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You will have to forgive me as I am very new to category theory - fifth of the way through Categories for a working mathematician. I'm interested in the following;

Let $F:A \to B$ and $G:A \to C$ be full functors, where $A$,$B$ and $C$ are groupoids and $B$ has a single object. Let $G$ be such that $Gf=Gf'$ if and only if $Ff=Ff'$ where $f$ and $f'$ are morphisms of $A$. What is the relationship of $G$ to $F$?

If $C$ is also a category with a single object then $C$ and $B$ are isomorphic?

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Your first question is difficult to answer. Is there some motivating problem? Your second question is impossible to answer – what is $D$? –  Zhen Lin May 13 '12 at 12:26
    
Sorry, I've edited the question now. Hopefully it's more answerable. Not really any motivating problem, I was just hoping there is some compact way of writing such a $G$ in terms of $F$ and revealing the properties of $G$. –  Alexander May 13 '12 at 12:45
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$D$ doesn't show up in your first question. –  Simon Markett May 13 '12 at 12:48
    
Doh! $D$ is now $B$. –  Alexander May 13 '12 at 15:03
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I still don't think you can say anything much. The hypotheses don't really control what happens in $B$ and $C$ outside the images of $F$ and $G$ (resp.) – I don't think it is even true in general that there is a functor $H : C \to B$ such that $H G \cong F$. It is absolutely not true that $B$ and $C$ have to be isomorphic when they're both groups: you can see this even in elementary group theory. (Let $A$ be the trivial group, let $B$ be a cyclic group of order 2, and let $C$ be a cyclic group of order 3...) –  Zhen Lin May 13 '12 at 18:21

1 Answer 1

up vote 1 down vote accepted

Surprisingly, even under all the hypotheses, the answer to your first question is essentially negative.$\DeclareMathOperator{\ob}{ob}$ $\DeclareMathOperator{\Hom}{Hom}$Let $\mathbb{A}$ be two disjoint copies of the infinite cyclic group $\mathbb{Z}$, let $\mathbb{B}$ be a single copy of $\mathbb{Z}$, and let $\mathbb{C}$ be a connected pair of copies of $\mathbb{Z}$. More precisely:

  • $\ob \mathbb{A} = \{ A_1, A_2 \}$, $\Hom(A_1, A_1) = \mathbb{Z}$, $\Hom(A_2, A_2) = \mathbb{Z}$, $\Hom(A_1, A_2) = \emptyset$, $\Hom(A_2, A_1) = \emptyset$.

  • $\ob \mathbb{B} = \{ B \}$, $\Hom(B, B) = \mathbb{Z}$.

  • $\operatorname{ob} \mathbb{C} = \{ C_1, C_2 \}$, $\Hom(C_1, C_1) = \mathbb{Z}$, $\Hom(C_2, C_2) = \mathbb{Z}$, $\Hom(C_1, C_2) = \mathbb{Z}$, $\Hom(C_2, C_1) = \mathbb{Z}$, composition given by addition.

Now, there is a "bad" functor $F : \mathbb{A} \to \mathbb{B}$ constructed as follows: on $A_1$, it acts as the identity, but on $A_2$, it acts as the outer automorphism $x \mapsto -x$. Let $G : \mathbb{A} \to \mathbb{C}$ be the obvious functor that acts as the identity on morphisms. I claim there is no functor $H : \mathbb{C} \to \mathbb{B}$ such that $H G \cong F$. Indeed, if $H$ were such a functor, then there would have to be an integer $n$ such that $$n + x = -x + n$$ for all $x \in \mathbb{Z}$, which is patent nonsense. (Basically, two points in the same connected component of a groupoid can only be related by an inner automorphism.)


Happily, the answer to your second question is affirmative. Something slightly stronger is true:

Proposition. If $\mathbb{A}$ is category, $\mathbb{B}$ and $\mathbb{C}$ are both categories with one object, $F : \mathbb{A} \to \mathbb{B}$ and $G : \mathbb{A} \to \mathbb{C}$ are full functors, and $F f = F f'$ if and only if $G f = G f'$, then there is a unique functor $H : \mathbb{C} \to \mathbb{B}$ such that $H G = F$ and $H$ is an isomorphism.

The proof is obvious and has more to do with algebra than category theory. As an additional exercise, you might want to try to explain why this can be done without the axiom of choice.

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Thank you. This was very helpful. –  Alexander May 13 '12 at 20:48
    
Could you recommend any good books on groupoids and action groupoids - preferably from a categorical perspective? –  Alexander May 13 '12 at 20:59
    
No, I'm afraid I'm not familiar with that subject. I get the sense that groupoids are studied more by topologists and geometers than category-theorists. –  Zhen Lin May 13 '12 at 23:28
    
you might find Topology and Groupoids by R. Brown interesting from the categorial point of view. In general: go to wikipedia Groupoid page and check the references by Ronald Brown –  magma May 14 '12 at 13:24

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