symmetric closed monoidal structure on CGTop
Dear all, In the category CGTop of compactly generated Hausdorff topological spaces, the product provides a symmetric closed monoidal structure. Question: Is there another symmetric closed monoidal structure on CGTop ? Thanks in advance. pg. PS : Fact1 : On the category Top of topological spaces, there is exacly one symmetric closed monoidal structure. See Borceux vol 2 Proposition 7.1.6 and @article {MR87k:54012, AUTHOR = {Pedicchio, Maria Cristina and Solimini, Sergio}, TITLE = {On a ``good'' dense class of topological spaces}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {42}, YEAR = {1986}, NUMBER = {3}, PAGES = {287--295}, ISSN = {0022-4049}, CODEN = {JPAAA2}, MRCLASS = {54B30 (18B30 18D10 18D15 54A10)}, MRNUMBER = {87k:54012}, MRREVIEWER = {D. Pumpl{\"u}n}, } 30-Jul-2002 18:56:10 -0300,2498;000000000000-00000024
Dear colleagues, my new preprint is now available in Category Theory section of http://au.arxiv.org/ You also can download it from my home page http://www.math.mq.edu.au/~mbatanin/papers.html the .ps file is preferable because it contains all the pictures. Some of the pictures are missing from dvi file. Below is the abstract Title: The Eckmann-Hilton argument, higher operads and E_n-spaces Authors: M.A.Batanin Comments: 52 pages, 19 figures Subj-class: Category Theory; Algebraic Topology MSC-class: 18D05, 18D50, 55P48 \\ The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is `the same' as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an n-operad A in the author's sense there exists a symmetric operad S^n(A) called the n-fold suspension of A such that the category of one object, one arrow , . . . , one (n-1)-arrow algebras of A is equivalent to the category of algebras of S^n(A). Moreover, under some mild conditions, we present an explicit formula for S^n(A) which involves taking the colimit over a remarkable categorical E_n-operad. In the case, where A is contractible in an appropriate sense, this formula provides us with an action of the E_n-operad on algebras of S^n(A). \\ 30-Jul-2002 18:56:10 -0300,1039;000000000000-00000025
Date: Fri, 26 Jul 2002 15:56:09 +0200 (MEST) From: Philippe Gaucher <gaucher@math.u-strasbg.fr> Hello,
Philippe Gaucher wrote:
In the category CGTop of compactly generated Hausdorff topological spaces, the product provides a symmetric closed monoidal structure.
Question: Is there another symmetric closed monoidal structure on CGTop ?
Yes, There is at least one more: the "componentwise" structure like in Top. Define the tensor product X#Y as the cartesian product endowed with the final topology with respect to all maps X->X#Y, x|->(x,y) for all (fixed) y in Y and all maps Y->X#Y, y|->(x,y) for all (fixed) x in X. Then X#Y is compactly generated because cg spaces are closed under final structures. Moreover, X#Y is Hausdorff because its topology is finer than the product topology. Obviously, this gives a symmetric monoidal structure on CGTop. A map X#Y->Z is continuous, iff it is separately continuous (i.E. contiuous in each variable). In order to define function spaces, first proceed as in Top; i.e for spaces X,Y consider the set of all continiuous maps from X to Y with the topology of pointwise convergence, i.e the topology as a subspace of the X-th cartesian power of Y. Unfortunately, this topology need not be compactly generated, but if we switch to its cg modification, everything works, and we obtain a symmetric monoidal closed structure. This structur does not coincide with the cartesian structure because there are separately continuos but non-continuous real functions on the product of the real line with itself. I do not know whether there are more than two symmetric monoidal structures on CGTop. Greetings Reinhard 29-Jul-2002 17:09:04 -0300,1556;000000000001-00000028
participants (3)
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Michael Batanin -
Philippe Gaucher -
Reinhard.Boerger＠FernUni-Hagen.de