[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572988-105":53,"doc-detail-572988-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","cheat-sheet-on-recollements","Cheat Sheet on Recollements","","This document provides a concise mathematical summary of recollements, a central concept in homological algebra and category theory. It delineates the structural framework of a recollement involving three categories—S, T, and U—and the functors connecting them. The text covers the four adjoint pairs, the fully faithful requirements for the functors, the vanishing composition conditions, and the (co)unit transformations that facilitate the decomposition of objects within the category T. This technical resource serves as an essential reference for understanding the interplay between these functors and the resulting distinguished triangles that characterize the triangulated structure of recollements. By simplifying the complex definitions into a manageable cheat sheet format, it assists researchers and students in quickly identifying the core axioms and properties necessary for working with these categorical constructions.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/cheat-sheet-on-recollements/572988/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/cheat-sheet-on-recollements/572988.png","ImageObject",442,249,{"name":88,"@type":89},"Blue Pony","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does a recollement consist of in category theory?","Question",{"text":108,"@type":109},"A recollement consists of three categories (S, T, and U) connected by six functors forming three adjoint pairs, satisfying specific conditions regarding full faithfulness and composition.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What are the primary conditions for the functors in a recollement?",{"text":113,"@type":109},"The functors to the category T must be fully faithful, and the composition of functors from left to right must satisfy the condition j* o i* = 0, which further implies i* o j* = 0 and i! o j* = 0.",{"name":115,"@type":106,"acceptedAnswer":116},"How are objects in category T decomposed in a recollement?",{"text":117,"@type":109},"Objects in T are decomposed using the four remaining (co)unit transformations related by the existence of distinguished triangles, which relate components coming from categories S and U.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572988,1792462774,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},962090760266,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Cheat sheet on recollements  \nPieter Belmans  \nSummary  \nA recollement looks like this:  \n(1)  \nS  \nT  \nj!  \nj   \nj􀀃  \nU  \nIf you can remember the shape of (1), you only need to remember:  \n1. each pair of adjacent arrows is an adjoint pair;  \n2. the composition from left to right satis􀀛es j􀀃 􀀎 i􀀃 = 0;  \n3. functors to T are fully faithful;  \n4. the four remaining (co)unit transformations are used to decompose objects ofT.  \nWith more details: each arrow is left adjoint to the arrow below, i.e. we are given 4 adjunctions:  \n(2)  \ni 􀀃 a i 􀀃 j! a j􀀃  \ni 􀀃 a i ! j􀀃 a j􀀃 ;  \nsuch that  \n1. i 􀀃 , j! and j􀀃 are fully faithful, which tells us that we have the following four isomorphisms for the appropriate (co)units:  \n(3)  \nϵ i 􀀃 ai􀀃 : i 􀀃 􀀎 i 􀀃 ) idS η i􀀃ai! : idS ) i ! 􀀎 i 􀀃ηj!aj􀀃 : idU ) j􀀃 􀀎 j!ϵj􀀃 aj􀀃 : j􀀃 􀀎 j􀀃 ) id T  \n2. the (only) composition from left to right satis􀀛es  \n(4) j􀀃 􀀎 i 􀀃 = 0;  \nwhich tells us by adjunction that  \ni 􀀃 􀀎 j􀀃 = 0  \n(5)  \ni ! 􀀎 j􀀃 = 0 :  \n3. the four remaining (co)unit transformations are related to eachother by the existence of distinguished triangles decomposing every object in a piece coming from S and T:  \nϵ i􀀃ai ! ηj􀀃 aj􀀃  \ni 􀀃 􀀎 i ! (T )  T  T  T  j􀀃 􀀎 j􀀃 (T ) (6)  \nϵj! aj􀀃 η i 􀀃 ai􀀃  \nj! 􀀎 j􀀃 (T )  T  T  T  i 􀀃 􀀎 i 􀀃 (T )  :","cbCaiowps3qoqfVk","https://ap.wps.com/l/cbCaiowps3qoqfVk","pdf",351692,"English","# Summary of Recollements\n## Definitions and Adjunctions\n## Functor Properties and Compositions\n## Distinguished Triangles","[{\"question\":\"What does a recollement consist of in category theory?\",\"answer\":\"A recollement consists of three categories (S, T, and U) connected by six functors forming three adjoint pairs, satisfying specific conditions regarding full faithfulness and composition.\"},{\"question\":\"What are the primary conditions for the functors in a recollement?\",\"answer\":\"The functors to the category T must be fully faithful, and the composition of functors from left to right must satisfy the condition j* o i* = 0, which further implies i* o j* = 0 and i! o j* = 0.\"},{\"question\":\"How are objects in category T decomposed in a recollement?\",\"answer\":\"Objects in T are decomposed using the four remaining (co)unit transformations related by the existence of distinguished triangles, which relate components coming from categories S and U.\"}]","Cheat Sheet on Recollements | PDF",1791930395]