$\mathbf{Pos}$ is the category where the objects are partially ordered sets (posets) and the arrows are monotone functions. That is, as far as I've understood, an arrow $h$ between posets $P$ and $Q$ such that if in $P$ we have $a\leq b$, then in $Q$ we have $h(a)\leq h(b)$.

An initial object is an object $I$ such that, for any other object $A$ in the category, there is exactly one arrow from $I$ to $A$. So in $\mathbf{Pos}$, it can only contain the empty set, because the empty set can only get mapped to the empty set. Furthermore, posets are defined on pairs of elements, so the initial object must be $(\emptyset,\emptyset)$, which is exactly what's written in these my lecture notes.

But what's the terminal object? In order for the arrow to be unique, I would need a one-element set. But a poset must have at least two elements. So why isn't the terminal object $(\bullet,\bullet)$, where the unique arrow into it sends any element in the domain to $\bullet$? The aforementioned notes say it's $({\bullet},({\bullet,\bullet}))$.

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    $\begingroup$ Since every poset is denoted by $(P, \le)$, where $P$ is the underlying set and $\le \subseteq P \times P$ is the order relation, the notation $(\bullet, (\bullet,\bullet))$ means $( \{ \bullet \} , \{ (\bullet ,\bullet)\} )$. $\endgroup$ – Crostul Nov 28 '15 at 17:41
  • $\begingroup$ Then shouldn't the initial object be denoted as $({\emptyset},({\emptyset,\emptyset}))$? $\endgroup$ – man_in_green_shirt Nov 28 '15 at 17:46
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    $\begingroup$ Nope. You have that the only relation over the emptyset is the empty relation $\emptyset \times \emptyset = \emptyset$. $\endgroup$ – Crostul Nov 28 '15 at 17:53

A set with one element is indeed a poset.

Any one-element set $A = \{a\}$, with the partial order $R = \{(a,a)\}$ (i.e. $a \leq a$) is a partially ordered set.
Then for any poset $P$ there exists a unique map \begin{align} !:~&P\longrightarrow \{a\} \\ &p \longmapsto ~~a \end{align} which is indeed monotone. Thus a one-element set is a terminal object in Pos.

  • $\begingroup$ So why the notation $({\bullet},({\bullet,\bullet}))$ for a one-element set? $\endgroup$ – man_in_green_shirt Nov 28 '15 at 17:30
  • $\begingroup$ I don't understand that notation, and the notes you hyperlinked necessitate username and password to access. $\endgroup$ – Andrea Nov 28 '15 at 17:34
  • $\begingroup$ Oh ok, I managed to access them by turning on 'private browsing' so I thought they'd be publicly available. I'll remove the link then $\endgroup$ – man_in_green_shirt Nov 28 '15 at 17:37

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