### 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(g*_{1}g_{2}) = φ(g_{1})φ(g_{2}) for any pair of elements g_{l} and g_{2} 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, ...