Example 2

Let Σ be a Species of structures in a theory $\mathscr{T}$ stronger than set theory and let $\sigma (x, y, s, u)$ be a term of $\mathscr{T}$ satisfying conditions (MO$_I$), (MO$_{II}$) and (MO$_{III}$) of E, IV, p. 11. Let S be a set; suppose that every element of S is a pair $(x, s)$, where $x$ is a set and $s$ a structure of species Σ on $x$. Let F be the set of mappings $f$ such that there exist $(x, s)$ and $(y, t)$ in S such that $f$ is a $\sigma$-morphism of $x$, endowed with the structure $s$, into $y$, endowed with the structure $t$; one then sets $o(f) = (x, s)$ and $t(f) = (y, t)$. Endowed with the law of composition given by $m(f, g) =g\circ f$, the quiver $(S,F, o, t)$ is a category. In this context, the elements of S are rather called objects.

Let C be a category.

For every vertex $a$ of C, the set $C_{a,a}$, endowed with the law of composition $(f, g)\mapsto f g$, is a monoid with identity element $e_a$.

A map $f$ of C is said to be invertible if there exists a map $g$ of C such that $o(g) =t(f),t(g) =o(f)$ and such that $f g$ and $gf$ are identity elements at $o(f)$ and $t(f)$ respectively. Such a map $g$ is unique (if $f g=e_{o(f)}$ and $g'f=e_{t(f)}$, one has $g=e_{t(f)}g= (g'f)g=$ $g'(f g) =g'e_{o(f)}=g'$) and is called the inverse of $f$; the inverse of an invertible map $f$ is denoted by $f^{-1}$.