From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2326 Path: news.gmane.org!not-for-mail From: Jpdonaly@aol.com Newsgroups: gmane.science.mathematics.categories Subject: Re: Function composition of natural transformations? Date: Wed, 4 Jun 2003 15:44:01 EDT Message-ID: <15f.2152bf37.2c0fa601@aol.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018577 3568 80.91.229.2 (29 Apr 2009 15:22:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:57 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jun 5 16:12:02 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 05 Jun 2003 16:12:02 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19O05V-0007Kq-00 for categories-list@mta.ca; Thu, 05 Jun 2003 16:07:49 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 70 Xref: news.gmane.org gmane.science.mathematics.categories:2326 Archived-At: For function composition, I just use the standard small circle \circ. So the function composite of natural transformations \tau and \sigma (if it exists) is \tau\circ\sigma. It is advisable to give up subscripting as a way of denoting values of (fully extended) natural transformations: The value of \tau at morphism a is just \tau(a). I would not use juxtaposition or any other generic means (e.g., a centered dot) of denoting composition in a general category for function composition or, for that matter, for any other composition which already has a specified composition symbol, but I do denote pointwise ("vertical") composition generically. Here is an example of how this goes---a line proof of the interchange law for function and pointwise composition: {\noindent\bf Proposition (Interchange Law):} When $\nu\mu\circ\tau\sigma$ is defined for natural transformations $\nu$, $\mu$, $\tau$ and $\sigma$, then so is $(\nu\circ\tau)\cdot(\mu\circ\sigma)$, and $$\nu\mu\circ\tau\sigma=(\nu\circ\tau)\cdot(\mu\circ\sigma).$$ \medskip {\noindent\bf Proof:} The void cases are trivial. Assume that $\nu\mu\circ\tau\sigma$ is defined. Then surely $\nu\circ\tau$ and $\mu\circ\sigma$ are defined, and $$\dom(\nu\circ\tau)=\dom\nu\circ\dom\tau=\cod\mu\circ\cod\sigma=\cod(\mu\circ \sigma),$$ so $\nu\circ\tau$ composes pointwise with $\mu\circ\sigma$. Calculate as follows at an $a$ in the common domain category of both sides of the interchange formula: $$\eqalign{[(\nu\cdot\mu)\circ(\tau\cdot\sigma)](a)=&\nu[\tau(\cod a)\cdot\sigma(a)]\cdot\mu(\dom[\tau(\cod a)\cdot\sigma(a)])\cr &\cr =&\nu(\tau(\cod a))\cdot(\dom\nu)[\sigma(a)]\cdot\mu[\dom\sigma(a)]\cr &\cr =&(\nu\circ\tau)(\cod a)\cdot(\cod\mu)[\sigma(a)]\cdot\mu[\dom\sigma(a)]\cr &\cr =&(\nu\circ\tau)(\cod a)\cdot\mu(\sigma(a))\cr &\cr =&(\nu\circ\tau)(\cod a)\cdot(\mu\circ\sigma)(a)\cr &\cr =&[(\nu\circ\tau)\cdot(\mu\circ\sigma)](a).\cr}$$ So the two sides of the interchange equation have the same intertwining function. Checking domain functors, $$\eqalign{\dom(\nu\mu\circ\tau\sigma)&=\dom\nu\mu\circ\dom\tau\sigma\cr &=\dom\mu\circ\dom\sigma\cr &=\dom(\mu\circ\sigma)\cr &=\dom(\nu\circ\tau)(\mu\circ\sigma);\cr}$$ similarly, $\cod(\nu\mu\circ\tau\sigma)=\cod(\nu\circ\tau)(\mu\circ\sigma)$. Thus the two natural transformations are equal. In this, \dom and \cod are defined by \def\dom {\hbox{\rm dom }} \def\cod {\hbox{\rm cod }} and respectively represent the domain and the codomain function on the implicit category. The proof uses the following formulas for pointwise composition in terms of fully extended natural transformations (i.e., in terms of their intertwining functions \pi and \tau): (\pi\cdot\tau)(a)=\pi(a)\cdot\tau(\dom a)=\pi(\cod a)\cdot\tau(a) which I can't help mentioning as an aside shows that evaluation of fully extended natural transformations at a morphism intertwines evaluation at its domain object with evaluation at its codomain object. (And, incidentally, codomains are on the left in my notations, domains on the right.) If I haven't explained something necessary here, I hope that you can nevertheless see that the above line proof represents a moderately massive amount of diagram drawing and chasing and would fit convincingly on the page of a textbook. I hope that this addresses your request. The only examples which I know are all in my personal set of notes which I set up PCTex32 over the last dozen years and which come out at about 200 pages. This is probably a little too much to drop on you all at once. I am, however, anxious to answer any further questions which you may have.