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Higher algebra is the study of algebraic structures that arise in the setting of higher category theory. Higher algebra generalizes ordinary algebra, or algebra in the setting of ordinary category theory. Ordinary categories have sets of morphisms between objects, and elements of a set are either equal or not. Higher categories, on the other hand, have homotopy types of morphisms between objects, typically called mapping spaces. Sets are examples of homotopy types, namely the discrete ones, but in general it doesn’t quite make sense to ask whether or not two “elements” of a homotopy type are “equal”; rather, they are equivalent if they are represented by points that can be connected by a path in some suitable model for the homotopy type. But then any two such paths might form a nontrivial loop, leading to higher automorphisms, and so on. The notion of equality only makes sense after passing to discrete invariants such as homotopy groups.
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