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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeNov 21st 2013
   • (edited Nov 21st 2013)
   • PermaLink
   Author: porton
   Format: MarkdownItex<a href="http://math.stackexchange.com/questions/569458/explicit-formula-for-exponential-objects-in-category-of-digraphs/569824">See this page</a> for the formulation of my problem. Please help to find explicit formulas for exponential object, exponential transpose, and evaluation. I am a category theory novice and have a trouble with this problem. I can't make these formulas to coincide: $\varepsilon \circ ( \sim f \times 1_{\Alpha}) = f$; $\varepsilon \circ \sim ( g \times 1_A) = g$. Solving this problem is a step toward proving that <b>Fcd</b> and <b>Rld</b> are cartesian closed categories. It is very important. Please help.

   See this page for the formulation of my problem.

   Please help to find explicit formulas for exponential object, exponential transpose, and evaluation.

   I am a category theory novice and have a trouble with this problem. I can’t make these formulas to coincide:

   $\varepsilon \circ ( \sim f \times 1_{\Alpha}) = f$;

   $\varepsilon \circ \sim ( g \times 1_A) = g$.

   Solving this problem is a step toward proving that Fcd and Rld are cartesian closed categories. It is very important. Please help.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 21st 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexYour second equation doesn't make sense. Let's start again. For $f: Z \times A \to B$, let $\tilde{f}: Z \to Mor(A, B)$ be the exponential transpose (or currying) of $f$, and let $\epsilon_{A, B}: Mor(A, B) \times A \to B$ be the evaluation map. The assignment $f \mapsto \tilde{f}$ is to be a bijection $$\hom(Z \times A, B) \stackrel{\cong}{\to} \hom(Z, Mor(A, B))$$ whose inverse takes $g: Z \to Mor(A, B)$ to $\epsilon_{A, B} \circ (g \times 1_A)$. The first equation $\epsilon_{A, B} \circ (\tilde{f} \times 1_A) = f$ says the composite $$\hom(Z \times A, B) \to \hom(Z, Mor(A, B)) \to \hom(Z \times A, B)$$ is an identity. The equation that says the composite $$\hom(Z, Mor(A, B)) \to \hom(Z \times A, B) \to \hom(Z, Mor(A, B))$$ is the identity is $\widetilde{\epsilon_{A, B} \circ (g \times 1_A)} = g$. You should use this instead of your second equation. Try seeing if you can establish these equations. But let me say that the cartesian closed structure on $\mathbf{Dig}$ (products, exponentials, exponential transpose, evaluation map) is inherited from the cartesian closed structure of the larger category of directed graphs, where a directed graph consists of a triple $(E, V, \langle s, t\rangle: E \to V \times V)$. Here the category of directed graphs may be identified with the category of functors $F: (\bullet \stackrel{\to}{\to} \bullet) \to Set$; this is a presheaf category, where there is a well-known description of the cartesian closed structure. (If you don't like this, then try the description given in Berci's answer. But maybe my observation will become useful to you later, when the objects of study become more complicated.)

   Your second equation doesn’t make sense.

   Let’s start again. For $f: Z \times A \to B$, let $\tilde{f}: Z \to Mor(A, B)$ be the exponential transpose (or currying) of $f$, and let $\epsilon_{A, B}: Mor(A, B) \times A \to B$ be the evaluation map. The assignment $f \mapsto \tilde{f}$ is to be a bijection

   $$\hom(Z \times A, B) \stackrel{\cong}{\to} \hom(Z, Mor(A, B))$$

   whose inverse takes $g: Z \to Mor(A, B)$ to $\epsilon_{A, B} \circ (g \times 1_A)$. The first equation $\epsilon_{A, B} \circ (\tilde{f} \times 1_A) = f$ says the composite

   $$\hom(Z \times A, B) \to \hom(Z, Mor(A, B)) \to \hom(Z \times A, B)$$

   is an identity. The equation that says the composite

   $$\hom(Z, Mor(A, B)) \to \hom(Z \times A, B) \to \hom(Z, Mor(A, B))$$

   is the identity is $\widetilde{\epsilon_{A, B} \circ (g \times 1_A)} = g$. You should use this instead of your second equation.

   Try seeing if you can establish these equations. But let me say that the cartesian closed structure on $\mathbf{Dig}$ (products, exponentials, exponential transpose, evaluation map) is inherited from the cartesian closed structure of the larger category of directed graphs, where a directed graph consists of a triple $(E, V, \langle s, t\rangle: E \to V \times V)$. Here the category of directed graphs may be identified with the category of functors $F: (\bullet \stackrel{\to}{\to} \bullet) \to Set$; this is a presheaf category, where there is a well-known description of the cartesian closed structure. (If you don’t like this, then try the description given in Berci’s answer. But maybe my observation will become useful to you later, when the objects of study become more complicated.)

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeNov 22nd 2013
   • (edited Nov 22nd 2013)
   • PermaLink
   Author: porton
   Format: MarkdownItexFrom Awodey's book it follows that $\epsilon(p;q)=\left(-\left(1_{\operatorname{MOR}(A;B)}\right)\right)(p;q)=1_{\operatorname{MOR}(A;B)}(p)(q)=p(q)$. But this formula does not make sense when $p$ and $g$ are edges of a digraph, because edges are not functions. Please help me to understand where my error is.

   From Awodey’s book it follows that $\epsilon(p;q)=\left(-\left(1_{\operatorname{MOR}(A;B)}\right)\right)(p;q)=1_{\operatorname{MOR}(A;B)}(p)(q)=p(q)$.

   But this formula does not make sense when $p$ and $g$ are edges of a digraph, because edges are not functions.

   Please help me to understand where my error is.

4. • CommentRowNumber4.
   • CommentAuthorporton
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: porton
   Format: MarkdownItexSee also <a href="http://math.stackexchange.com/questions/576382/category-theory-where-is-my-error">this question</a> about my error.

   See also this question about my error.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 24th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexVictor, I have written some things out [here](http://ncatlab.org/toddtrimble/show/digraph+stuff) which gives some concrete details on what you wanted from #1, from a point of view I was hinting at in #2.

   Victor, I have written some things out here which gives some concrete details on what you wanted from #1, from a point of view I was hinting at in #2.

6. • CommentRowNumber6.
   • CommentAuthorporton
   • CommentTimeNov 24th 2013
   • PermaLink
   Author: porton
   Format: MarkdownItexYour "here" link requires a password.

   Your “here” link requires a password.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 24th 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSorry; I created the link straight out of finishing my edits. Try [here](http://ncatlab.org/toddtrimble/published/digraph+stuff) instead.

   Sorry; I created the link straight out of finishing my edits. Try here instead.