[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657563-105":59,"doc-detail-657563-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","from-dynamical-systems-to-categories","From dynamical systems to categories","","A personal expository essay connects the mechanics of dynamical systems to the core constructions of category theory. Starting from time actions on a phase space, it reviews monoid-like axioms for flows and illustrates non-invertible versus invertible examples, then introduces a refined arrow-based viewpoint using Hom(x,−). This leads to the definition of a category, followed by functors as morphisms between dynamical-system structures.",{"@graph":69,"@context":122},[70,84,105],{"@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/from-dynamical-systems-to-categories/657563/",{"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/from-dynamical-systems-to-categories/657563.png","ImageObject",300,407,{"name":92,"@type":93},"Stanford","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does the document define a dynamical system in terms of time and phase space?","Question",{"text":112,"@type":113},"It presents sets for phase space and time, then defines a flow that moves points according to time, satisfying identity, composition, and associativity laws.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does Hom(x,−) represent and why is it introduced?",{"text":117,"@type":113},"Hom(x,−) is the set of possible moves from a point x, visualized as arrows. It captures situations where dynamics are not homogeneous and the available future depends on the starting point.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the essay motivate the definition of a category from dynamical systems?",{"text":121,"@type":113},"By interpreting time as arrows between space points and using composition rules for arrows, it arrives at the category axioms: objects, hom-sets, identity morphisms, associative composition, and optional invertibility leading to groupoids.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657563,1792982832,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":52},2336477552062,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","From dynamical systems to categories  \nJean-Baptiste Stiegler  \nOctober 20, 2026  \nI wanted to write this text to share a fun thought I had. It is also a pretext to (finally) try out Typst. The goal is to show how a simple abstraction of dynamical systems leads to the definition of a category. And even to a 2-category!  \nBut bear in mind that all of this is useless, surely not new, and I’m no abstract nonsense expert.  \nContext  \nIn June 2025, out of curiosity, I attended a workshop about mathematics and philosophy, with an emphasis on category theory. During one talk, a philosopher discussed about the concept of time in mathematics. From what I remember, he mainly talked of discrete and continuous times (ℤ and ℝ) . Although he seemed sympathetic, the content didn’t satisfy me much. As a dynamicist, it questioned what time meant to me…  \nUsual dynamical systems  \nLet’s try to give a definition of a dynamical system. From my experience, the layman will think of differential equations, but of course it’s much richer.  \nLet 􀀃 and 􀀄 be sets, which physicists could call a phase space and the time. A dynamical system is away to move points 􀀅 ∈ 􀀃 according to the flow 􀀇 of time 􀀈 ∈ 􀀄 :  \n􀀇 : (􀀄 (, 􀀃􀀅)  􀀃􀀇 􀀑 (􀀅)) .  \nThe notation 􀀇􀀑 underlies that time is acting on space. Three axioms are imposed.  \n1. There exists a time 0 ∈ 􀀄 when nothing moved yet:  \n􀀇0 = id .  \n2. There exists a composition law for the flow of time: if 􀀈 , 􀀈′ ∈ 􀀄 , then  \n􀀇􀀑+􀀑′ = 􀀇􀀑 ∘ 􀀇􀀑′.  \n3. The flow composition is associative : if 􀀈 , 􀀈′, 􀀈″ ∈ 􀀄 , then  \n􀀇􀀑+􀀑′ +􀀑″ = 􀀇􀀑 ∘ 􀀇􀀑′ ∘ 􀀇􀀑″.  \nHence, we recover the concept of a monoid action. Time can be discrete or continuous (􀀄 = ℕ or ℝ+ ), invertible (groups like ℤ and ℝ), and more generally any monoid (even non-commutative nor “onedimensional”, like ℤ2 , ℂ1 or the free group 􀀢2 ) . Out of simplicity (and to match most examples), we’ll stick to the additive notation.  \nWe could also enrich the phase space 􀀃 with more structure (topology, measure), but that would take us too far.  \nSome examples  \nWe can start with the easiest example, 􀀄 acting on itself by translation:  \n􀀇 : (􀀄(􀀈×, 􀀄􀀅)  􀀄􀀈+ 􀀅) .  \n1 Complex time differential equations are a thing!  \nTake 􀀄 = ℕ or ℝ, we can represent moves with arrows, invertibles or not.  \n+4  \n+2  \n…  \n0 1 2 3 4 5 6 7 8 9  \nFigure 1: With 􀀄 = ℕ, non-invertible time.  \n−1  \n− √2 0 1 􀀭 􀀭 + 227 …  \nFigure 2: With 􀀄 = ℝ, invertible time.  \nIf the time is not one-dimensional, the graph can look more complicated (too much to draw it by myself) .  \nFigure 3: The free group 􀀢2 = ⟨􀀲 , 􀀳⟩ acting on itself, represented by its Cayley graph.  \nAllowed “time” at a given point (arrows)  \nAll previous examples were homogeneous. They consisted of monoids acting on themselves, and so each point looked the same. But often, this isn’t the case. We need a more subtle definition of dynamical systems than the one given earlier.  \nIf 􀀅 ∈ 􀀃 , let’s call Hom(􀀅 , −) the set of possible times/moves doable from there, diagrammatically represented by arrows.2 This notation will make sense later.  \nTake for example the space 􀀃 = [0, 1 [, with continuous time flowing towards the right. Think of a differential equation whose solution explodes in finite time. The set of possible moves from a point 􀀅 is  \nHom (􀀅 , −) = [0, 1 − 􀀅[ .  \nThey are all different, as sketched in the next figure.  \n2To match classical denominations, we could call the ends of arrows starting from 􀀅 the future orbit, and the sources of arrows ending at 􀀅 the past orbit.  \n+ 34 + 12  \nX  \n0  ~~ ~~34 1  \nFigure 4: The flow on the right gives different sets of arrows, for instance Hom(0, −) = [0, 1 [ and  \nHom ( 34 , −) = [0, 14 [ .  \nMonoidoid (category)  \nIn the previous example, we saw that time can be represented diagrammatically by arrows connecting points of space. Composition of arrows have a monoid-like structure, similar to many monoids interconnected. Categoricians call this a “monoidoid”, less ridiculously referred as… a category!  \nDe","cbCaiuCNo6rHTMUC","https://ap.wps.com/l/cbCaiuCNo6rHTMUC","pdf",119471,"English","# Context\n# Usual dynamical systems\n# Some examples\n## Allowed “time” at a given point (arrows)\n# Monoidoid (category)\n# Morphism of dynamical systems (functors)\n# Conclusion","[{\"question\":\"How does the document define a dynamical system in terms of time and phase space?\",\"answer\":\"It presents sets for phase space and time, then defines a flow that moves points according to time, satisfying identity, composition, and associativity laws.\"},{\"question\":\"What does Hom(x,−) represent and why is it introduced?\",\"answer\":\"Hom(x,−) is the set of possible moves from a point x, visualized as arrows. It captures situations where dynamics are not homogeneous and the available future depends on the starting point.\"},{\"question\":\"How does the essay motivate the definition of a category from dynamical systems?\",\"answer\":\"By interpreting time as arrows between space points and using composition rules for arrows, it arrives at the category axioms: objects, hom-sets, identity morphisms, associative composition, and optional invertibility leading to groupoids.\"}]","From dynamical systems to categories | PDF"]