# ct.category theory – Why does each functor define an invariant, but is not every invariant always functorial? Examples?

In category theory, each functor defines an invariant, but not all invariants are functional

Why ?

Can you give some examples when

• a functor is an invariant
• an invariant is a functorial
• an invariant is not functorial

I also need another example:

– a functor is not an invariant $$=>$$ it's absurd and false but I need you to show me the underlying contradiction

Idea

For $$A$$ a monoid equipped with an action on an object $$V$$, a invariant of action is a generalized element of $$V$$ which is taken by the action to itself, hence a fixed point for all operations in the monoid.

A action of a category $$C$$ on a tray $$S$$ nothing is more than one functor $$rho: C to$$ Together.