⎯ΘωΘ⟶<p><a href="https://mathstodon.xyz/tags/First" class="mention hashtag" rel="tag">#<span>First</span></a> we <a href="https://mathstodon.xyz/tags/reformulate" class="mention hashtag" rel="tag">#<span>reformulate</span></a> <a href="https://mathstodon.xyz/tags/ordinary" class="mention hashtag" rel="tag">#<span>ordinary</span></a> <a href="https://mathstodon.xyz/tags/colimits" class="mention hashtag" rel="tag">#<span>colimits</span></a> in the <a href="https://mathstodon.xyz/tags/language" class="mention hashtag" rel="tag">#<span>language</span></a> of <a href="https://mathstodon.xyz/tags/tensor" class="mention hashtag" rel="tag">#<span>tensor</span></a> <a href="https://mathstodon.xyz/tags/products" class="mention hashtag" rel="tag">#<span>products</span></a>, in a <a href="https://mathstodon.xyz/tags/way" class="mention hashtag" rel="tag">#<span>way</span></a> that <a href="https://mathstodon.xyz/tags/suggests" class="mention hashtag" rel="tag">#<span>suggests</span></a> <a href="https://mathstodon.xyz/tags/more" class="mention hashtag" rel="tag">#<span>more</span></a> <a href="https://mathstodon.xyz/tags/general" class="mention hashtag" rel="tag">#<span>general</span></a> <a href="https://mathstodon.xyz/tags/weighted" class="mention hashtag" rel="tag">#<span>weighted</span></a> colimits.</p><p><a href="https://mathstodon.xyz/tags/Assume" class="mention hashtag" rel="tag">#<span>Assume</span></a> for the <a href="https://mathstodon.xyz/tags/moment" class="mention hashtag" rel="tag">#<span>moment</span></a> that the <a href="https://mathstodon.xyz/tags/receiving" class="mention hashtag" rel="tag">#<span>receiving</span></a> category C has all <a href="https://mathstodon.xyz/tags/coproducts" class="mention hashtag" rel="tag">#<span>coproducts</span></a> and <a href="https://mathstodon.xyz/tags/coequalizers" class="mention hashtag" rel="tag">#<span>coequalizers</span></a>. As is <a href="https://mathstodon.xyz/tags/well" class="mention hashtag" rel="tag">#<span>well</span></a> <a href="https://mathstodon.xyz/tags/known" class="mention hashtag" rel="tag">#<span>known</span></a>, it <a href="https://mathstodon.xyz/tags/follows" class="mention hashtag" rel="tag">#<span>follows</span></a> that C <a href="https://mathstodon.xyz/tags/has" class="mention hashtag" rel="tag">#<span>has</span></a> <a href="https://mathstodon.xyz/tags/all" class="mention hashtag" rel="tag">#<span>all</span></a> colimits; the <a href="https://mathstodon.xyz/tags/proof" class="mention hashtag" rel="tag">#<span>proof</span></a> is we can <a href="https://mathstodon.xyz/tags/write" class="mention hashtag" rel="tag">#<span>write</span></a> <a href="https://mathstodon.xyz/tags/down" class="mention hashtag" rel="tag">#<span>down</span></a> a <a href="https://mathstodon.xyz/tags/formula" class="mention hashtag" rel="tag">#<span>formula</span></a> for the colimit of F:J→C: as a <a href="https://mathstodon.xyz/tags/coequalizer" class="mention hashtag" rel="tag">#<span>coequalizer</span></a> of a <a href="https://mathstodon.xyz/tags/pair" class="mention hashtag" rel="tag">#<span>pair</span></a>:</p><p>∑j,k∈Ob(J)hom(j,k)×F(j) [[→ →]] ∑k∈Ob(J)F(k) [[→]] colimJF</p>