Higher Algebra

Tools from Higher Algebra

Exercises for Higher Algebra

(1) Let R be a division ring. Prove that every module over R is free. You will need to use Zorn’s lemma: Recall that a partially order set (=poset) S is a set with a relation x ≤ y defined between some pairs of elements x, y ∈ S, such that: (i) x ≤ x ; (ii) x ≤ y and y ≤ x implies x = y ; (iii) x ≤ y , y ≤ z ⇒ x ≤ z . A chain in S is a subset T ⊂ S such that for all t, t ′ in T , either t ≤ t ′...

Higher Algebra – MATH 570 Course

Higher Dimensional Algebra Vii: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to F...

Higher Algebra I (math 570) Course Notes