Groupoid (category theory)

In category theory, a groupoid is a category in which all morphisms are isomorphisms. Groupoids can be considered as a generalization of groups . Namely, the category corresponding to the group $G$ ${\ displaystyle G}$ $G$ , has exactly one object and one arrow for each element $g$ ${\ displaystyle g}$ $g$ of $G$ ${\ displaystyle G}$ $G$ . Arrow composition is defined as the multiplication of the corresponding elements in a group. It can be seen that in this case each arrow is an isomorphism. Thus, the set of arrows of a groupoid can be considered as some set with a partially defined binary operation of multiplication, so that for each element there are left and right inverses, as well as left and right units for multiplication.

Groupoids naturally replace symmetry groups in category theory and arise when classifying classes of isomorphic objects.

Examples

Any category that is a group is a groupoid.
Let be $C$ ${\ displaystyle C}$ Is an arbitrary category, and $D\hookrightarrow C$ ${\ displaystyle D \ hookrightarrow C}$ - a subcategory whose objects coincide with the objects $C$ ${\ displaystyle C}$ , and morphisms are all kinds of isomorphisms in $C$ ${\ displaystyle C}$ . Then $D$ ${\ displaystyle D}$ - groupoid.
Let be $X$ ${\ displaystyle X}$ Is a linearly connected topological space . Then its fundamental groupoid $\Pi _{1}(X)$ ${\ displaystyle \ Pi _ {1} (X)}$ Is a 2-category whose objects are all points from $X$ ${\ displaystyle X}$ , and the arrows from $x\in X$ ${\ displaystyle x \ in X}$ at $y\in X$ ${\ displaystyle y \ in X}$ correspond to all kinds of (geometric) paths from $x$ ${\ displaystyle x}$ at $y$ ${\ displaystyle y}$ :

f\colon [0;1]\to X,~f(0)=x,\;f(1)=y

{\ displaystyle f \ colon [0; 1] \ to X, ~ f (0) = x, \; f (1) = y}

Two functions

f

{\ displaystyle f}

and

g

{\ displaystyle g}

set the same path if exists

s:[0;1]\to [0;1]

{\ displaystyle s: [0; 1] \ to [0; 1]}

, so that

f=g\circ s

{\ displaystyle f = g \ circ s}

or

g=f\circ s

{\ displaystyle g = f \ circ s}

. The composition of the arrows is set by the composition of the paths:

fg(t)={\begin{cases}f(2t),\;0\leqslant t\leqslant 1/2\\g(2t-1),\;1/2\leqslant t\leqslant 1\end{cases}}

{\ displaystyle fg (t) = {\ begin {cases} f (2t), \; 0 \ leqslant t \ leqslant 1/2 \\ g (2t-1), \; 1/2 \ leqslant t \ leqslant 1 \ end {cases}}}

2-morphism from

f

{\ displaystyle f}

at

g

{\ displaystyle g}

Is a homotopy from

f

{\ displaystyle f}

at

g

{\ displaystyle g}

. A fundamental groupoid is a categorization of a fundamental group . Its advantage is that the space does not require the choice of a marked point, so that there are no problems with the non-canonical isomorphism of fundamental groups at different points or with spaces that have several connected components. The fundamental group of loops from a point

x\in X

{\ displaystyle x \ in X}

arises as a group of 2-isomorphic automorphisms of an object

x\in \Pi _{1}(X)

{\ displaystyle x \ in \ Pi _ {1} (X)}

.

Category of vector bundles of rank $n$ ${\ displaystyle n}$ over a contractible space with non-degenerate mappings naturally forms a groupoid. This remark underlies the introduction of the concept of jerba (which is a special case of the stack ), which is a structure on the category of sheaves of a given type. Jerbas are geometric objects classified by cohomology groups $H^{2}(X,{\mathcal {G}})$ ${\ displaystyle H ^ {2} (X, {\ mathcal {G}})}$ where ${\mathcal {G}}$ ${\ displaystyle {\ mathcal {G}}}$ - a bunch of groups on $X$ ${\ displaystyle X}$ . The concept is especially important in the case of non-Abelian groups. ${\mathcal {G}}$ ${\ displaystyle {\ mathcal {G}}}$ .

Groupoid (category theory)

Examples

See also

More articles: