Talk:Lawvere's fixed-point theorem

Terminal / Inital Object

Near the end of section Statement it says, in brackets:

| where 1 is a terminal object in C

Shouldn't it be an initial object since the morphisms b being discussed have 1 as their domain? 188.23.53.170 (talk) 16:36, 19 March 2025 (UTC)

Good question, but the answer is "no". The fact that 1 is a terminal object does not mean that there can't be morpshisms with it as domain. In fact, morphisms $b:1\to B$ can be thought of as a generalized notion for speaking of the "elemenents" of an object $B$ (of course, $B$ is not necessarily a set and therefore may have no elements per se; but see Set elements as morphisms from terminal objects). This is the intuition for defining a fixed point of a morphism $f:B\to B$ as a morphism $b:1\to B$ such that $f\circ b=b$, as it is analogous to something like "$f(b)=b$ for $b\in B$." JoaoFrancisco1812 (talk) 21:30, 28 May 2025 (UTC)