generalized element

generalized element (plural generalized elements)

1. (category theory) A morphism whose codomain is some specified object.
   • 1995, Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, →ISBN, page 17:

     Thinking of ${\displaystyle x\in _{A}B}$ as a generalized element, for any ${\displaystyle f:B\rightarrow C}$, we may write ${\displaystyle f(x)}$ for the composite ${\displaystyle f\circ x}$. In this notation the first domain–codomain axiom reads as follows: for any ${\displaystyle x\in _{A}B}$ and ${\displaystyle f:B\rightarrow C}$ there is a well-defined ${\displaystyle f(x)\in _{A}C}$; that is, at each stage A, f takes A-elements of B to A-elements of C.

• generic element