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# Exercise 92 - Chapter 3

edited June 2018

Show that the limit formula in Theorem 3.90 works for products. See Example 3.89.

Theorem 3.90

Let $$\mathcal{J}$$ be a category presented by the finite graph $$(V, A, s, t)$$ together with some equations, and let $$D : \mathcal{J} \rightarrow \textbf{Set}$$ be a set-valued functor. Write $$V = \lbrace v_1 , \cdots , v_n \rbrace$$. The set $$\lim_{\mathcal{J}} D := \lbrace (d_1 , \cdots , d_n ) | d_i \in D(v_i ) \text{ and for all } a : v_i \rightarrow v_j \in A \text{, we have } D(a)(d_i ) = d_j \rbrace$$ together with the projection maps $$p_i : (\lim_{\mathcal{J}} D) \rightarrow D(v_i )$$ given by $$p_i (d_1 , \cdots , d_n ) := d_i$$, is a limit of $$D$$.

Example 3.89

Products are limits where the indexing category consists of two objects $$v, w$$ and no arrows, $$\mathcal{J} = [ \overset{v}{\bullet} \; \overset{w}{\bullet} ]$$.

There are only id morphisms in $$\mathcal{J}$$ which is isomorphic to $$\textbf{2}$$.
$$V_{\mathcal{J}} = \lbrace v, w \rbrace \\ A_{\mathcal{J}} = \lbrace 1_v, 1_w \rbrace$$ $$X \times Y = \lim_{\mathcal{J}} D := \lbrace (D(v), D(w)), (D(1_v), D(1_w)) \rbrace$$ $$p_v : X \times Y \mapsto D(v) \\ p_w : X \times Y \mapsto D(w);$$
Comment Source:There are only id morphisms in \$$\mathcal{J} \$$ which is isomorphic to \$$\textbf{2} \$$. $V_{\mathcal{J}} = \lbrace v, w \rbrace \\\\ A_{\mathcal{J}} = \lbrace 1_v, 1_w \rbrace$ $X \times Y = \lim_{\mathcal{J}} D := \lbrace (D(v), D(w)), (D(1_v), D(1_w)) \rbrace$ $p_v : X \times Y \mapsto D(v) \\\\ p_w : X \times Y \mapsto D(w);$