of the category of objects as is the specification of objects in themselves. This discovery was made in the 1940s in connection with a particular problem in algebraic topology. In the 1950s, further struggle with particular problems in algebraic topology, functional analysis, algebraic geometry, and so on, led to the development of Abugan categories,

5:00 adjoint functors, an extremely ubiquitous structure, bicon, and rotundi. And in the past 30 years, numerous further concepts have been developed, such as topos, two-dimensional categories, hybrid categories, monads, etc., etc. The subject matter of mathematics is often said to consist of, or in any case in large part consists of, the study of space and its quantity and the mutual relationships of space and quantity. I believe we can already, on top of the theory of categories, make a very sharp distinction between space and quantity, and as well, in a calculable way, with their relationship. Uses or properties in mathematical thought. On the one hand, a space is an arena for becoming. One location may become another location. One state may become another state. One temperature may become another temperature. So each space is an arena for possible becoming. On the other hand, each space is a domain of a variable quantity. I would like to try to explain Again, in mathematical form, we'll talk about the theory of categories, these aspects. First, there are many categories of spaces. There are topological spaces, you could say continuous, truly continuous spaces. There are smooth or differentiable spaces. There are combinatorial. And each of these has many refinements, ramifications. So there are, in fact, there are many categories of spaces.

7:30 There is at least one property which is not possessed by categories of quantity. And I will try to explain this in terms of the categorical notion of product and co-product. A great many mathematical categories have both products and co-products. But the categories differ in the relationship between products. Distinguish formally two kinds of relationships between products and co-products, which I'll call distributive categories and linear categories, as a first determination of the distinction between space and quantity. Now, I will speak only about finite products and co-products, but let me remark that those of you who may have heard of the concept of topos, Topos is a much richer structure than the distributive category, but in particular, every topos is a distributive. However, there are many other examples of distributive categories, and I wanted to include all those in the general discussion. So, the category which has finite products and finite co-products. Finite includes empty and binary, and these two cases suffice. A terminal object is also called a terminal object. A terminal object in a category is an object to which there is exactly one map from any other object. Similarly, an empty coproduct is just the reverse or so-called dual. A coterminal object or empty coproduct is an object which has a unique map from it to any other object in the category. I will often speak briefly of a terminal object by calling it one and of a coterminal or empty coproduct by calling it zero. On the other hand, the binary products and coproducts are explained by the following universal property that in order to map a given object into a product, it's equivalent to map it into each of the factors of that product.

10:00 This equivalence is mediated by the special maps known as projections and diagonals. Dually, the coproducts are characterized by the fact that a map from a coproduct into an arbitrary object is determined by a family of maps, one from each summand in the coproduct. And again, this equivalence is mediated by injections. Now, in any category having both finite products and coproducts, there is a canonical math from a coproduct of products into a product of coproducts, namely, or perhaps if I say sum and product, it would be more clear, from a sum of products into a product of sums. Here I'm imagining precisely the ordinary distributive law of elementary algebra, which says, for example, that A times B plus A times C is equal to A times the sum B plus C. But note I only said that there's a canonical map. This canonical map exists in any category. However, it's precisely in the distributive categories that this canonical map is an isomorphism, and hence, because this map is an isomorphism, I used to say an invertible map, we may say that the distributive law holds for products and for coproducts considered as sums. So, this is the first determination that the categories of spaces are distributed. are defined by the requirement that the products and co-products actually coincide, they're actually the same. So it turns out that only the most trivial category that could be both distributive and linear, in other words these are essentially disjoint categories of categories.

12:30 It was pointed out by Professor McLean in 1950 that in any linear category there is a unique commutative and associative addition operation on the maps. The math would have a given domain and a given co-domain, and moreover, composition distributes over this addition. In other words, linear categories are precisely the general context in which the basic formalism known as linear algebra can be interpreted. Now, so, in what way can a space act as the domain of variable quantity? Well, I think there are basically two ways, two ways in which a space can act as the domain of variable quantity. These two ways, which in a certain fashion reflect already the idea of space and quantity themselves, have been traditionally called in philosophy extensive and intensive quantities. So the extensive quantities are in some sense quantities of space, whereas by contrast the intensive quantities essentially act as ratios of extensive ones. So one might say that the intensive quantities are quantities of quantity. What are the features which distinguish the extensive and intensive? Well, first let me give an example. I should mention here the names of Grossmann and Maxwell, because although, as I said, these terms, extensive and intensive, have been known in philosophy for centuries, they have somehow got forgotten in the 20th century mathematics and physics, or almost. For which the subtitle was Theory of Extensive Quantities. He made great progress in correcting the imbalance which existed between the study of intensive and extensive quantities.

15:00 Maxwell, in his work on physical chemistry, managed to get the extensive quantities accepted as such within the particular science of thermodynamics. But even though students still learn these terms in thermodynamics, they are not used in other parts of physics, even though by rights they should be, even in all parts of mathematics. Of course, in thermodynamics, the typical examples are things like volume, mass, energy, entropy are extensive, whereas things like pressure, density, temperature are intensive. This is mentioned in thermodynamics because the mere fact of making a distinction is a useful guide. For example, if you have an equation, both sides should be both extensive or both intensive, and so on. So, the striking mathematical feature which distinguishes extensive from intensive is the fact that a given type of extensive quantity is a covariant functor. I'll give an example. For example, during this month of September, I'm making a tour of Spain, so my particular path could be represented as a map in a spatial category between a space which is taken as a spatialization of time into the space which represents Spain, a particular map, another trip would be another map. There is an extensive quantity known as duration. Under the map, which is given by my particular tour, this duration transforms into another extensive quantity on the co-domain of that, namely on the country itself. This extensive quantity on the country itself still has dimensions of time, but it's defined not on time but on the country.

17:30 Tour in every particular part of Spain is a certain total amount of time, so it's a variable quantity of extensive nature, but it arose through the covariant functorality, sometimes called the push forward of the extensive quantity duration. Another familiar example, of course, is if you have a body like a cloud in the sky placed in space at a certain instant in time. All of these terms are placed in space in a certain way. Then the extensive quantity known as mass, which is varying over the cloud, is pushed forward and becomes an extensive quantity on space. The cloud moves, you have a new extensive quantity on space, but still it's the push forward of the fixed mass distribution on the cloud itself. So abstractly, or to extract the essence of this idea, I say that a type of extensive quantity is a co-variant, co-product-preserving functor from a distributive category, that is a spatial category, to a linear category. The fact that it should be coproduct-preserving describes the fact that if a space can be represented as the disjoint sum, a coproduct, of two other spaces, then to specify a distribution over it, an extensive quantity over it, is the same as to specify a distribution over each of the summands, in general. And, of course, coproducts in the linear category are the same as products. I gave the example of push forward of mass along a placement and push forward of duration along a tour. There are some sort of canonical examples of this covariant functoriality of extensive quantity which should also be mentioned.

20:00 In particular, remember the terminal space. Space plays the role of an abstract point if there is space. So if I evaluate the extensive quantity type on this terminal space, I do obtain a certain linear object. This linear object should be considered as the object of the one point in the iteration. By contrast, the value of this type in some typical space, like the body or the time or the country and so on, will consist of the variable expenses. But now remember that terminal space is characterized by the fact that any space has a unique map to it, so therefore, in particular, the covariant punctuality applies to that. It means that there is a linear map from the variable extensive quantity delivered in any given space into the constant, and this map is known as total, the total value of any extensive quantity. For example, a cloud has a variable mass, different parts have different mass, and so on. But, in particular, it has a total mass, and the total mass is constant. It no longer varies over the cloud. On the other side, a map from a terminal space to a given space is not unique. In fact, a map from a terminal space to a given space is essentially a point. If I apply the extensive quantity type to that, once again I get a linear map. But now it goes from the constant quantities into the variable extensive quantities. This is known as the Dirac measure. The Dirac measure concentrates on that point. It may have different weights, but the weight that it has is determined by the constant, any constant extensive quantity. For example, if I place the body in space, say I place the cloud in the space of the Earth.

