Gabriel topologies on coherent quantales

AbstractThe set of Gabriel topologies on a coherent quantale ordered under inclusion is a frame (studied by Rosenthal, Simmons and others). The set of all those Gabriel topologies that are inaccessible by directed joins (we call such topologies compact) is a subframe of it. When the quantale under consideration is commutative the frame of compact topologies is coherent. Several notions of spectra in ring theory appear as instances of this construction. When the quantale is non-commutative and coherent and its finite elements are closed under (right) implication then the frame of compact topologies is locally compact and compact. We present an interpretation of the notion of compact Gabriel topology on a coherent quantale in terms of deductively closed sets of formulae for a system of prepositional logic without the contraction and possibly the exchange rule (but admitting weakening). Our local compactness (and the subsequent spatiality) results for the frame of compact topologies correspond to a completeness theorem for such a system

Limits of small functors

For a small category K enriched over a suitable monoidal category V, the free completion of K under colimits is the presheaf category [K*,V]. If K is large, its free completion under colimits is the V-category PK of small presheaves on K, where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PK.Comment: 17 page

A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Delta^n may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations. This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.Comment: 81 pages. Supersedes arXiv:math/0411344 and arXiv:math/0411345. To appear in Advances in Mathematics. Version 2: tiny errors correcte

19. Yüzyılın Başlarındaki Osmanlı Müzik Stilinin Efterpi Müzik Koleksiyonunun İncelenmesi Aracılığı İle Araştırılması

Thesis (M.A) -- İstanbul Technical University, Institute of Social Sciences, 2018Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 2018In the current research, we study the musical style of the early 19th century through a primary source of that time, the Efterpi musical collection. This source provides us with vital information regarding particular socio-musical features and it gives us the opportunity to acquire new knowledge concerning the musical performance of this period. Furthermore, Efterpi musical collection has not been studied adequately by the current scholarship and this research reveals many unknown elements of the musical style as well as the mindset of the Ottoman musicians in the beginning of 19th century.Mevcut araştırmada, 19. yüzyılın başlarındaki müzik stilini o zamanın ana kaynağı olan Efterpi müzik koleksiyonuyla inceliyoruz. Bu kaynak bize belirli sosyo-müzikal özellikler hakkında önemli bilgiler sağlamaktadır ve bize bu dönemin müzikal performansı ile ilgili yeni bilgiler edinme fırsatı vermektedir. Ayrıca, Efterpi müzik koleksiyonu mevcut akademisyenler tarafından yeterince çalışılmamıştır ve bu nedenle bu araştırma, 19. yüzyılın başlarında Osmanlı müzisyenlerinin zihniyetinin yanı sıra müzik tarzının pek bilinmeyen unsurlarını ortaya koymaktadır.Yüksek LisansM.

NOTIONS OF FLATNESS RELATIVE TO A GROTHENDIECK TOPOLOGY

ABSTRACT. Completions of (small) categories under certain kinds of colimits and exactness conditions have been studied extensively in the literature. When the category that we complete is not left exact but has some weaker kind of limit for finite diagrams, the universal property of the completion is usually stated with respect to functors that enjoy a property reminiscent of flatness. In this fashion notions like that of a left covering or a multilimit merging functor have appeared in the literature. We show here that such notions coincide with flatness when the latter is interpreted relative to (the internal logic of) a site structure associated to the target category. We exploit this in order to show that the left Kan extensions of such functors, along the inclusion of their domain into its completion, are left exact. This gives in a very economical and uniform manner the universal property of such completions. Our result relies heavily on some unpublished work of A. Kock from 1989. We further apply this to give a pretopos completion process for small categories having a weak finite limit property. 1. Introduction—Basic concept

Notions Of Flatness Relative To A Grothendieck Topology

Completions of (small) categories under certain kinds of colimits and exactness conditions have been studied extensively in the literature. When the category that we complete is not left exact but has some weaker kind of limit for finite diagrams, the universal property of the completion is usually stated with respect to functors that enjoy a property reminiscent of flatness. In this fashion notions like that of a left covering or a multilimit merging functor have appeared in the literature. We show here that such notions coincide with flatness when the latter is interpreted relative to (the internal logic of) a site structure associated to the target category. We exploit this in order to show that the left Kan extensions of such functors, along the inclusion of their domain into its completion, are left exact. This gives in a very economical and uniform manner the universal property of such completions. Our result relies heavily on some unpublished work of A. Kock from 1989. We further apply this to give a pretopos completion process for small categories having a weak finite limit property