# Tag Info

## Hot answers tagged functors

Accepted

### Example of a functor that doesn't reflect isomorphism

A very generic counterexample: Take for $\mathcal{D}$ the category with one object $O$ and one morphism $f$. There is a functor $F$ from any category to $\mathcal{D}$ that sends every object to $O$ ...
• 1,903
Accepted

### Can the map sending a presentation to its group be considered as a functor?

Yes, yes, yes, to all three questions. And it can be done very generally and very nicely for universal algebra $\$ [or even for first order structures]. Let's fix an algebraic signature consisting of ...
• 90.8k
Accepted

### What is the difference betwen equivalence and isomorphism of functors in categories.

Well, the actual difference between the two statements is that for an equivalence of categories, we only require that that the composites $F \circ G$ and $G \circ F$ are naturally isomorphic to the ...
• 5,458

• 18.5k
Accepted

### Forgetful functor from complete metric space to metric space

This forgetful functor doesn't really "forget" any structure the way most forgetful functors do: it just forgets the fact that your metric spaces happen to be complete. This doesn't change the ...
• 331k
There is indeed a category $E$ with no objects (and therefore no morphisms). However, if a category $A$ has at least one object then there does not exist any functor $A\to E$. Let's think about sets ...