Intuitionism's Limits: if C is a category sufficiently complex to demonstrate that some C-arrow f:a-->b is monic and B is a subcategory of C containing just f (and the requisite identity arrows), do we still know that f is monic? Should we? (Or, in other words, which view *should* dominate: Intuitionism, Realism, the category theoretic...?) What if C is something (semi?)fundamental like a category of all sets and functions, or a category of categories? I suppose the answer is that monicity is relative to a category, but what supports this as a claim? And doesn't it seem to contradict the reasonable realist claim that we can somehow know f in B to be monic? (Or am I missing something straightforward: that properties can be granted to f by its relationship to C via an inclusion functor?) This goes to the issue of the adequacy of category theory as a foundation in more than the simply technical sense. (I could be using the term "realism" incorrectly too: I take it to be a positon, in maths at least, that mathematical entities can have collections of properties beyond the constraints of a given theoretical context.) William James
William James <wjames@arts.adelaide.edu.au> writes:
I suppose the answer is that monicity is relative to a category, but what supports this as a claim?
It seems to me that category theory takes the sensible viewpoint that mathematical entities (e.g. objects and morphisms) only become interesting through their relationship with other entities. Every arrow looks just like every other arrow if we consider it in isolation. Every arrow in a category C is an image of the what James Dolan calls the "walking arrow" --- the nonidentity morphism in the free category C0 on a single morphism --- under some functor F: C0 -> C. Studying an arrow in isolation is just like studying the walking arrow, which is completely dull. The fun begins only when we have a bunch of arrows and start composing them. This is one reason why I think n-category theory should be useful in physics problems like quantum gravity, where it only makes sense to speak of where or when an event occurs relative to other events, not with respect to some spacetime manifold of fixed geometry. For some of the technical apsects of how this might go, see: John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105. Louis Crane, Clock and category: is quantum gravity algebraic?, J. Math. Phys. 36 (1995), 6180-6195. These both appeared in a special issue on diffeomorphism-invariant physics.
To William James, You must be using a non-standard (philosophical?) definition of "monic", since it is obvious using the standard (mathematical) definition that a monic remains a monic in any subcategory containing it (to get unnecessarily technical, because it's given by a universally quantified Horn sentence). Could you tell us your definition? (For the record: f: A -> B is monic iff for all x,x':X -> A it x f x' f is the case that X -> A -> B = X -> A -> B implies x = x'.)
If a and b are *two* objects, then, in the category consisting solely of those two objects, their respective identity maps, and one further map from a to b (and nothing more), that map is both monic and epic. Once embedded in another category, however, that map may easily fail to remain monic, may easily fail to remain epic, may remain one but not the other -- there's no telling. And if a = b instead, and f and the identity on a are the only *two* maps there are, then clearly f *may* be idempotent, hence neither monic nor epic; then again, f *may* be involutory, hence a true isomorphism. I think true realism requires that one pay strict attention to the definitions, refraining from free-associations with the vibrations of the terms defined. Cheers, -- Fred
Intuitionism's Limits: if C is a category sufficiently complex to demonstrate that some C-arrow f:a-->b is monic and B is a subcategory of C containing just f (and the requisite identity arrows), do we still know that f is monic? Should we? (Or, in other words, which view *should* dominate: Intuitionism, Realism, the category theoretic...?) What if C is something (semi?)fundamental like a category of all sets and functions, or a category of categories?
Whoops! The question is trivialised by using monicity as the relevant property. Reconsider it in terms of say f as an isomorphism, or of f holding some property in C that B lacks the resources to demonstrate. I'm thinking aloud on this question: constructive maths should say that of f in B there is no demonstration forthcoming, so judgment will be withheld on whether or not f has the property; a category theorist might say that category theory does not dwell on elements and that, in context, B is no different from any isomorph of 2, so there positively is no further property of f to be had other than that which can be demonstrated in any isomorph of 2. This is more than Intuitionism will allow. Might I, then, go on to say that the philosophies of constructive mathematics and category theory really are different? William James (if I'm digging a hole, I want it to be big)
Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST) From: William James <wjames@arts.adelaide.edu.au> (I grant you the original question would have been more recognisable given better use of language: "...philosophies of constructive mathematics and *of* category theory...") Your original question was "Which view should dominate?", where "the category theoretic view" was one of your options. (You had several questions but this one seemed the most central.) If you are asking whether the primary expression of structure should be in terms of relations between elements or transformations of objects, then I would answer this as follows. The analogous question for physics is whether energy and matter consist of particles or waves. The consensus in physics today is that both energy and matter can be viewed more or less equally accurately, if not equally insightfully, as either particles or waves. Which offers more insight depends on the circumstances. The corresponding position for mathematics would be that structure can be expressed more or less equally well in elementary or transformational terms, and that which approach gives more insight depends on the circumstances. The extent to which this is not the consensus in mathematics today is less a reflection on either approach than on the conceptual health of mathematics relative to that of physics. Vaughan Pratt
participants (5)
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Fred E J Linton -
John Baez -
Peter Freyd -
Vaughan Pratt -
William James