You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Lucius Gregory Meredith (Apr 10 2020 at 15:19):i'm curious if there has been any work on an alternate presentation of category theory that uses slightly different building blocks than morphisms. In particular, i'm thinking about monads, and specifically comprehensions as an alternate building block.
Lucius Gregory Meredith (Apr 10 2020 at 15:24):Set comprehensions are essentially the basis of the set theoretic foundations of mathematics. Monads generalize set comprehensions. Likewise, comprehensions are at the core of some of the most active areas of programming language design and practice. One has only to look at SELECT-FROM-WHERE to Haskell's do-notation to Scala's for-notation to XQuery's FLWOR expressions to the last 15 years of language integrated query to see how fertile and useful this notion is; and to see that monads lie at the heart of it.
Lucius Gregory Meredith (Apr 10 2020 at 15:27):My intuition is that while monads currently require notions of morphisms to express, we might be able to turn the situation on its head and express notions of morphisms via monads, much in the same way as we view extensional set-theoretic views of functions as particular kinds of sets of ordered pairs.
John Baez (Apr 10 2020 at 18:15):Hi! By the way, a whole stream on this topic seems excessive. This sort of thing would typically be a "topic" within some stream.
Lucius Gregory Meredith (Apr 10 2020 at 18:22):Thanks for the comment! You mean on alternative presentations of category theory? It seems pretty broad to me. The arrows only approach is certainly worth discussion in its own right, and there may yet be other presentations. i, for one, would like to see an effective presentation in terms of a domain equation, or something like that, as well. So, i see this as extremely broad with lots of subtopics.
John Baez (Apr 10 2020 at 18:24):Any topic could in theory become very broad if enough people wrote about it in enough different ways.
John Baez (Apr 10 2020 at 18:24):If "alternative presentations" means "alternative presentations of category theory" I think it'd fit snugly in theory: category theory.
John Baez (Apr 10 2020 at 18:25):But @Christian Williams is the organizer here. Carry on!
Lucius Gregory Meredith (Apr 10 2020 at 18:26):In terms of an effective presentation, i have spent some time developing a domain equation/type theoretic inspired approach to graphs that can be used as a basis for a wide range of category theoretic phenomena. (https://github.com/rchain/ladl/blob/master/ToGL.md)
Notification Bot (Dec 02 2021 at 12:02):This topic was moved here from #theory: alternate presentations > alternate presentation of category theory by Morgan Rogers (he/him)
Morgan Rogers (he/him) (Dec 02 2021 at 12:04):This thread may well stay dead, but the otherwise inactive stream it originated from has been deleted, so I moved this here for posterity.