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The contravariant functor $C(-)$ given by $$ \hom_{Top}(-,\mathbb{R}):cCW\to Rng $$ where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, faithfulness and fullness of this functor?

edited Oct 28 2010 at 15:55

Dmitri Pavlov
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asked Oct 28 2010 at 14:01

roger123
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Surjective on objects: Definitely not, how do you get mathbb{Z} or worse a noncommutative ring? Full: How do you induce the zero map between two rings of continuous functions with a continuous function between the spaces? Faithful: This is the only interesting one. I am guessing that it is faithful. Examining the proof that it is injective on objects (looking at the MaxSpec construction), should point you in the right direction I think. Maybe your question is more interesting if you restrict your attention to R-modules? – Steven Gubkin Oct 28 2010 at 14:27

Steven: the zero map shouldn't really count: it's not a ring homomorphism here. (I assume rings have identity and the identity is preserved, a standard convention for commutative rings.) – KConrad Oct 28 2010 at 14:38

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Rng is a strange choice of target category. You want at least commutative R-algebras and you actually get a commutative Banach algebra or, even better, a commutative C*-algebra over R with trivial involution. – Qiaochu Yuan Oct 28 2010 at 15:57

The Gelfand-Naimark-Theorem gives an answer. But it does not tell you how to see whether a space is a CW by looking at its function algebra. – Johannes Ebert Oct 28 2010 at 17:55

Dmitri Pavlov · Accepted Answer · 2010-10-28 15:53:01Z UTC

Corollary 4.1.(i) in Johnstone's book Stone Spaces (electronic version: http://gen.lib.rus.ec/get?nametype=orig&md5=C26F62F69C32101307213F1960F85BA3) states that the category of realcompact spaces is dual to the full subcategory of the category of commutative rings consisting of rings of the form C(X). The functor C implements the duality.

The category of compact CW-complexes embeds into the category of realcompact spaces as a full subcategory, hence the functor C is fully faithful.

The functor of continuous functions from compact CW-spaces to the reals

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