⏚ Antoine Chambert-Loir<p>A small <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> thread about <a href="https://mathstodon.xyz/tags/barycenter" class="mention hashtag" rel="tag">#<span>barycenter</span></a>, <a href="https://mathstodon.xyz/tags/torsor" class="mention hashtag" rel="tag">#<span>torsor</span></a>, and <a href="https://mathstodon.xyz/tags/extension" class="mention hashtag" rel="tag">#<span>extension</span></a>.</p><p>A torsor for a group object (in a topos), or a principal homogeneous space under a group, or a principal bundle wrt a group (in differential geometry) are more or less the same thing: you are given a group G, an object P with a — right — free and transitive action of G on P ((p,g)↦p⋅g) which tells you that P and G are very close: you have some p ∈ P, or you have such a p “locally”.</p><p>This is slightly complicated, I admit, but…</p>