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Recall the two definitions of equivalence of categories:

Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} \to \mathbb{C}$ are functors and $\alpha : G F \Rightarrow \textrm{id}_\mathbb{C}$ and $\beta : F G \Rightarrow \textrm{id}_\mathbb{D}$ are natural isomorphisms.

Definition 2. A weak equivalence of categories is a fully faithful functor $F : \mathbb{C} \to \mathbb{D}$ that is also essentially surjective on objects, i.e. for each object $d$ in $\mathbb{D}$ there exists an object $c$ in $\mathbb{C}$ and an isomorphism $d \to F c$ in $\mathbb{D}$.

Assuming the axiom of choice, one can show that a weak equivalence $F : \mathbb{C} \to \mathbb{D}$ extends to an equivalence of categories $(F, G, \alpha, \beta)$. On the other hand, it is clear that this principle is equivalent to the axiom of choice: indeed, if we have some family of non-empty sets $( X_i : i \in I )$, we may form the category $\mathbb{C}$ whose object set is $\coprod_{i \in I} X_i$, such that there exists a unique arrow between two objects if and only if they come from the same set $X_i$; taking $\mathbb{D}$ to be the discrete category on the indexing set $I$, we have an evident projection functor $F : \mathbb{C} \to \mathbb{D}$, and by construction it is a weak equivalence; on the other hand, any functor $G : \mathbb{D} \to \mathbb{C}$ fitting into an equivalence $(F, G, \alpha, \beta)$ must yield a family $( x_i : i \in I )$ where $x_i = G i \in X_i$. Notice also that the categories $\mathbb{C}$ and $\mathbb{D}$ constructed here are groupoids (indeed, for that matter, setoids).

Question. Suppose $\mathbb{C}$ is a connected groupoid, i.e. between any two objects there exists at least one (iso)morphism between them. Then, if $\mathbb{D}$ is the full subcategory of $\mathbb{C}$ spanned by any single object, then the inclusion $G : \mathbb{D} \hookrightarrow \mathbb{C}$ is automatically a weak equivalence. Do I need the axiom of choice to extend $G$ to an equivalence of categories?

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It seems like the extension you want is equivalent to the proposition that if $(X_i:i\in I)$ is a family of sets with the same cardinality, then there is a section $f:I\to\coprod_{i\in I} X_i$. I have no idea how this relates to the axiom of choice though. By the way, you should probably mention that your categories are assumed to be small (i.e. $\operatorname{ob}(\mathbb{C})$ "is" a set). –  Colin McQuillan Jun 13 '12 at 21:36
@ColinMcQuillan: I think it's a little bit more sophisticated than that. It is well-known that there are models of set theory in which there exists an uncountable set that is a countable union of pairs. I feel that a (connected) groupoid has more structure than this, but this is just an intuition... –  Zhen Lin Jun 13 '12 at 21:45
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1 Answer 1

The following are equivalent over ZF.

AC6. The Cartesian product of a set of non-empty sets is non-empty.

AC7. The Cartesian product of a set of non-empty sets of the same cardinality is non-empty.

GWE. Every weak equivalence from a group to a small connected groupoid is an equivalence of categories.

The numbering of AC6 and AC7 is taken from Rubin's "Equivalents of the Axiom of Choice", where they are shown to be equivalent. That AC6 implies GWE is a special case of the proposition that under the axiom of choice, every weak equivalence of categories is an equivalence.

It remains to show that GWE implies AC7. Let $X_i$ be a family of non-empty sets of the same cardinality $X$. Let $\mathbb{D}$ be the permutation group of $X$ considered as a category with one object. Let $\mathbb{C}$ the the groupoid whose objects are indexed by $I$ and whose morphisms $i\to j$ correspond to bijections $X_i\to X_j$. Pick $i\in I$ and let $G$ send $X$ to $X_i$ according to some bijection of $X$ to $X_i$; this is a weak equivalence $\mathbb{D}\to\mathbb{C}$. By GWE there is an equivalence $(F,G,\alpha,\beta)$.

I'll argue that $F$ must be given by bijections $f_j:X_j\to X$. This part is a bit sketchy. Let $\pi$ be a transposition of $X_j$, that is, a permutation that exchanges two elements. There is a bijection $g:X_i\to X_j$, so $g^{-1}\pi g$ is a transposition, so $F(g^{-1}\pi g)$ is a transposition, so $F(\pi)=F(g)F(g^{-1}\pi g)F(g):X\to X$ is a transposition. An isomorphism $S_{X_i}\to S_X$ that sends transpositions to transpositions must be induced by a bijection of the underlying sets (I think).

Since $X$ is non-empty there exists $x\in X$. For each $j\in I$ let $y_j$ be the the unique element $y_j\in X_j$ such that $f_j(y_j)=x$. Then $y\in\prod_j X_j$. So $\prod_j X_j$ is non-empty as required.

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Why is $F(g^{-1} \pi g)$ a transposition? Certainly, it is an involution, but not all involutions are transpositions. (For example, in the degenerate case $I = 1$, $X = 6$, we could take $F$ to be the exceptional automorphism of $S_6$ sending transpositions to triple transpositions.) –  Zhen Lin Jun 13 '12 at 23:01
Zhen Lin, $GF(g^{-1}\pi g)=\alpha_{X_i}^{-1} g^{-1}\pi g \alpha_{X_i}$ is conjugate to a transposition, and $G$ is just an inclusion. –  Colin McQuillan Jun 14 '12 at 6:51
Also, your comment gives a much quicker proof along these lines: we can assume $X$ does not have cardinality $6$ (for example, replace all the $X_i$'s by their squares $X_i\times X_i$ - it suffices to choose an element of $X_i\times X_i$). Since $S_X$ has no outer automorphisms, $F$ is given by bijections $X_j\to X$. –  Colin McQuillan Jun 14 '12 at 6:59
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