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Peiyuan Zhu (Dec 09 2022 at 02:07):How to understand presheaf action as "boundary extraction"?
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David Egolf (Dec 09 2022 at 06:09):I'm reading that chapter too these days! :smile:
(available here: https://direct.mit.edu/books/oa-monograph/5460/Sheaf-Theory-through-Examples )
I think the author goes on to elaborate in the remainder of that page what they meant. Is there a more specific question you had about their discussion?
You might also find this exercise relevant, from Riehl's "Category theory in context":
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Nathaniel Virgo (Dec 09 2022 at 09:36):I'm reading it too.
For this particular point, the books "Conceptual Mathematics" by Lawvere and Schanuel, and "Generic Figures and their Gluings" by Reyes, Reyes and Zolfaghari both go into a lot more detail.
Nathaniel Virgo (Dec 09 2022 at 09:39):But basically the point is that the vertex-figure is supposed to be the "boundary" of a loop-figure, because that's where it starts and ends. Then similarly in the next paragraph, the boundaries of the edges of a graph are vertices. Then the presheaf action gives you a set of functions that take figures and return their boundaries, such as the function that takes an edge of a graph and returns its head.