A Horizontal Categorification of Gelfand Duality

There is no need to explain why the notions of geometry and space are fundamental both in mathematics and in physics. Typically, a rigorous way to encode at least some basic geometrical content into a mathematical framework makes use of the notion of a topological space, i.e. a set equipped with a topological structure. Although being just a preliminary step in the process of developing a more sophisticated apparatus, this way of thinking has been very fruitful for both abstract and concrete purposes. In a very important development, I. M. Gel'fand looked not at the topological space itself but rather at the space of all continuous functions on it, and realized that these seemingly different structures are in fact essentially the same. In slightly more precise terms, he found a basic example of anti-equivalence between certain categories of spaces and algebras (see for example [Bl,Theorems II.2.2.4,II.2.2.6] or [L,Section 6]). Since on the analytic side C(X; C) is a special type of a Banach algebra called a C*-algebra, the study of possibly non-commutative C*-algebras has been often regarded as a good framework for "non-commutative topology". The duality aspect has been later enforced by the Serre-Swan equivalence [K,Theorem 6.18] between vector bundles and suitable modules (see also [FGV] for a Hermitian version of the theorem and [T1, T2, W] for generalizations involving Hilbert bundles). By then, breakthrough results have continued to emerge both in geometry and functional analysis, based on Gel'fand's original intuition, for about four decades. In connection with physical ideas, L. Crane-D. Yetter [CY] and J. Baez-J. Dolan [BD] have recently proposed a process of categorification of mathematical structures, in which sets and functions are replaced by categories and functors. From this perspective, in this paper, we wish to discuss a categorification of the notion of space extending and merging together Gel'fand duality and Serre-Swan equivalence. On one side of the extended duality we have a horizontal categorification (a terminology that we introduced in [BCL2, Section 4.2]) of the notion of commutative C*-algebra, namely a commutative C*-category, or commutative C*-algebroid (see definition 2.1), whilst the corresponding replacement of spaces, the spaceoids (see definition 3.2), are supposed to parametrize their spectra. Spaceoids could be described in several different albeit equivalent ways. In this paper we have decided to focus on a characterization based on the notion of Fell bundle. Originally Fell bundles were introduced in connection with the study of representations of locally compact groups, but we argue that they come to life naturally on the basis of purely topological principles. Rather surprisingly, to the best of our knowledge, the notions of commutative C*-category and its spectrum have not been discussed before, despite the fact that (mostly highly noncommutative) C*-categories have been somehow intensively exploited over the last 30 years in several areas of research, including Mackey induction, superselection structure in quantum field theory, abstract group duality, subfactors and the Baum-Connes conjecture. At any rate, we make frequent contact with the related notions that can be found in the literature, hoping that our approach sheds new light on the subject by approaching the matter from a kind of unconventional viewpoint. Of course, once we have a running definition, it seems quite challenging in the next step to look for some natural occurrence of the notion of spaceoid in other contexts. For instance, we are not aware of any connection with the powerful concepts that have been introduced in algebraic topology to date. Also, the appearance of bundles in the structure of the spectrum suggests an intriguing connection to local gauge theory but we have not developed these ideas yet. Some of our considerations have been motivated by a categorical approach to non-commutative geometry [BCL2], and it is rewarding that some of its relevant tools (e.g., Serre-Swan theorem, Morita equivalence) appear naturally in our context. More structure is expected to emerge when our categories are equipped with a differentiable structure. In the case of usual spaces, in the setting of A. Connes' non-commutative geometry [C], this has been achieved by means of a Dirac operator, and then axiomatized using the concept of spectral triple.

Here below we present a short description of the content of the paper. In section 2 we mention, mainly for the purpose of fixing our notation, some basic definitions on C*-categories. Section 3 opens recalling the notion of a Fell bundle in the case of involutive inverse base categories and then proceeds to introduce the definition of the category of spaceoids that will eventually subsume that of compact Hausdorff spaces in our duality theorem. The construction of a small commutative full C*-category starting from a spaceoid is undertaken in section 4, while the spe

…(Full text truncated)…

This content is AI-processed based on ArXiv data.