F.W. Lawvere: WHY ARE WE CONCERNED? III
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Fri, 31 Mar 2006 09:15:28 -0500 (EST)
WHY ARE WE CONCERNED? III
The second main misconception about category theory
Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. Since such perceptions hold back the learning of category theory, and hence facilitate its misuse as a mystifying shield, they are among our concerns. We need to deal with the size preoccupation head on.
Experience has shown that we cannot build up or construct mathematical concepts from nothing. On the contrary, centuries of experience become concentrated in concepts such as "there must be a group of all rotations" and we then place ourselves conceptually within that creation; we state succinctly the properties which that creation as a structure seems to have, and then develop rigorously the consequences of those properties taken as axioms. The notion of category arose in that way, and in turn serves as a powerful instrument for guiding further such developments. Placing ourselves conceptually within the metacategory of categories, we routinely make use of the leap which idealizes the category of all finite sets as an object. The question is, what more? Of course we make use of the experience of those who have labored to justify mathematics, and it is fortunate that ultimately our results are compatible with theirs. (Mac Lane's use of the term metacategory is not
mysterious; it simply refers to the universe of discourse of any model, in the special context where the elements of such a model are themselves called categories and functors. In the spirit of algebra, we do not concentrate on the cumulative hierarchy which might have been used to present the metacategory, but rather on the mathematical category itself.)
The supposed size problems of category theory are often concentrated in functor category formation. For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness). That existence statement is compatible with standard set theory, although it is often presumed to be
incompatible.
In the original 1945 exposition of category theory, it was the Goedel-Bernays account of the cumulative hierarchy (see posting II) that was cited as probably relevant (in case the problem of justifying category theory should come up). As a result, category theorists have been worried about supposed "illegitimacies" that might arise from violating the Goedel-Bernays rules (which in essence stemmed from von Neumann). These rules expressed an expediency which was a very effective trick at the time, identifying two kinds of membership relation and truncating the
content at a plausible level. The Goedel-Bernays theory is well known to have the same logical strength as the Zermelo-Fraenkel system. An important advantage is that the greater expressive power of Goedel-Bernays permits it to be finitely axiomatizable, whereas Zermelo-Fraenkel is not; the greater expressive power concerns an element V of any model in which all small sets of the model can be embedded (just as another smaller element captures all finite sets). But the greater expressive power still allows mutual relative consistency: To every model of Goedel-Bernays, a model of Zermelo-Fraenkel can be constructed in a fairly straightforward
manner: just take the small elements; in the converse direction there are two procedures (left and right adjoint?): given a model of Zermelo-Fraenkel, one can take all definable subsets of it, or just all subsets, and in either case a model of Goedel-Bernays apparently results. Because these mutual interpretations are hypothetical, relatively weak assumptions are required on the background category of sets taken as the recipient of models. In fact, with only slightly stronger assumptions on the background category one can construct, for any model of Zermelo-Fraenkel, a model of what set theorists use daily as BG+, which contains as elements not only V but W = V^V, V^W etc.
Our practice is consistent with the minimal assumptions of professional set theorists: For any model of BG+ the presented metacategory of categories is both cartesian closed (in the usual elementary sense) and also has an object S of small sets. (Those facts strongly augment well-known properties, such as the existence of the first four finite ordinals and their adequacy in the metacategory relative to
the sub-metacategory of discrete categories; of course these same ordinals also co-represent one of the "2-category" structures on the metacategory).
The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it. Of course functor categories may no longer enjoy the same enrichment, just as functor categories starting from finite sets may not have finite hom-sets; but that is no reason to avoid considering them, and functionals on them, etc. when such considerations serve mathematics.
It is of special interest to note that the restrictive "law" (under which categorists have been chafing) was already repealed forty years ago by Goedel and Bernays themselves. In their correspondence of 1963, it appears that they had been informed that a student of Eilenberg was working on a project to base set theory and mathematics on category theory; their immediate response was that mathematics will have to consider finite types over the class of small sets. (The relative consistency was presumably obvious to them.)
Even though most set-theorists have themselves maintained clarity on the distinction, the identification of two kinds of membership in a formalized theory may have fostered in the minds of others a confusion between smallness (of a class or set) and existence as an element of the (meta)universe. Certainly, the specific meaning of smallness needs to be clarified (although for some purposes it can be taken as a parameter). There is a way of specifying smallness that is directly related to fundamental space/quantity dualities (rather than to imagined "building up" by stronger and stronger closure properties).
Just as Dedekind finite sets X are characterized by the condition that a natural map X --->Hom(Q^X, Q) is an isomorphism, so indications from the study of rings of continuous functions and other branches of analysis strongly suggest that all small sets X should satisfy the same sort of isomorphism, with the truth-value space Q being replaced by the real line (in both cases, Hom refers to the binary algebraic operations on the object Q). There is the possibility to assume that conversely all sets X satisfying that isomorphism are small i.e. that, like the Dedekind-finite sets, they belong to a single uniquely-determined category S. That possibility in itself would imply no commitment concerning the existence or non-existence of super-huge objects in the metacategory "beyond" S, S^S, etc. Such an axiom would be somewhat stronger than ZF, but much weaker than the standard discussions of contemporary set theorists.
