# nLab identity-on-objects functor

Identity-on-objects functors

# Identity-on-objects functors

## Definition

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 $Obj$.

## Equivalence-invariance

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.