Why "thin groupoids" are not ubiquitous?

Question

Google search for "thin groupoid" finds surprisingly few (namely 7) pages. But "thin groupoid" is a term to denote an important notation of a groupoid with every loop being the identity. I met it twice in my work. Why is it missing?

For example there are isomorphisms between filters and ideals. The groupoid (having just two objects) expressing this isomorphisms is thin, that is going from a filter to an ideal and then back to a filter is an identity.

Maybe an other term is used by mathematicians?

I draw a diagram with edges being isomorphisms between classes of objects and loops of isomorphisms are identities. Is there any term for this?

My question: What is a widespread notation related to this if any, or why there may be no such notation?

Two examples, from the theory of filters and theory of funcoids:

From this online article (an expanded above mentioned example about filters and ideals):

From this online article:

Note that all arrows in the above diagrams are isomorphisms (and moreover they are bijections, as they are isomorphisms of the category $\mathbf{Set}$).

See also this my question.

Answer 1

You seem to be describing a setoid.

Also, every such category is equivalent to a set (assuming the category is small); typically we only care about categories up to equivalence, so in such situations this notion doesn't give anything new.

However, as an aside, I think setoid is a better notion than set for many purposes, even outside the context of category theory. (e.g. when actually doing arithmetic, the ring of integers modulo some integer $m$ is usually treated in a way that resembles a setoid more than any other formalism)

I would use the term "setoid" to refer to any category equivalent to a setoid; I'm not sure if that matches up with common practice.

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In your examples, however, you're not really asking about a kind of groupoid. You are specifically interested in things that aren't part of the groupoid structure, namely the actual identity of the morphisms.

The intent of caring about the actual identity of the morphisms seems to be for ZFC-style reasoning: you are using the identity of the morphisms to implicitly define a functor from your groupoid to your category of interest.

For example, one of your examples is essentially just a labelled diagram: the usual way to represent labelled diagrams in category theory is a functor $J \to \mathcal{C}$, where $J$ is an abstract category with the right shape.

As such, I think the natural way to express the structure you're interested in is via a functor whose domain is a "thin groupoid" and whose codomain is your category of interest.

As such, existing terminology for the structure you're interested in is quite unlikely to be form "groupoid satisfying properties".

Answer 2

As was explained by Hurkyl, thin groupoids are known as setoids. They appear in algebraic geometry. Whereas algebraic stacks are categories fibered in groupoids, schemes are categories fibered in setoids. For example, the fiber of $\mathbb{P}^n$ over $S$ is the category of $(\mathcal{L},s_0,\dotsc,s_n)$, where $\mathcal{L}$ is a line bundle on $S$ and $s_0,\dotsc,s_n$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$. This is a setoid. Usually one replaces this (any?) setoid by its set of isomorphism classes.

Many other examples of setoids appeared in my research. The most basic one is the following: If $R,S$ are commutative rings, then the category of cocontinous tensor functors $\mathsf{Mod}(R) \to \mathsf{Mod}(S)$ is equivalent to the discrete category of ring homomorphisms $\hom(R,S)$. So in particular that category is a setoid. (The reason is basically that $R$ generates $\mathsf{Mod}(R)$ and that a cocontinous tensor functor is fixed on the unit object.)