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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.8 Categories

Eugenia Cheng

When we study GROUPS [I.3 §2.1] or VECTOR SPACES [I.3 §2.3], we pay particular attention to certain classes of maps between them: the important maps between groups are the group HOMOMORPHISMS [I.3 §4.1], and the important maps between vector spaces are the LINEAR MAPS [I.3 §4.2]. What makes these maps important is that they are the functions that “preserve structure”: for example, if φ is a homomorphism from a group G to a group H, then it “preserves multiplication,” in the sense that φ(g1g2) = φ(g1)φ(g2) for any pair of elements gl and g2 of G. Similarly, linear maps preserve addition and scalar multiplication.

The notion of a structure-preserving map applies far more generally than just to these two examples, ...

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