Compact Hausdorff and C^*-algebra "objects" in a category. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:21:27Z http://mathoverflow.net/feeds/question/14888 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14888/compact-hausdorff-and-c-algebra-objects-in-a-category Compact Hausdorff and C^*-algebra "objects" in a category. Andrew Stacey 2010-02-10T10:06:58Z 2010-02-13T05:41:20Z <p>This is yet more on "algebraic objects in functional analysis".</p> <p>Since <a href="http://ncatlab.org/nlab/show/compactum" rel="nofollow">Compact Hausdorff spaces</a> are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable category representing functors from that category to CompHaus.</p> <p>An obvious such functor is the spectrum of a unital $C^*$-algebra. This seems to imply that $\mathbb{C}$ is a <em>compact Hausdorff object</em> in the category of unital $C^*$-algebras. So:</p> <blockquote> <p><strong>Question 1</strong>: Is this right?</p> </blockquote> <p>Followed by the obvious:</p> <blockquote> <p><strong>Question 2</strong>: Are there any other interesting "Compact Hausdorff" objects in other categories?</p> </blockquote> <p>Similarly, $C^\ast$-algebras is algebraic, and whilst Banach spaces isn't algebraic then it embeds in an algebraic theory (of <a href="http://ncatlab.org/nlab/show/totally+convex+space" rel="nofollow">totally convex spaces</a>). Again, to any compact Hausdorff space one can assign its $C^\ast$-algebra of continuous functions to $\mathbb{C}$. This suggests that $\mathbb{C}$ is a "$C^\ast$-algebra" object in CompHaus - except that $\mathbb{C}$ is not a compact Hausdorff space. However, we have a way out due to the way that $C^\ast$-algebras are algebraic: it's the unit ball that we should be thinking of and this is continuous functions to the closed unit disc in $\mathbb{C}$, which is compact Hausdorff. Thus $\{z \in \mathbb{C} : |z| \le 1\}$ seems to be a $C^\ast$-algebra object in Compact Hausdorff spaces. Again:</p> <blockquote> <p><strong>Question 3</strong>: Is this right?</p> </blockquote> <p>and</p> <blockquote> <p><strong>Question 4</strong>: Are there any other interesting "$C^\ast$-algebra" objects in other categories?</p> </blockquote> <p>and</p> <blockquote> <p><strong>Question 5</strong>: Are there any "Banach space" objects (or "totally convex space" objects) floating around anywhere?</p> </blockquote> http://mathoverflow.net/questions/14888/compact-hausdorff-and-c-algebra-objects-in-a-category/14891#14891 Answer by Sridhar Ramesh for Compact Hausdorff and C^*-algebra "objects" in a category. Sridhar Ramesh 2010-02-10T10:35:30Z 2010-02-13T05:41:20Z <p>Question 1: If I understand you correctly, you're proposing that $\mathbb{C}$ should be a compact Hausdorff object in some category because it represents a functor from that category to the category CH of compact Hausdorff spaces (in something like the sense that the functor $Hom(-, \mathbb{C})$ into Set factors through the forgetful functor from CH to Set). But I don't see why this should be sufficient to make $\mathbb{C}$ a compact Hausdorff object.</p> <p>That is, presumably, from the approach of functorial semantics, a compact Hausdorff object in a category C should be a product-preserving functor from L to C, where L is the dual of the Kleisli category for the ultrafilter monad on Set (that is, L is the Lawvere theory whose category of (Set-)models is the category of compact Hausdorff spaces). I can see how, more generally, for any Lawvere theory L and category C, every C-model of L (i.e., a product-preserving functor F from L to C) induces a representable functor Hom(-, F(1)) from C to Set which factors through the forgetful functor from Set-models of L to Set. But it's not obvious to me that the converse of this holds as well (that every representable functor from C to Set with this factorization property arises from some C-model of L).</p> <p>Perhaps I'm missing something and your reasoning for $\mathbb{C}$ being a compact Hausdorff object is something more than this. Perhaps I'm hopelessly confused. But, tentatively, I think the answer to question 1 is "No" or at least "Not necessarily".</p> <p>(Edit: As seen below, the correspondence does go both ways, so the last line is retracted, leaving the second-to-last line...)</p> http://mathoverflow.net/questions/14888/compact-hausdorff-and-c-algebra-objects-in-a-category/14895#14895 Answer by Tom Leinster for Compact Hausdorff and C^*-algebra "objects" in a category. Tom Leinster 2010-02-10T11:54:25Z 2010-02-10T11:54:25Z <p>This "answer" doesn't even get as far as answering question 1, but I'll go ahead anyway. </p> <p>All I want to say is how I think "compact Hausdorff space object" should be defined. This should be equivalent to what Sridhar said, though I haven't stopped to think about it. </p> <p>Let $\mathcal{E}$ be a category with small products. A compact Hausdorff object in $\mathcal{E}$ should be an object $X$ of $\mathcal{E}$ together with, for each set $I$ and ultrafilter $U$ on $I$, a function $\xi_U: X^I \to X$ satisfying some axioms that I'm too lazy to write down, but will explain a bit in a moment. </p> <p>When $\mathcal{E} =$ <strong>Set</strong>, you can think of $\xi_U$ as specifying the $U$-limit of each $I$-indexed family of points of $X$. (That there's exactly one limit point is the compact Hausdorff property.) One axiom tells you what happens when $U$ is the principal ultrafilter on some $i \in I$: then $\xi_U$ sends a family <strong>x</strong> to $x_i$. A second says something about limits of limits. A third (and I think there are only three) says something about what happens when you have a map $I \to J$. </p> <p>This formulation doesn't come out of thin air, you won't be surprised to hear---there's a systematic process for taking a (suitable kind of) monad on <strong>Set</strong> and producing a definition of its "algebras" in <em>any</em> category with products. But I won't go into that now.</p> http://mathoverflow.net/questions/14888/compact-hausdorff-and-c-algebra-objects-in-a-category/14951#14951 Answer by Dave Penneys for Compact Hausdorff and C^*-algebra "objects" in a category. Dave Penneys 2010-02-10T22:03:37Z 2010-02-10T22:03:37Z <p>The "Bohrification" paper arXiv:0905.2275 may be relevant to Question 4. As I understand, they discuss the notion of $C^\ast$-algebra objects in a given topos.</p>