74,397 research outputs found
Completing categorical algebras : Extended abstract
Let Ī£ be a ranked set. A categorical Ī£-algebra, cĪ£a for C, for short, is a small category C equipped with a functor ĻC : C n each Ļ ā Ī£n , n ā„ 0. A continuous categorical Ī£-algebra is a cĪ£a which C; has an initial object and all colimits of Ļ-chains, i.e., functors N each functor ĻC preserves colimits of Ļ-chains. (N is the linearly ordered set of the nonnegative integers considered as a category as usual.) We prove that for any cĪ£a C there is an Ļ-continuous cĪ£a C Ļ , unique up to equivalence, which forms a āfree continuous completionā of C.
We generalize the notion of inequation (and equation) and show the inequations or equations that hold in C also hold in C Ļ . We then find examples of this completion when ā C is a cĪ£a of finite Ī£-trees ā C is an ordered Ī£ algebra ā C is a cĪ£a of finite A-sychronization trees ā C is a cĪ£a of finite words on A.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en InformĆ”tica (RedUNCI
The Combinatorics of Iterated Loop Spaces
It is well known since Stasheff's work that 1-fold loop spaces can be
described in terms of the existence of higher homotopies for associativity
(coherence conditions) or equivalently as algebras of contractible
non-symmetric operads. The combinatorics of these higher homotopies is well
understood and is extremely useful.
For the theory of symmetric operads encapsulated the corresponding
higher homotopies, yet hid the combinatorics and it has remain a mystery for
almost 40 years. However, the recent developments in many fields ranging from
algebraic topology and algebraic geometry to mathematical physics and category
theory show that this combinatorics in higher dimensions will be even more
important than the one dimensional case.
In this paper we are going to show that there exists a conceptual way to make
these combinatorics explicit using the so called higher nonsymmetric
-operads.Comment: 23 page
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