22:30 For example, in the space of the Earth, there is the magnetic field of the Earth, which is a variable quantity, but an intensive one, variable of the Earth. Due to the placement of the cloud in space, There is an induced variable quantity over the cloud itself, which is still the magnetic field of the Earth, but as the cloud feels it at that particular placement. Again, if I change the placement, the reduced magnetic field changes. But notice this is a contradictory correspondence. The placement went this way, but the transfer of the magnetic field went backwards. An example of an interesting intensive quantity in Spain is the frequency with which a given language can be heard. Well, due to my particular viewer, that intensive quantity on the geography is transformed into an intensive quantity varying through the month of September. The frequency with which I can hear a given language. The other aspect about intensive quantities makes them different from extensive is that they can be multiplied, that they're multiplication of intensive quantities, and this multiplication is also preserved, not only in the new era, but multiplication is also preserved by the backward inducing. Again, we can, apart from these more interesting examples about placements of bodies and tours through countries, We can again isolate some standard trivial, but playing an important role, examples of the contravariance, contrariality of engines, again with the help of the terminal space. So any space has a unique map to the terminal space. Therefore, if I have a kind of intensive quantity, there is a map going backward, which is the science of every constant intensive quantity of that type. A special case of the variable one. What is the special case? Well, these are known as the constant. The constant variable quantity is considered as special variable ones.

25:00 On the other hand, if I have a point of a space that is a map from the terminal space to it, again I have an induced backward map which is known as the evaluation. So an intensive quantity can be, in particular, can be evaluated at any point and the result is a constant quantity. One must be careful. Intensive quantities are not necessarily determined by the ensemble of all their values and points, but at least they all have, they all do admit this aspect. Now, the relationship between extensive and intensive quantities. I referred to it before as ratio. However, we know it's very important in teaching mathematics to realize that, I mean, for example, when we talk about the domain of rational functions. The definition of derivatives is very important to realize that division, i.e. ratio, is an inverse process, a process fraught with difficulty and particularity, inverse to a simpler one of multiplication. If one is dealing with constant quantities, it's perhaps sufficient to tell the students you can't divide by zero. It becomes much more complicated. So as a general policy, one should consider multiplication as basic and division as derived from it. So the relationship between intensive and extensive quantity stated more exactly is that given intensive quantity bearing over a certain space and also an extensive one in the same space, you can multiply them and get a new extensive one. Volume is a variable quantity in the room. Density of the air is a variable intensive quantity, but the product of the two is mass, which is again a variable extensive quantity.

27:30 Now, this strange but fundamental action or multiplication of a contravariant intensive quantity on the covariant extensive ones satisfies its own version of punctuality, naturality, or its kind of homogeneity. Which is so important and so basic that in different fields it has different names. In algebraic topology, it's called the projection formula. In group representation theory, it lies at the base of Frobenius' reciprocity. In quantum mechanics, it is called the canonical commutation relation. While in subjective logic, it is often submerged into a side condition on variables with the validity of a rule of entrance for existential quantification in the time of conjunction. So, to sum up then, spatial categories are distributive categories. Linear categories are quantitative. But there are two kinds of quantity. Extensive, which is co-variant in linear, and intensive, which is contra-variant and multiplicative as well as linear, and the relation between the two is that the intensive acts on the extensive in some kind of multiplication, which might in some cases... Now how can such systems of extensive and intensive quantities be constructed on a given spatial or distributive category? Well, typically the intensive quantities may be what's called representable. That is to say, there's a particular space, like the real number, that's sort of considered as a spatial object. Such that the intensive quantities of arbitrary variable word arbitrary x are just the maps from the categories from x into this fixed object. So this representability automatically implies the contravariant nature because composing with spatial maps will pull back these kinds of quantities. Now, one of the important lines for defining extensive quantities is by declaring them to be just linear functionals, smooth linear functionals, on the intensive quantities.

