categories and their associated string diagrams. >> Abstract: Several typographical errors corrected 2012-05-13. >> endobj The scheme has many merits, but sacrifices useful type stream 1 More recently, simplicial sets have found many applications in higher category theory because simplicial sets which satisfy a certain horn lifting criterion (see Def- inition 5.7) provide a model for (1; 1)-categories, in which every cell of dimension greater than one is invertible. Nevertheless, the exposition is sufficiently detailed to make it clear what is quantum version of a marginal probability distribution is a reduced density operator. www.paoloperrone.org/phdthesis.pdf This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. Form the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Preprints and early-stage research may not have been peer reviewed yet. xڝV�n�0��+|�vE����I�"-��=E�M�,ɕ���ww)�JB��Drg�����4��*>U9�k���I6���� fg0ͿYL^]q5�s#�t��/�_�~]��\. We give a construction which formalizes the idea of "evaluating an expression partially": for example, "2+3" can be obtained as a partial evaluation of "2+2+1". Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. Every joint probability distribution on a finite set can be modeled as a rank one density operator. %���� It is hoped that this will be The parallels are not coincidental, and a common blueprint is found in category theory. This construction can be given for all monads on a concrete category, and it is linked to the famous bar construction, of. 10, 2005. to explicitly prove many standard results in our proposed string diagram based �����o8��/u� traditional proofs by diagram pasting retain the vital type information, but Reprint published on 2012-09-19, revised 2013-09-22 and 2020-04-15. We apply the latter to both ordinary and higher categories. We describe representable functors the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. Part IV introduces quasi-categories and homotopy coherence. 1 0 obj << They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. Basic Category Theory TOM LEINSTER arXiv:1612.09375v1 [math.CT] 30 Dec 2016 University of Edinburgh. Reprint published on 2005-04-23. PDF | These notes were originally developed as lecture notes for a category theory course. BASIC CONCEPTS OF ENRICHED CATEGORY THEORY G.M. Our approach is to proceed primarily by example, systematically applying useful not just to mathematicians, but also to physicists, computer scientists, 2 0 obj << ISBN 978-1-107-04424-1 (hardback). For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables. 3 0 obj << combine the strengths of these two perspectives, we propose the use of string /ProcSet [ /PDF /Text ] We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. These notes were originally developed as lecture notes for a category theory course. that require awkward bookkeeping in more traditional notation. In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. © 2008-2020 ResearchGate GmbH. Mathematical category theory was originally invented to describe the most general sort of. rules for many category theoretic concepts. ���_�)csSȠs�S�N���*e�\�әQ&�P��xl�}:˹L\�T �����.�qQdE�ÐF}�. Transmitted by Steve Lack, Ross Street and RJ Wood. PROFESSOR OF PURE MATHEMATICS, UNIVERSITY OF SYDNEY Received by the editors 2004-10-30. 1 First published as Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. Other assets of these notes are: • Thorough explanation of the Yoneda lemma and its signiﬁcance, with both intuitive All the documents are freely available in the arXiv. It is shown here that the basic structure is the same for both kinds of categories, and the topological structures of pointed sets (psychological categories), loop spaces and reduced suspensions (memory and learning), the wedge sum (the coproduct in a categorical dialectical pair expressing Riegel's Dialectical Psychology), and the smash product (succession of thoughts) have the nature of "trains of thought." ResearchGate has not been able to resolve any citations for this publication. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. Theorems, interpretations and concrete examples are given for monads as well as for comonads. These higher level applications will not be discussed here. retain the type information whilst pursuing a calculational form of proof. %PDF-1.4 xڝVM��6���Qj���ͦ�M��9=82mXI�>Z��̐\ym9 D�3o�3?�Y�Yɋ�Usǲ�>����2%�XK� Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. All rights reserved. >> All content in this area was uploaded by Paolo Perrone on Dec 26, 2019. The theory is then illustrated with an experiment that exploits these ideas. via open sets, via closed sets, via neighborhoods, via convergent ﬁlters, and via closure operations. We develop string ��o��=� V#��>�'�KF�����v��ѯ����_VY�x9z|�g� pL�U!�A[��dk� Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, July 2002. Join ResearchGate to find the people and research you need to help your work. informal treatment of topological notions, and have omitted most proofs. By contrast, KELLY, F.A.A. This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. graphical techniques to many aspects of category theory. �5�|�}�����x��q�e%l��z_�a���1eaY ��B:�|�������7�pӎ` �e��2�~�Z�v6�`�~��y��σ�B���,�H�t���q��;Y0b��Np���b�c߭�i7UHw�xu�bt�\.�- IE?9��U)�njwTp����,za^��C��=�K0%.�l���g������q-E^.���=���9�w_{b*#��"�� ����O�J�P���ɫ-�S7���_.�E�Iɧ纊ůQT�hV�%�7dEB������iL��6���c�sB�A��1g����p��(�������*��Ua؋���ō� ����2Ⱦ�>�]��K�?���!Z���7Bq{�`�2�4TR�zn&85�1�U}&Ը����(��}x�qR+��dբn�5��;�?���!��SZ�KHo���qŷ+I�)c���8�5i.w��������:�p��p��LS��5Ԋq�N��"pw�u�D����k;N�k���m;�)T��ǊP0 ��F�3&��I0�4��f�zL'L?���o���E@킛aGK.~D*��5聨4��+����M��U����qK��6�T��pR1�#T�kH����u �1����+2��7�Q�&�����d�4�}�L���u8��#��e �����R���. Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Let us brieﬂy recall the arrow notation for functions between sets.1 A /Contents 3 0 R Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). 2010 Mathematics Subject Classi cation: 18-01,68-01. /Length 933 Key words and phrases: Category theory, computing science. Category theory can explain what the notion of “natural” really means. poorly express the reasoning and development of categorical proofs. /Filter /FlateDecode There are natural functors between the categories of topological spaces and simplicial sets called the total singular complex functor and geometric realization, which form an adjoint pair and give a Quillen equivalence between the usual model structures on these categories. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction. Basic Category Theory Jaap van Oosten. monads, Kan extensions, limits and colimits. It is the most general thing in mathematics, and seems well suited to the description of something as wide-ranging as mind itself.

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