From: Steve Harris Subject: Re: NEWBIE: Category Theory & Closure Operators Date: Wed, 03 Feb 1999 16:35:48 +0000 Newsgroups: sci.math,sci.logic,comp.lang.functional Vasili, You ask, where do closure operators fit into category theory? As it happens, I have had occasion to address this very concretely in my work in relativity, where my current project is to define completion operators of various sorts for spacetimes. Here is how I abstracted the notion of completeness, which I surmise is at least similar to what you are looking for in closure: One has a category X of objects and a subcategory X_0 of "complete" objects. Let U: X_0 --> X be the "forgetful functor" (i.e., inclusion), and I the identity functor on X. A completion operator C consists of the following: a functor C: X --> X_0 a natural transformation i: I --> UC the fact that X (via i) is left adjoint to U Just in case we're not using the same terminology: By a natural transformation i I mean that for each object A in X, i_A : A --> C(A) is a map in X, and for f: A --> B in X, C(f)i_A = (i_B)f. (Conceptually: Any object is embedded in its completion, and maps in X extend to maps of the completed objects.) By left adjoint, I mean that for any objects A in X and B in X_0, and any map f: A --> U(B), there is a unique map f_0 : C(A) --> B with (U(f_0))i_A = f. (More simply, ignoring U: For any f: A --> B with B already complete, there is a unique f_0 : C(A) --> B with (f_0)i_A = f.) Conceptually: Any map into a complete object extends uniquely to the completion of the domain. The important point is that left adjoints are unique up to natural equivalence (a natural transformation consisting of isomorphisms): See Corollary IV.1 in Maclane's Categories for the Working Mathematician. This is what makes it possible to speak of the functor C as *the* completion (or closure) operator with respect to the notion of completion embodied by the subcategory C_0. For closure, I think you would do very much the same thing, where you call an object A in X closed if C(A) = A, i.e, C is a closure operator. In my work, the problem was that I was starting with a collection of objects X' (spacetimes) and a sort of completion operator C (addition of the future causal boundary), but for A in X', C(A) was not in X' (a topological or causal space, but not a spacetime). I had to expand my category X' and suitably generalize my operator C so as to achieve a category X with functor C so that C actually took X to X (actually, to the subcategory X_0). My solution, for the case of "future causal completion", is dealt with (in excruciatingly categorical formuation) in "Universality of the Future Chronological Boundary", J. Math. Phys 39 (1998), p. 5427-5445. Steve Harris ============================================================================== From: leinster@dpmms.cam.ac.uk (Tom Leinster) Subject: Re: NEWBIE: Category Theory & Closure Operators Date: 3 Feb 1999 18:34:49 GMT Newsgroups: sci.math,sci.logic,comp.lang.functional In article <7984dm$4kq$1@nnrp1.dejanews.com>, bhalchin@my-dejanews.com writes: |> |> OK .. this time I will try to be less vague. Closure operators show up all |> over the place, e.g. in topology, universal algebra, measure theory (sigma |> algebra generation) and are used for generation. Now category theory |> generalizes many concepts that we find in various parts of math, e.g. the |> notion of a product of two objects (in topology, we have product spaces & the |> class of topologies is a category with continuous maps as arrows/morphisms). |> |> The question: since closure operators are so ubiquitous and we have the |> claims of category theory, WHERE do closure opertaors fit into category |> theory?? As left adjoints. For instance, fix a topological space X, let P(X) be the poset of all subsets of X, and let C be the poset of all closed subsets of X. There are order-preserving functions Cl: P(X) ---> C I: C ---> P(X), closure and inclusion respectively. If we view P(X) and C as categories (in which a morphism is an inclusion) and Cl and I as functors, then Cl is left adjoint to I. The same goes for when X is an algebra (group, ring, ...), C is the poset of all subalgebras, and Cl(A) is the subalgebra generated by a subset A of X. More hand-wavingly, closure operators assign to a given thing the smallest nice thing containing it. This is exactly what your typical left adjoint does. Take, for instance, the functor which assigns to any set S the (real) vector space whose basis is S. Then what this functor does is to produce the "smallest vector space containing S". Incidentally, I'm not sure who it is who makes "the claims of category theory" referred to above, but I don't know of any category theorists who claim that it can encompass all mathematical ideas... Tom ============================================================================== From: "Michael Deckers" Subject: Re: NEWBIE: Category Theory & Closure Operators Date: 5 Feb 1999 18:11:39 GMT Newsgroups: sci.math,sci.logic,comp.lang.functional > > The question: since closure operators are so ubiquitous and we have the > claims of category theory, WHERE do closure opertaors fit into category > theory?? > A closure operator on a partially ordered set is a special case of a monad = triple in a category. A pair of adjoint functors produces a monad (and a comonad), and a monad gives rise to many pairs of adjoint functors. This is dicussed extensively in: Michael Barr, Charles Wells: Toposes, Triples and Theories. Grundlehren der Mathematischen Wissenschaften, Band 278. Springer 1985. ISBN 0387961151. Saunders MacLane, Ieke Moerdijk: Sheaves in Geometry and Logic: A First Introduction to Topos Theory --------------------------------------------------------------------------- Michael H Deckers email: Michael.Deckers@mch6.siemens.de --------------------------- end of article --------------------------------