notation. hom-set, hom-class [zhang2021type, 1.2] [tt-0003]

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\(\operatorname {Hom}\) in \(\operatorname {Hom}_{{\cal C}}(X, Y)\) is short for homomorphism, since an arrow in category theory is a morphism (i.e. an arrow), a generalization of homomorphism between algebraic structures.

This notation could be unnecessarily verbose, so when there is no confusion, we follow [leinster2016basic] and [zhang2021type] to simply write \(X,Y \in \operatorname {Ob}({\cal C})\) as \[X,Y \in {\cal C}\] and \(f \in \operatorname {Hom}_{{\cal C}}(X, Y)\) as \[f \in {\cal C}(X, Y)\]

In some other scenarios, when the category in question is clear (and it might be to too verbose to write out, e.g. a functor category), we omit the subscript of the category and write just \[\operatorname {Hom}(X, Y)\]

In general, collection \(\operatorname {Ob}({\cal C})\) and \(\operatorname {Arr}({\cal C})\) are not neccessarily sets, but classes. In that case, \(\operatorname {Hom}_{{\cal C}}(X, Y)\) is called a hom-class.

Later, we will also learn that \(\operatorname {Ob}\) and \(\operatorname {Arr}\) are representable functors.