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.
Gurkenglas (Sep 17 2021 at 20:31):An arrow has a source and a target. A path of arrows has each target equal to the next source. A function witnesses "Each X induces one Y.". A functor witnesses "Each path of X induces one Y.". A category is just a functor where X happens to be Y, and doesn't deserve particular attention, right? Except that the functor law only works right if the source and target categories are available to participate in the supposedly-unique inducing.
And then one could presumably do the same generalization to make natural transformations, rather than functors, the basic building block. Can you recommend prior work?
Morgan Rogers (he/him) (Sep 18 2021 at 09:28):Gurkenglas said:
A category is just a functor where X happens to be Y, and doesn't deserve particular attention, right? Except that the functor law only works right if the source and target categories are available to participate in the supposedly-unique inducing.
It sounds like you're conflating the different presentations of the theory of categories. There is a one-sorted "just arrows" presentation and a two sorted "objects and arrows" presentation of (1-)categories. This can be lifted to 2-categories; there are 1, 2 and 3-sorted presentations of 2-categories, and perhaps you're describing what these look like for the 2-category of categories.
Morgan Rogers (he/him) (Sep 18 2021 at 09:31):There can be advantages to considering the fewer-sorted presentations, but in practice people don't do it very often because the objects and morphisms have distinct interpretations. If you want to see the 1-sorted presentation for 1-categories written out, look up Freyd and Scedrov's book, Categories, Allegories. I don't know where to send you for the 2-categorical one, though.
Sam Speight (Sep 18 2021 at 10:42):I believe Street's 'The algebra of oriented simplexes' contains a 1-sorted definition of strict -category.
James Deikun (Sep 18 2021 at 11:30):Section 2.1 of Emily Riehl's Complicial sets, an overture, which I recently happen to have read, recalls this definition in modern notation.
James Deikun (Sep 18 2021 at 11:36):(You'll also see it in Dominic Verity's Complicial Sets along with a veritable compendium of classical results on locally finitely presentable categories and other interesting things ... but if you want the original it's at https://researchers.mq.edu.au/en/publications/the-algebra-of-oriented-simplexes ...)
James Deikun (Sep 18 2021 at 13:06):This construction seems to me to work because the globe category is the Cauchy completion of a one-object semicategory (in this case an inversive semigroup) ...
James Deikun (Sep 18 2021 at 13:08):... making the two of them Morita equivalent, which means they have the same category of comodels (the category of globular sets).