The following are taken from elementary categories, elementary toposes by McLarty.
Definition: A group is a structure with a unit, $e,$ a multiplication, $x,y,$ and an inverse operation, $x^{-1},$ satisfying these equations:
unit: $\quad$ $x\cdot e =x$ $\quad$ and $\quad$ $e\cdot x=x$
inverse: $\quad$ $x\cdot x^{-1} =e$ $\quad$ and $\quad$ $x^{-1}\cdot x=e$
associativity: $\quad$ $(x\cdot y) \cdot z=x \cdot (y\cdot z)$
More precisely, define a group in $C$ to be an object $G$ of $C$ together with arrows
$e:1\rightarrow G$ $\quad$ $m:G\times G\rightarrow G$ $\quad$ $\_^{-1}:G\rightarrow G$
such that these diagrams commute:
Notation 1: We will use the notation $A\times B, p_1, p_2$ to name the one we are talking about, or draw it:
$\textbf{Notation 2:}$ We will use $\langle f,g \rangle$ to name the unique arrow induced by $f:T\rightarrow A$ and $g:T\rightarrow B$, so the following diagram commutes by definition:
![[Image]](https://i.stack.imgur.com/Yn6hn.jpg)
$\textbf{Questions:}$
$\textbf{(1)}$ In the definition for Groups above, how does one draw the commutative diagram $x\circ e=x$, the arrow for $e$ is $e:1\rightarrow G$, would $x\circ e=x$ be $x\circ e:1\rightarrow G\rightarrow G,$ where $x$ is the map that goes from $G$ to $G$?
$\textbf{(2)}$ In the first commutative diagram where the notation $\langle e,G \rangle$ appears, what does that notation mean? The first time that the angle bracket notation makes an appearance in McLarty's text is in Notation 2 above, following right after Notation 1. Thank you in advance.
(1) It is the exact same as the first triangle given, except the horizontal arrow would be $\langle G, e \rangle$. (I can infer from this definition that the author is using $G$ as notation for the identity map $1_G : G \to G$.) Chasing an element $x \in G$ around the triangle, "over then down" is $$ (m \circ \langle G, e \rangle)(x) = m(x,e) $$ and the diagonal would be $G(x)=x$, so this commutativity forces $m(x,e)=x$, which is the category way of saying $xe=x$ (right identity).
(2) I guess my response above answers this. $\langle e, G \rangle$ is the function that sends $x$ to the pair $(e,x)$.