http://phck.net/everything/#blog notes by philip hackney Hugo en-us Sun, 02 Aug 2026 00:00:00 +0000 http://phck.net/post/edgewise.html Sun, 02 Aug 2026 00:00:00 +0000 http://phck.net/post/edgewise.html In which I ask an LLM to generalize the Bergner–Osorno–Ozornova–Rovelli–Scheimbauer characterization of 2-Segal objects in terms of edgewise subdivision. Spoiler: it does a good job. http://phck.net/post/spheres.html Sun, 26 Jul 2026 00:00:00 +0000 http://phck.net/post/spheres.html We explain two results. Theorem: For n≥1, Δⁿ/∂Δⁿ is 2n-Segal, but not lower (2n−1)-Segal. If n≥2, then it is upper (2n−1)-Segal if and only if n is odd. Theorem: For n≥1, ∂Δⁿ⁺¹ is (n+1)-Segal but neither upper nor lower n-Segal. http://phck.net/post/against-active-inert.html Wed, 13 May 2026 00:00:00 +0000 http://phck.net/post/against-active-inert.html Naively, one might hope that, for any algebraic copattern, presheaves which send active-inert pushout squares to pullbacks might be a good notion of generalized decomposition space. We push back on this a bit with some pointed examples, including one related to double categories. http://phck.net/post/2-segal-spaces-and-delta-s.html Fri, 23 May 2025 00:00:00 +0000 http://phck.net/post/2-segal-spaces-and-delta-s.html For ΔS presheaves, there is no difference between the Segal and the 2-Segal conditions. http://phck.net/post/cfgmc_question.html Fri, 02 Jan 2015 00:00:00 +0000 http://phck.net/post/cfgmc_question.html We present a statement about whether certain full subcategories of (cofibrantly generated) model categories inherit a model structure. The question is about situations where it applies, and there are a couple updates based on comments from Patchkoria and Riehl; later we learned this theorem is due to Intermont–Johnson.