From: ram@zedat.fu-berlin.de   
      
   Peter Fairbrother  wrote or quoted:   
   >Yikes! I know a bit about lattices as groups (I am a cryptologist), but   
   >not as abstract algebra. Though I can read that, but not grok it. I will   
   >maybe get a book ..   
      
     In the meantime, it came to my mind, there might be a generalization   
     of the concept in category theory. So, I asked the chatbot and got:   
      
   |In classical topology, continuity of a function between   
   |topological spaces means the preimage of every open set is open.   
   |   
   |In lattice theory and denotational semantics, continuity   
   |often refers to preserving directed suprema or certain   
   |order-theoretic limits.   
   |   
   |Category theory abstracts these ideas by defining a /functor/   
   |to be /continuous/ if it preserves limits. More precisely:   
   |   
   |  A functor F: C --> D is called /continuous/ if it preserves all   
   |  small limits that exist in C. That means for any diagram D: J --> C   
   |   whose limit exists in C, the image under F of this limit   
   |  is isomorphic to the limit of the composed diagram F o D in D.   
   |   
   |  This notion extends to various categories including posets,   
   |  where a poset is continuous if it is a continuous category in   
   |  the sense of having particular filtered colimits and adjoints   
   |  related to ind-objects. The classical notion of continuous   
   |  posets fits as a special case of continuous categories.   
   |   
   |  Thus, continuity in category theory generalizes both continuity   
   |  in topological spaces and continuity of functions on lattices (as   
   |  used in denotational semantics and domain theory). It unifies these   
   |  via the preservation of universal constructions like limits.   
   |   
   |  Continuous categories and continuous functors form a rich   
   |  theory that includes locally finitely presentable categories,   
   |  Grothendieck toposes, and higher-categorical analogs, showing   
   |  deep connections across topology, order theory, and semantics.   
   |   
   |Hence, continuity is abstracted by category theory as the   
   |preservation of limits by functors, which generalizes the   
   |classical and order-theoretic notions of continuity.   
   .   
      
     "Continuity in category theory", with or without quotation   
     marks, can be given to web search engines to learn more . . .   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)