Category Theory
Zulip Server
Archive

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.

Stream: learning: questions

Topic: Formalizing "formal" Torsor sums?

Alex Kreitzberg (Sep 12 2025 at 23:29):

In any affine space, or any Torsor, a list of points $a, b, c, d$ in the space, has a corresponding list of differences $b - a, c - b, d - c$, which you can add together to get a final distance $(b - a) + (c - b) + (d - c) = d - a$.

Often I'm thinking about the left hand side of such a sum as an actual image of the "zig zag" path something takes, which is then "the same" as the net vector traveled on the right side of that equation.

In fact, this is a special case of a similar sort of reasoning often done with categories. For example the expression $f \circ g = h$ is often displayed as a diagram where an explicit $g, f$ path is understood to be "the same" as a visually seperate arrow $h$.

What are these "formal paths" that we eventually reduce? How should I think about them or define them? Are there any fun things I can do with these "formal expressions"?

Mike Shulman (Sep 13 2025 at 05:29):

Well, a formal path in $C$ can be regarded as a morphism in $TC$, where $T$ is the free-category monad on quivers, and the operation taking it to its composite is the $T$-algebra structure $TC \to C$. Similarly for formal sums in any (abelian) group/monoid. I don't see anything special about the case when it's the displacement group of a heap.