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Exercise 92 - Chapter 3

edited June 2018 in Exercises

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

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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, \( \mathcal{J} = [ \overset{v}{\bullet} \; \overset{w}{\bullet} ] \).

Comments

  • 1.
    edited May 2018

    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} \\). \[ V_{\mathcal{J}} = \lbrace v, w \rbrace \\\\ A_{\mathcal{J}} = \lbrace 1_v, 1_w \rbrace \] \[ X \times Y = \lim_{\mathcal{J}} D := \lbrace (D(v), D(w)), (D(1_v), D(1_w)) \rbrace \] \[ p_v : X \times Y \mapsto D(v) \\\\ p_w : X \times Y \mapsto D(w); \]
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