Abstract
We present the notion of abstract gestures and show how it encompasses Mazzola’s notions of gestures on topological spaces and topological categories, the notion of diagrams in categories, and our notion of gestures on locales. A relation to formulas is also discussed.
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Our translation.
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Any category of presheaves on a small category is a Grothendieck topos. In fact, given a category of presheaves on a small category \(\mathscr {C}\), it is a category of sheaves if we consider on \(\mathscr {C}\) the trivial topology, whose unique covering sieve for each object of \(\mathscr {C}\) is the maximal sieve.
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See the theorem at [8, p. 47], which holds for cocomplete categories. This theorem remains valid if we only assume the existence of the colimits involved in the definition of L.
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A topological space X is said to be locally compact if for each point \(x\in X\) and each open neighborhood U of it, there is a compact neighborhood of x contained in U. In the case when X is a Hausdorff space, this definition is equivalent to saying that each point in X has a compact neighborhood. In this way, every compact Hausdorff space is locally compact.
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Or core-compact, according to the terminology in [3].
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This name is due to Milnor, who first studied the geometric realization in the context of algebraic topology, though for simplicial sets instead of digraphs. However, in [10], this object is called spatialization.
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A digraph is locally finite if each vertex is the tail or head of only finitely many arrows.
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I borrowed this idea from Octavio Agustín-Aquino.
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That is, local rings: all non-invertible elements form a two-sided ideal.
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The author ignores whether or not such a left adjoint exists.
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Though probably the more interesting objects to study are the pointwise Kan extensions and hence the realizations and gesture objects defined by means of (co)limits as above.
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Which was not precisely a diamond since it is noticed there that there is a possible framework for formulas for each field k.
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Arias, J.S. (2017). Abstract Gestures: A Unifying Concept in Mathematical Music Theory. In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_14
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