30:00 This is the point of view supported in the topological context by the classical degrees representation theorem, and in another context, which I'll come to in a moment, by the universal coefficient theorem. It's also at the basis for Laurence Schwartz's determination of the concept of smooth, the smooth category, which is justified as being the so-called dual space or linear function. These functions will automatically arise in the general context that I have described, because if you take an intensive quantity, F, and an extensive quantity, mu, multiply them together to get another extensive quantity, its total is the integral of F described in mu. The usual integral is the total as a product. Extensive. So, in particular, each extensive quantity mu gives rise to a functional which assigns to every intensive f the integral, the total value of the product. Now, again, this functional does not always determine the extensive quantity. And it may also be that not every reasonable functional comes from an intensive f. It depends on the particular kind of extension of quantum compression. However, in certain contexts, there is the reach theorem, for example, which shows that the linear functionals do determine the underlying extensive quantities, and therefore dis-identification is possible in those cases. In such a case where the extensive quantities can be identified with functionals on the intensive ones, The fundamental canonical commutation relation or projection formula, whatever it's called, is automatic because in that case the action of the intensive on extensive is just defined by integrating the product of the intensive upon it.

32:30 On the other hand, there's also an opposite situation where this automatic validity of the fundamental canonical commutation relation holds. Namely, in which we assume given notions of extensive quantity and define appropriate notions of intensive in terms of that. Namely, if we have some, let's say in general, two notions of extensive quantity, i.e. two pro-variant additive functors in the distributed category, we might say, well since the role of intensive quantities is supposed to be However, as a ratio, or really as a transformation of one kind of extensive quantity into possibly another one, we could just consider the natural transformations between those types considered as functions. However, said like that, this would just amount to some notion of constant intensive quantity because there's no particular space at that dimension. A construction which was implicitly already given in the Eilenberg and McLean's 1945 paper, but which began to be very much emphasized by Rotendieck around 1960, is the following. If you're given a category, for example a distributive category, and also a particular object in it, you can construct another distributive category whose objects are the maps going to that particular object, and whose maps are commutative triangles above these. This category could be thought of as families of spaces parametrized by a given... the category of objects are families of spaces parametrized by a given base space or perhaps namely that the objects in the family are sort of the fibers of these maps or you could consider them as generalized subsets, generalized sub-objects which might have multiplicity. In any case, under both of those guises, this construction of a new category associated with a particular object plays a very important role. Now if we, given a particular space X, if we consider this category of spaces over it, it has an obvious forgetful counter to the original category of spaces.

35:00 And we take the domain of the map and forget about the map. Now, if we compose that forgetting process with two extensive quantity types, we can now consider natural transformations between those composites. These are now natural transformations from one extensive quantity type to another, which vary over a given space. In fact, the naturality in this guise is precisely equivalent to the fundamental projection formula or canonical connotation relation. According to the second doctrine, which defines intensive in terms of extensive, we simply define it intensive to mean everything that satisfies the projection point. And this is also a viable doctrine. So we have two extreme cases where one type of quantity can be defined in terms of the other. In general, they both have their own role to play. In particular, notice that in my definition of linear category, I did not say that the addition was a moving group. That, of course, is a very important case, but an opposite case is for the addition of quantities. By the way, see, the quantities in general are the mass in which this addition of quantities might be inappropriate. Now, already the Grossmann brothers were aware that the... The actual subject matter of logic is idempotent and variable quantities, so indeed the quantities, I mentioned for example that quantities, intensive quantities might be representable by fixed objects like the real numbers, but also idempotent quantities may be representable by so-called truth value objects. This is one of the axioms of topology theory. So that as a special case in quantity, we have the input, additively the input, which, in another way, are the actual subject matter of logic.

