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**11 - 19**of**19**### HOMOTOPICAL EQUIVALENCE OF COMBINATORIAL AND CATEGORICAL SEMANTICS OF PROCESS ALGEBRA

, 711

"... Abstract. It is possible to translate a modified version of K. Worytkiewicz’s combinatorial semantics of CCS (Milner’s Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turn ..."

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Abstract. It is possible to translate a modified version of K. Worytkiewicz’s combinatorial semantics of CCS (Milner’s Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller’s privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra. Contents

### Homology, Homotopy and Applications, vol. 10(3), 2008, pp.193–221 DIAGRAMS INDEXED BY GROTHENDIECK CONSTRUCTIONS

"... Let I be a small indexing category, G: I op → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. ..."

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Let I be a small indexing category, G: I op → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G.

### DOI: 10.1016/j.top.2003.10.008 A remark on K-theory and S-categories

, 2013

"... It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [20]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, ..."

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It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [20]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories?”

### ASPHERICITY STRUCTURES, SMOOTH FUNCTORS, AND

"... This paper was translated from French by Jonathan Chiche. Abstract. The aim of this paper is to generalize Grothendieck’s theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories. ..."

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This paper was translated from French by Jonathan Chiche. Abstract. The aim of this paper is to generalize Grothendieck’s theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories.