Fundamental groupoid of a space

Let $X$ be a topological space. Its fundamental groupoid ${\mathrm{\Pi }}_1(X)$ is the following category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

• The objects are the points of $X$.
• A morphism $x\to y$ is an endpoint-fixed path-homotopy class $[\alpha ]$ of paths $\alpha :[0,1]\to X$ with $\alpha (0)=x$ and $\alpha (1)=y$, in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints.
• If $\alpha :x\to y$ and $\beta :y\to z$, composition is $[\beta ]\circ [\alpha ]=[\alpha \ast \beta ]\mathrm{.}$ Thus the path written first is traversed first, while the categorical composite has the usual right-to-left notation.
• The identity at $x$ is the class of the constant path $c_x$, and the inverse of $[\alpha ]$ is the reversed path class $[\bar{\alpha }]$.

The endpoint-fixed concatenation calculations in Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation prove that composition is independent of representatives, associative, and unital, and that reversal gives a two-sided inverse. Hence every morphism is an isomorphism, so this category is a groupoid.

No connectedness, local connectedness, basepoint, or universal cover is part of the definition. If $X=\mathrm{\varnothing }$, both its object and morphism classes are empty. A one-point space still has all of its endpoint-fixed loop classes; contractibility is not inserted into the definition.

With the displayed categorical composition, the multiplication in the categorical automorphism group at $x$ is opposite to the library's published first-loop-first multiplication on the same underlying loop classes. The next proposition records the canonical reversal isomorphism; silently identifying the two products would reverse every later monodromy formula.