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The note introduces preliminary definitions, characterizes co/ends as particular co/limits, and derives results from that characterization. Later sections highlight examples across Algebra, Algebraic Topology, Category Theory, and extensions to higher-dimensional category theory. Appendices sketch further results relying on co/end calculus and are designed as guided exercises.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/fosco-loregian-the-coend-my-only-cofriend/296959/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/fosco-loregian-the-coend-my-only-cofriend/296959.png","ImageObject",300,407,{"name":92,"@type":93},"Lute","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the main goal of the survey in the introduction?","Question",{"text":113,"@type":114},"To familiarize readers with co/end calculus in category theory by collecting examples of its applications, while clarifying that the material is not presented as original work.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the note characterize co/ends?",{"text":118,"@type":114},"After giving preliminary definitions, it characterizes co/ends as particular co/limits and then derives multiple results directly from that characterization.",{"name":120,"@type":111,"acceptedAnswer":121},"In which mathematical areas are co/end calculus applications discussed?",{"text":122,"@type":114},"The later sections present examples that use co/end calculus as an abstract method for explicit computations across Algebra, Algebraic Topology, and Category Theory, including higher-dimensional generalizations.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},296959,1790275239,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},137454149569,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","THIS IS THE (CO)END, MY ONLY (CO)FRIEND  \nFOSCO LOREGIAN†  \nAbstract. The present note is a recollection of the most striking and useful applications of co/end calculus. We put a considerable effortin making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory as well as some generalizations to higher dimensional category theory.  \nThe appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu (端楔術, du¯anxi¯e shù; literally “the art [of handling] terminal wedges”), can regard basically every statement as a guided exercise.  \nContents  \nIntroduction. 2  \n1. Dinaturality, extranaturality, co/wedges. 6  \n2. Yoneda reduction, Kan extensions 20  \n3. The nerve and realization paradigm. 24  \n4. Weighted limits. 35  \n5. The theory of relators. 46  \n6. Operads using coends. 55  \n7. Higher dimensional coend calculus. 63  \nAppendix A. Promonoidal categories 91  \nAppendix B. Fourier transforms via coends. 93  \nReferences 94  \nKey words and phrases . end, coend, dinatural transformation, operad, profunctor, relator, Kan extension, weighted limit, nerve and realization, promonoidal category, Yoneda structure.  \n1  \n2 F. LOREGIAN  \nIntroduction.  \nThe purpose of the present survey is to familiarize its readers with what in category theory is called co/end calculus, gathering a series of examples of its application; the author would like to stress clearly, from the very beginning, that the material presented here makes no claim of originality: we put a special care in acknowledging carefully, where possible, each of the countless authors whose work was a source of inspiration in compiling this note. Among these, every erroneous or missing attribution must be ascribed to the mere ignorance of the author, which invites everyone he didn’t acknowledge to contact him.  \nThe introductory material appearing in section 1 is the most classical and comes almost verbatim from the classical reference [ML98]; the nerverealization formalism is a patchwork of various results, scattered in the (algebraic) topology literature; these results are often presented as a deus ex machina, mysterious machineries leaving the reader without any clue about why they work so well. The reformulation of operads using coends comes verbatim from [Kel05]; the chapter discussing the theory of relators (or profunctors, or distributors; in French it is common to call them distributeurs, following an idea of J. Bénabou; we deviate from this unfortunate use) comes from the work of J. Bénabou, and I immensely profited from some notes taken by T. Streicher, [Béner]; during May 2016 I had the pleasant opportunity to meet Thomas, and I fell in love with his mathematical style, near to the craftsmanship of certain watchmakers, but also vital and passionate. I hope this text is better, in view of those pleasant days of friendly mathematics.  \nChapter 4 on weighted co/limits is taken almost verbatim from [Rie14 , II.7]: only a couple of implicit conceptual dependencies on co/end calculus have been made explicit, and some examples left as exercises there are discussed in full detail; the fact that we follow so faithfully the exposition of [Rie14] must be interpreted not as an act of plagiarism, but instead ad an implicit invitation to get acquainted with such a wonderful book. I count Emily as one of the most fierce supporters of Category Theory among the“new generation”, and I’m proud to be considered a good mathematician by her.  \nA subsequent section discusses co/end calculus in hi","cbCaipZvzniFJXTo","https://ap.wps.com/l/cbCaipZvzniFJXTo","pdf",1166042,98,"English","# Introduction\n# Dinaturality, extranaturality, co/wedges\n# Yoneda reduction, Kan extensions\n# The nerve and realization paradigm\n# Weighted limits\n# The theory of relators\n# Operads using coends\n# Higher dimensional coend calculus\n# Promonoidal categories\n# Fourier transforms via coends\n# References","[{\"question\":\"What is the main goal of the survey in the introduction?\",\"answer\":\"To familiarize readers with co/end calculus in category theory by collecting examples of its applications, while clarifying that the material is not presented as original work.\"},{\"question\":\"How does the note characterize co/ends?\",\"answer\":\"After giving preliminary definitions, it characterizes co/ends as particular co/limits and then derives multiple results directly from that characterization.\"},{\"question\":\"In which mathematical areas are co/end calculus applications discussed?\",\"answer\":\"The later sections present examples that use co/end calculus as an abstract method for explicit computations across Algebra, Algebraic Topology, and Category Theory, including higher-dimensional generalizations.\"}]","FOSCO LOREGIAN - The (Co)End, My Only (Co)Friend | PDF",1789739181,247]