*Thin groupoid* is an important but a heavily overlooked concept.

When I did Google search for “thin groupid” (with quotes), I found just $latex {7}&fg=000000$ (seven) pages (and some of these pages were created by myself). It is very weird that such an important concept was overlooked by the mathematical community.

By definition of *thin category*, thin groupoid is a groupoid for every pair $latex {A}&fg=000000$, $latex {B}&fg=000000$ of objects of which there are at most one morphism $latex {A \rightarrow B}&fg=000000$.

I recall that a groupoid is a category all morphisms of which are isomorphisms. Moreover, in all examples below objects are sets and (iso)morphisms are isomorphisms of $latex {\mathbf{\mathrm{Set}}}&fg=000000$ that is bijections.

So, roughly, “thin groupoid” means: Between every two sets in consideration there is considered at most one bijection. In other words, all objects in consideration are equivalent up to an isomorphism.

**1. Equivalent definitions of thin groupoid **

**Theorem 1** * The following definitions of thin groupoid are equivalent:*

- a groupoid with at most one morphism $latex {A \rightarrow B}&fg=000000$ for given objects $latex {A}&fg=000000$, $latex {B}&fg=000000$;
*a groupoid with each cycle of morphisms being identity.*

**Proof:** The only thing we need to prove (as all the rest is obvious) is that for thin groupoid each cycle of morphisms is identity. But really, composition of a cycle of morphisms is an endomorphism, but because our category is thin, there are be just one such morphism, the identity morphism. $latex \Box&fg=000000$

“Each cycle of morphisms is identity” intuitively means: Every object is equivalent to itself in exactly one way.

**2. Examples **

** 2.1. Filters, ideals, etc. **

For a lattice $latex {\mathfrak{Z}}&fg=000000$ I denote meets and joins correspondingly as $latex ({\sqcap})&fg=000000$ and $latex ({\sqcup})&fg=000000$.

Filters and ideals are well known concepts:

*Filters* are subsets $latex {F}&fg=000000$ of $latex {\mathfrak{A}}&fg=000000$ such that:

- $latex {F}&fg=000000$ does not contain the least element of $latex {\mathfrak{A}}&fg=000000$ (if it exists).
- $latex {A \sqcap B \in F \Leftrightarrow A \in F \wedge B \in F}&fg=000000$ (for every $latex {A, B \in \mathfrak{Z}}&fg=000000$).

*Ideals* are subsets $latex {F}&fg=000000$ of $latex {\mathfrak{A}}&fg=000000$ such that:

- $latex {F}&fg=000000$ does not contain the greatest element of $latex {\mathfrak{A}}&fg=000000$ (if it exists).
- $latex {A \sqcup B \in F \Leftrightarrow A \in F \wedge B \in F}&fg=000000$ (for every $latex {A, B \in \mathfrak{Z}}&fg=000000$).

I also introduce *free stars* and *mixers*:

*Free stars* are subsets $latex {F}&fg=000000$ of $latex {\mathfrak{A}}&fg=000000$ such that:

- $latex {F}&fg=000000$ does not contain the least element of $latex {\mathfrak{A}}&fg=000000$ (if it exists).
- $latex {A \sqcup B \in F \Leftrightarrow A \in F \vee B \in F}&fg=000000$ (for every $latex {A, B \in \mathfrak{Z}}&fg=000000$).

*Mixers* are subsets $latex {F}&fg=000000$ of $latex {\mathfrak{A}}&fg=000000$ such that:

- $latex {F}&fg=000000$ does not contain the greatest element of $latex {\mathfrak{A}}&fg=000000$ (if it exists).
- $latex {A \sqcap B \in F \Leftrightarrow A \in F \vee B \in F}&fg=000000$ (for every $latex {A, B \in \mathfrak{Z}}&fg=000000$).

I will denote $latex {\mathrm{dual}\, A}&fg=000000$ where $latex {A \in \mathfrak{Z}}&fg=000000$ the corresponding element of the dual poset $latex {\mathfrak{Z}^{\ast}}&fg=000000$. Also I denote

$latex \displaystyle \langle \mathrm{dual} \rangle X \overset{\mathrm{def}}{=} \left\{ \mathrm{dual}\, x \mid x \in X \right\} . &fg=000000$

It is easy to show that filters, ideals, free stars, and mixers are related by the bijections presented in the following diagram:

(where $latex {\neg}&fg=000000$ denotes set-theoretic complement).

This diagram is a $latex {4}&fg=000000$-elements thin groupoid (which is a subcategory of $latex {\mathbf{\mathrm{Set}}}&fg=000000$). These bijections are order isomorphisms if we define order in the right way.

In the case if $latex {\mathfrak{Z}}&fg=000000$ is a boolean lattice, there is also an alternative diagram (also a $latex {4}&fg=000000$-elements thin groupoid (which is a subcategory of $latex {\mathbf{\mathrm{Set}}}&fg=000000$)):

(here $latex {\langle \neg \rangle X \overset{\mathrm{def}}{=} \left\{ \bar{x} \mid x \in X \right\}}&fg=000000$).

See http://www.math.portonvictor.org/binaries/dual-filters.pdf for more information.

** 2.2. Funcoids **

Funcoids, funcoidal reloids, and filters on lattices $latex {\Gamma}&fg=000000$ (don’t worry if you don’t know meanings of these terms, see my Web site for a book on this topic) are isomorphic as presented by the following diagram which is also a thin groupoid. The isomorphisms preserve order and composition.

See my book and http://www.math.portonvictor.org/binaries/funcoids-are-filters.pdf for more information.