According to the nlab, the categorical trace of a 1-endomorphism $F:C\to C$ in a 2-category is the set hom$(1\_C, F)$ of global elements of $F$. If $F$ is a functor in the 2-category Cat, the categorical trace is a set of natural transformations that assign to each object of $C$ a coalgebra of $F$ such that the obvious square commutes.


Any functor can be considered a special kind of profunctor; given an endofunctor, we can compute the coend of the corresponding profunctor.


Both of these concepts are generalizations of the trace, which for a function counts the number of fixpoints. In what sense do these "count" the fixpoints of a functor? I don't see how the categorical trace of a functor relates to fixpoints at all.


Also, does the notion of what constitutes a fixpoint change? The coend, in particular, seems like it might count an object $c$ as a fixpoint of $F$ if it's in the same *endomorphism* class rather than the same *isomorphism* class as $Fc$.



In what sense do the categorical trace and coend count fixed points?
Simon Willerton explains it all very well here: <http://www.simonwillerton.staff.shef.ac.uk/ftp/TwoTracesBeamerTalk.pdf>


