An **identity-on-objects functor** $F: A\to B$

- Is between categories with the same set of objects, i.e. $Obj = ob(A) = ob(B)$, and
- Has as its underlying object function $F_{ob}: Obj \to Obj$ the identity function on $Obj$.

This simple definition appears to go against the principle of equivalence, since it mentions equality of objects, and indeed identity of *sets* of objects.

However, the notion of “two categories with an identity-on-objects functor between them”, which is much more commonly used than the notion of “an identity-on-objects functor between two previously given categories”, can be defined in a more invariant way. Explicitly, it consists of one set $Ob$ of objects, two families of arrows $A(x,y)$ and $B(x,y)$ for $x,y\in Ob$, each with composition and identities making them into a category structure, plus a family of functions $A(x,y) \to B(x,y)$ commuting with these category structures.

More abstractly, this can be defined as a category enriched over the arrow category $Set^\to$. See also M-category and F-category.

Last revised on September 22, 2017 at 04:13:32. See the history of this page for a list of all contributions to it.