# Chapter 1 Problems

1.4.A. Suppose that the partially ordered set $\mathscr I$ has an initial object $e$. Show that the limit of any diagram indexed by $\mathscr I$ exists.

If $A_j\xrightarrow{F(m)} A_k$ is a diagram indexed by $\mathscr I$ then $A_e$ is its limit.

_Proof._ Let $W$ be an object with maps $f_i$ to each $A_i$ such that $f_j = F(m)\circ f_k$ whenever $m: j\to k$. Take $h$ to be the map $f_e : W\to A_e$. To show that it is unique consider 'a product' $f_e\times f_e': W \rightrightarrows A_e$. Since $e$ is initial, the map $A_e \to A_e$ must be the identity, so $1\circ f_e = f_e'$.

1.4.B. Show that in the category $Sets$,

$$S =\left\{ (a_i)_{i\in\mathscr I} \in \prod A_i : F(m)(a_j) = a_k \text{ for all } m \in \text{Mor}_{\mathscr I}(j,k) \in \text{Mor}(\mathscr I) \right\},$$

along with the obvious projection maps to each $A_i$, is the limit $\lim\limits_{\leftarrow} A_i$.

_Proof._ Again let $W$ be an object with maps $f_i$ to each $A_i$ such that $f_j = F(m)\circ f_k$ whenever $m:j\to k$. Take $h:W\to S$ to be the map defined by $h(w) = (f_i(w))_{\mathscr I}$. Denote the obvious projection maps $S\to A_i$ by $g_i$. Then $g_j \circ h (w) = g_j ((f_i(w))_{\mathscr I}) = f_j(w)$ so '$h$ commutes'. To show that $h$ is unique let $h'$ be another map $W\to S$. Suppose $h'(w) = (a_i)_{\mathscr I}$. Since $g_j \circ h' = f_j$ it must be the case that $a_j = f_j(w)$.