37:30 Now I would like to propose, again, something which is very, um, completely worked out. The logic is usually interpreted in an intensive fashion. That is, the intensive quantities, also called propositional functions, are in the first instance contra-variant. The contravariance in that case is known as substitution. Given a spatial map, you can substitute it into a propositional function and you get another propositional function. And of course that preserves and, that is to say, it's also more cryptic as well as linear. Now, how do these logical entities actually arise in mathematics? I say that they arise as descriptions of where quantities live. For non-indipotent quantities, for example, if you consider where is the magnetic field non-zero, then that becomes an indipotent quantity. Or in the extension case, one might speak of the populated part of Spain, the world. Well, the populated part has a propositional function which describes it, which is, of course, indipotent. How did it arise? The populated part is the part where the population is. Population itself is not integral. Add two populations, you get a bigger one. So what I'm trying to get across is it seems to me that logic itself is not really a starting point in mathematics. The important quantities, both intensive and extensive, arise as the supports of quantities which are not integral. Now, this relationship is, I think, pretty well understood in the intensive case. However, I failed to find any systematic discussion of the supports of extensive quantum mechanics. These supports of measures and distributions are discussed in various particular contexts in analysis, but the general logical theory on them seems to be lacking. Well, particularly the idea of the extensive aspect of logic also seems to be underdeveloped. It's usually just considered as sort of the adjoint to the intensive part, and as such it plays a very important role, but the separate role, for example, the notion of finite, finite subsets.

40:00 Finite subsets of a space are mapped co-variably from one space to another, so in some sense the concept of finite is a logical notion, but not a propositional function, because it's not preserved by substitution. You substitute a map into a finite, you may not get a finite anymore, depending on the nature of the map. So it seems to me that the extensive logic needs to be developed just as extensive properties in general along with the intensive logic and in particular the theory of not the roots of the equation, which is the intensive story, but the supports of a measure of distribution needs to be seen as a logical or philosophical relationship as well as the sort which arises in a particular context in analysis. Now, as we come back to the role of space as an arena for becoming, I already alluded in my example to the possibility that, well, in fact, the categories of space, those distributive categories, are categories of spaces. As opposed to a whole flock of other distributed categories, which I won't mention, namely categories of sheaves or covered spaces and so on, which are somehow elaborate descriptions of a single space, they are also distributed. Further support to the idea that distributed categories are non-spatial. But the categories of spaces, of spaces, distributed categories of spaces, tend to have... The following particular feature that they contain spaces or objects that can act as parameterizers of becoming. Now, the parameterizer of becoming is basically a connected, strictly bi-pointed object. The distributive categories, by the way, are perhaps the most general context in which some kind of concept of connectedness or components can be discussed.

42:30 These are the initial coterminal, not empty in other words, but also not the coproduct of smaller spaces, cannot be expressed as the sum of smaller spaces which is connected. On the other hand, strictly bi-pointed means that you can find two points, i.e. two maps in the terminal object, which are completely disjoint. So such a thing I call the parameterizer of becoming. And those distributed categories which contain an object with these simple abstract properties are quite qualitatively different from the others. So the process of becoming in a certain space is accompanied by a map from a parameterizer of becoming into that space. The idea is that the point of the space which is associated by the map to one of the two distinguished instances becomes The one which is associated to the other. This study of becoming is justified because the parametrizer is connected. Somehow they are really continuously becoming. This is not to negate the role of... Alright, now this possibility of internalizing to the spatial category the study of becoming through use of becoming parametrizers obviously has... A very important application, which is the study of particular dynamical processes, the fact that we can study dynamical systems, differential equations, and so forth, all depends on having, in some way or another, internalized to the category of spaces and becoming, truly becoming, parametrized. Of course, the rich mathematical content that a particular category may have permits a detailed study of the mathematical processes, which is very important. However, I want to concentrate here on the qualitative aspect, which remains after all such connected processes of becoming are imagined completed. That is to say, we look only at the possibility of becoming, not the actual details of becoming.

45:00 So in the Horowitz spirit, we arrive in this way at the so-called homotopy category of any such spatial distributed category. Homotopy category is one in which the objects may be considered the same, but the mass have become sorted into, well actually the mass are the components of the space of mass in the given category. So that, in particular, all spaces which can be contracted have become all isomorphic to the terminal space, and any two points which can become one another have become equal, and any map which can become another map has become equal to it. So there is an extremely rich qualitative structure left over after we have made this passage. The, all the remarks, all the general remarks about space, quantum mechanics, and extensive and intensive quantum