Post #1816214
2026-02-19 09:32 UTC
@dwarn This sounds very much like stuff Walter Taylor worked on in 70's [https://doi.org/10.4153/CJM-1977-054-9].
In the paper, he showed (among many other things) that topological semilattices have trivial homotopy groups, or as he phrased it:
THEOREM 6.2. Semilattices obey the law x = 1 in homotopy.
Replies (1)
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@jakub_et_al@mathstodon.xyz 2026-02-19 09:36
@dwarn There many more varieties of topological algebras that have that property. Universal algebra later established a condition SD(∧), i.e., varieties with *meet semidistributive congruence lattices*, that precisely describes those varieties. Essentially, Taylor techniques show that any equation, that is satified by the algebra, has to be satisfied by its homotopy groups. There are no non-trivial models for an SD(∧) variety in groups.