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John van de Wetering (Sep 13 2020 at 20:43):A monoid is a category with a single object. A preorder is a category with at most one arrow between each pair of objects.
In this sense, a monoid is very 'arrow-heavy' while a preorder is very 'object heavy'. Is there a sense in which this can be made rigorous in some kind of scale in which monoids and preorders occupy the two different extremes?
Perhaps the extreme end of the 'object heavy' side should actually be a discrete category, but anyway.
David Michael Roberts (Sep 13 2020 at 21:17):The inclusion of the category of posets into the category of categories has a left adjoint, namely sending a category to the poset with the same objects and an arrow between two objects iff there is one in the category (so: squash all the hom sets down to singletones). In the other direction, the inclusion of monoids into the category of pointed categories and strictly pointed functors has a right adjoint, taking only the pointed object and its endomoprhism monoid.
Peter Arndt (Sep 13 2020 at 23:10):As an addition to David's first answer, let me point out what gets forgotten by the poset squashing: The categories that become equivalent to the terminal category (i.e. that "have no poset part") are the so-called "strongly connected" categories, i.e. categories such that between any two objects there are morphisms in both directions. Clearly monoids are strongly connected, but not only monoids (e.g. the category of groups is strongly connected, and so is any of its full subcategories, because of the trivial homomorphisms).
Peter Arndt (Sep 13 2020 at 23:10):One can view this through topos theory. Instead of considering your categories as linked by functors and distinguished up to equivalence, you can consider them as linked by profunctors and distinguished only up to Morita equivalence: Small categories are called Morita equivalent if the presheaf categories are equivalent. This is the case if and only if they have equivalent idempotent completions.
These presheaf categories are toposes, and from a topos you can extract its "localic" (a locale is a particular poset) part: You take the so-called hyperconnected-localic factorization (see here: https://ncatlab.org/nlab/show/%28hyperconnected%2C+localic%29+factorization+system) of the unique map to the terminal topos .
A topos is called hyperconnected if and only if the second map of the factorization is an equivalence and localic if and only if the first part is an equivalence. In general none of the maps needs to be an equivalence, but in this sense the two phenomena are opposite ends of a spectrum.
For the map from a presheaf topos, , this hyperconnected-localic factorization is precisely given by inserting the poset squashing of . So a presheaf topos is hyperconnected if and only if the index category is strongly connected and localic if and only if the index category is a poset, and that is a "spectrum" picture like the one you were asking for - with posets at one end, but with more than monoids at the other end.
David Michael Roberts (Sep 14 2020 at 00:05):I also meant to add this is the sort of thing Lawvere likes to view with his Hegelian hat on: "unity of opposites" kind of thing
John van de Wetering (Sep 14 2020 at 08:21):@Peter Arndt That's a great answer, thanks!
Peter Arndt (Sep 14 2020 at 10:47):This topos view, looking at categories up to Morita equivalence, even sort of allows for the separation into a groupoid and a poset part, as I pointed out at this old MathOverflow question: https://mathoverflow.net/questions/19190/category-groupoid-x-poset
But you may lose sight of your original category if you go that far...
Peter Arndt (Sep 14 2020 at 10:54):In another answer to that MO-question, Tim Campion points out this article of Wells, where he decomposes a category into a group part and a non-group part: https://www.sciencedirect.com/science/article/pii/0021869380901301
There may well be a common refinement of all these views of small categories...
John Baez (Sep 14 2020 at 15:11):Thanks, Peter - I feel you finally gave me some intuition for the hyperconnected-localic factorization.
Morgan Rogers (he/him) (Sep 14 2020 at 21:27):Peter Arndt said:
There may well be a common refinement of all these views of small categories...
For one thing, we could consider refining the factorisation system that we're using. The surjection-inclusion factorisation doesn't take us very far for Grothendieck toposes over , but there are a bunch of factorisation systems for localic morphisms (cf a relatively obscure paper called "Factorization theorems for geometric morphisms, II" by Johnstone which covers several, and Bunge and Funk's Singular Coverings of Toposes which is dedicated to another, although that isn't a very accessible book) and there might be some factorization systems for hyperconnected morphisms in the pipeline somewhere.
In a different direction, you could add properties to your geometric morphisms and see what happens on each side! This is one of the things @Jens Hemelaer and I did in our paper together; it allows you to see a correspondence between properties of monoids and properties of spaces (such as the space of downward closed subsets of a poset, to keep it relevant to the discussion here!) which is entirely invisible at the level of small categories.
Once you've moved up to Grothendieck toposes, you could also replace "small categories" with "small sites" in order to get the full range of toposes up there, but if toposes aren't already your close friends that might not seem like such a natural thing to do. :sweat_smile:
Robert Seely (Sep 14 2020 at 22:42):Bunge and Funk's Singular Coverings of Toposes which is dedicated to another, although that isn't a very accessible book)
Available on Amazon.com, as well as on "the Russian site" (about which I will say no more! ;-) )