Abstract
Functional topology is concerned with developing topological concepts in a category endowed with certain axiomatically defined classes of morphisms (Clementino et al. 2004). In this paper, we extend functional topology to a monoidal framework, replacing categorical pullbacks by pullbacks relative to the monoidal structure (which itself replaces the product) or more generally relative to a relation on the category (Janelidze, Appl. Categ. Structures, 17(4),351–371, 2009). Our main application is to the opposite Woronowicz category of C ∗-algebras. In this category a natural class of proper morphisms yields the unital algebras as compact objects. When restricted to the commutative C ∗-algebras, we recover exactly the morphisms induced by proper continuous maps of locally compact Hausdorff spaces. We further endow this category with a factorization system and investigate the precise relation with the proper maps, building on an approach which we previously developed with the eye on the category of schemes (Lowen and Mestdagh, J. Pure Appl. Algebra 217(11), 2180–2197, 2013). We also show how our results for C ∗-algebras can naturally be adapted to the opposite Woronowicz category of nondegenerate algebras over a commutative ring.
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To Horst Herrlich: through hugely inspiring mathematics and moonlit memories your wonderful mind shines on.
The authors acknowledge the support of the European Union for the ERC grant No 257004-HHNcdMir and the support of the Research Foundation Flanders (FWO) under Grant No G.0112.13N.
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Lowen, W., Mestdagh, J. Tensor Functional Topology on Woronowicz Categories. Appl Categor Struct 24, 569–604 (2016). https://doi.org/10.1007/s10485-015-9423-x
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DOI: https://doi.org/10.1007/s10485-015-9423-x