************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere
************************************************************
Fri, 31 Mar 2006 09:15:28 -0500 (EST)
WHY ARE WE CONCERNED? III
The second main misconception about category theory
Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. Since such perceptions hold back the learning of category theory, and hence facilitate its misuse as a mystifying shield, they are among our concerns. We need to deal with the size preoccupation head on.
Experience has shown that we cannot build up or construct mathematical concepts from nothing. On the contrary, centuries of experience become concentrated in concepts such as "there must be a group of all rotations" and we then place ourselves conceptually within that creation; we state succinctly the properties which that creation as a structure seems to have, and then develop rigorously the consequences of those properties taken as axioms. The notion of category arose in that way, and in turn serves as a powerful instrument for guiding further such developments. Placing ourselves conceptually within the metacategory of categories, we routinely make use of the leap which idealizes the category of all finite sets as an object. The question is, what more? Of course we make use of the experience of those who have labored to justify mathematics, and it is fortunate that ultimately our results are compatible with theirs. (Mac Lane's use of the term metacategory is not
mysterious; it simply refers to the universe of discourse of any model, in the special context where the elements of such a model are themselves called categories and functors. In the spirit of algebra, we do not concentrate on the cumulative hierarchy which might have been used to present the metacategory, but rather on the mathematical category itself.)
The supposed size problems of category theory are often concentrated in functor category formation. For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness). That existence statement is compatible with standard set theory, although it is often presumed to be
incompatible.
In the original 1945 exposition of category theory, it was the Goedel-Bernays account of the cumulative hierarchy (see posting II) that was cited as probably relevant (in case the problem of justifying category theory should come up). As a result, category theorists have been worried about supposed "illegitimacies" that might arise from violating the Goedel-Bernays rules (which in essence stemmed from von Neumann). These rules expressed an expediency which was a very effective trick at the time, identifying two kinds of membership relation and truncating the
content at a plausible level. The Goedel-Bernays theory is well known to have the same logical strength as the Zermelo-Fraenkel system. An important advantage is that the greater expressive power of Goedel-Bernays permits it to be finitely axiomatizable, whereas Zermelo-Fraenkel is not; the greater expressive power concerns an element V of any model in which all small sets of the model can be embedded (just as another smaller element captures all finite sets). But the greater expressive power still allows mutual relative consistency: To every model of Goedel-Bernays, a model of Zermelo-Fraenkel can be constructed in a fairly straightforward
manner: just take the small elements; in the converse direction there are two procedures (left and right adjoint?): given a model of Zermelo-Fraenkel, one can take all definable subsets of it, or just all subsets, and in either case a model of Goedel-Bernays apparently results. Because these mutual interpretations are hypothetical, relatively weak assumptions are required on the background category of sets taken as the recipient of models. In fact, with only slightly stronger assumptions on the background category one can construct, for any model of Zermelo-Fraenkel, a model of what set theorists use daily as BG+, which contains as elements not only V but W = V^V, V^W etc.
Our practice is consistent with the minimal assumptions of professional set theorists: For any model of BG+ the presented metacategory of categories is both cartesian closed (in the usual elementary sense) and also has an object S of small sets. (Those facts strongly augment well-known properties, such as the existence of the first four finite ordinals and their adequacy in the metacategory relative to
the sub-metacategory of discrete categories; of course these same ordinals also co-represent one of the "2-category" structures on the metacategory).
The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it. Of course functor categories may no longer enjoy the same enrichment, just as functor categories starting from finite sets may not have finite hom-sets; but that is no reason to avoid considering them, and functionals on them, etc. when such considerations serve mathematics.
It is of special interest to note that the restrictive "law" (under which categorists have been chafing) was already repealed forty years ago by Goedel and Bernays themselves. In their correspondence of 1963, it appears that they had been informed that a student of Eilenberg was working on a project to base set theory and mathematics on category theory; their immediate response was that mathematics will have to consider finite types over the class of small sets. (The relative consistency was presumably obvious to them.)
Even though most set-theorists have themselves maintained clarity on the distinction, the identification of two kinds of membership in a formalized theory may have fostered in the minds of others a confusion between smallness (of a class or set) and existence as an element of the (meta)universe. Certainly, the specific meaning of smallness needs to be clarified (although for some purposes it can be taken as a parameter). There is a way of specifying smallness that is directly related to fundamental space/quantity dualities (rather than to imagined "building up" by stronger and stronger closure properties).
Just as Dedekind finite sets X are characterized by the condition that a natural map X --->Hom(Q^X, Q) is an isomorphism, so indications from the study of rings of continuous functions and other branches of analysis strongly suggest that all small sets X should satisfy the same sort of isomorphism, with the truth-value space Q being replaced by the real line (in both cases, Hom refers to the binary algebraic operations on the object Q). There is the possibility to assume that conversely all sets X satisfying that isomorphism are small i.e. that, like the Dedekind-finite sets, they belong to a single uniquely-determined category S. That possibility in itself would imply no commitment concerning the existence or non-existence of super-huge objects in the metacategory "beyond" S, S^S, etc. Such an axiom would be somewhat stronger than ZF, but much weaker than the standard discussions of contemporary set theorists.
************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere
************************************************************
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