Adjunctions between Groupoids and Hilbert spaces

Question

I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the category of groupoids of this type, ie, generated by such an adjunction? For instance, are there any interesting adjunctions between the category of finite groupoids and FDHilb?

This question is a bit vague, as has been pointed out. We can restrict the problem to exactly the group of finite groupoids, and the category of finite dimensional Hilbert spaces with all linear maps.

Answer 1

One suggested variant was to ask whether there are any interesting adjunctions between the category $FinGpd$ of finite groupoids and the category $FinHilb$ of finite-dimensional Hilbert spaces, which is still undefined. The answer is still no.

First let me suppose that the morphisms of $FinHilb$ are all linear maps. Then the automorphism group of $\mathbb C \in FinHilb$ is $\mathbb C^\ast$, a divisible abelian group. As such, it admits no nontrivial homomorphisms to any finite group. So if $F: FinHilb \to FinGpd$ is a functor, then the induced map $Aut(\mathbb C) \to Aut(F\mathbb C)$ is trivial; in particular, $F(1) = F(2)$ where $1,2: \mathbb C \to \mathbb C$ are the scalar maps. Moreover, if $F$ is a right adjoint, then it preserves abelian group objects, so we may subtract $1$ from $2$ to see that $F(0) = F(1)$. From this it follows that $F$ is constant at the terminal object. A similar argument using cogroup objects shows there are no interesting left adjoints $FinHilb \to FinGpd$.

On the other hand, if the morphisms of $FinHilb$ are chosen differently, then it's unlikely that $FinHilb$ has finite products / finite coproducts, so there can't be a nontrivial right / left adjoint functor $FinGpd \to FinHilb$.

The only exception I can think of is if one takes the morphisms of $FinHilb$ to be contractive linear maps, i.e. those maps $f$ such that $\|f(x)\| \leq \|x\|$. With this definition, $FinHilb$ still doesn't have finite products, so there can't be a right adjoint functor $FinGpd \to FinHilb$, but it has finite coproducts. But any right adjoint functor $FinHilb \to FinGpd$ will still identify all nonzero scalars for the same reasons as above, which I think already makes any such functor uninteresting.

Answer 2

I don't know what morphisms you intend for the category of finite-dimensional Hilbert spaces, but it doesn't actually matter. The answer is no, there are no interesting adjunctions between the category of groupoids and the category of finite-dimensional Hilbert spaces.

More generally,

If $C$ is a complete and cocomplete category and $D$ is a small category then there are typically no interesting adjunctions between $C$ and $D$.

More precisely,

Observation 1: Let $C$ be a category with all small coproducts and $D$ a small category. Then for any $c \in C, d \in D$ there is at most one morphism $L(c) \to d$.

Proof: For any set $S$, the coproduct $\amalg_{s \in S} c$ exists in $C$. So the coproduct $\amalg_{s \in S} L(c)$ exists in $D$. The result follows from the next observation.

Observation 2: Let $D$ be a small category and $d \in D$. Suppose that the coproduct $\amalg_{s \in S} d$ exists for any set $S$. Then there is at most one morphism $d \to d'$ for any $d' \in D$.

Proof: This is the same argument as the proof that no small category is cocomplete unless it is a poset: if there are two distinct morphisms $d \to d'$, then the cardinality of $Hom(\amalg_{s \in S} d, d')$ is at least $2^{|S|}$, which eventually exceeds the cardinality of $D$.

In your case, the only object of $FinHilb$ which has at most one morphism to any other object is $0$. So the only left adjoint functor $Gpd \to FinHilb$ is the constant functor at $0$.