12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Friday May 03, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Propositions As Types (view source)
Revision as of 13:36, 27 March 2009
, 13:36, 27 March 2009→Natural Transformation
===Natural Transformation===
===Natural Transformation===
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<p>'''Definition 2.1.''' Given functors <math>F, G : \mathcal{A} \rightrightarrows \mathcal{B},</math> a ''natural transformation'' <math>t : F \to G</math> is a family of arrows <math>t(A) : F(A) \to G(A)</math> in <math>\mathcal{B},</math> one arrow for each object <math>A\!</math> of <math>\mathcal{A},</math> such that the following square commutes for all arrows <math>f : A \to B</math> in <math>\mathcal{A}</math>:</p>
<pre>
<pre>
t(A)
F(A) o------------------>o G(A)
| |
| |
F(f) | | G(f)
| |
v v
F(B) o------------------>o G(B)
t(B)
</pre>
<p>that is to say, such that</p>
|-
|
|
| that is to say, such that
<p><math>G(f)t(A) = t(B)F(f).\!</math></p>
|-
|
|
<p>{Lambek & Scott, 8).</p>
</pre>
|}
===Graph (Review)===
===Graph (Review)===
