Timeline for Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?
Current License: CC BY-SA 4.0
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| Jul 23, 2023 at 17:14 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jul 23, 2023 at 17:13 | vote | accept | HDB | ||
| Jul 23, 2023 at 17:09 | comment | added | Tim Campion | In general, a span of sets $A \leftarrow B \to C$ is naturally thought of as an undirected bipartite graph, with vertices $A \amalg C$ and edges $B$. The pushout in spaces is the geometric realization of this graph, homotopy equivalent to a disjoint union of wedges of copies of $S^1$, and the pushout in sets is $\pi_0$ of this space. So the pushout is preserved by the inclusion from sets to spaces if and only if there are no (undirected) cycles in the bipartite graph. For the inclusion from discrete presheaves to space presheaves, this condition should be checked levelwise. | |
| Jul 23, 2023 at 17:00 | history | answered | Tim Campion | CC BY-SA 4.0 |