From: Vaughan Pratt <pratt@cs.stanford.edu>
To: categories@mta.ca
Subject: Re: Addendum
Date: Mon, 17 Jan 2000 13:18:18 -0800 [thread overview]
Message-ID: <200001172118.NAA06734@coraki.Stanford.EDU> (raw)
In-Reply-To: Your message of "Sun, 16 Jan 2000 22:37:40 PST." <200001170637.WAA03764@coraki.Stanford.EDU>
>How about the positive integers with * as sum, with 1->3 as the only
>nonidentity arrow? The unique cubical coalgebra is 1 but there is no
>square coalgebra.
Oops, that gets associativity at the cost of * no longer being a functor
(pointed out to me by Peter Selinger).
Ok, let me dig myself in deeper by making my example more complicated.
Instead of 1->3, put an arrow from i to j whenever i <= j <= 2i.
Now every i is a square coalgebra but no i is a cubical coalgebra.
Now adjoin a new object oo (infinity), with x+oo = oo+x = oo for all x,
and the identity at oo as the only new arrow. oo is both a square *and*
a cubical coalgebra. Since it is disconnected from the other square
coalgebras there can't be a final such. But oo is the only cubical
coalgebra, with only one self-map, making it a final cubical coalgebra.
Vaughan
next prev parent reply other threads:[~2000-01-17 21:18 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-01-15 18:07 Addendum Peter Freyd
2000-01-17 6:37 ` Addendum Vaughan Pratt
2000-01-17 21:18 ` Vaughan Pratt [this message]
2000-01-18 19:47 ` Addendum Vaughan Pratt
2000-01-18 9:49 ` Addendum Dr. P.T. Johnstone
2004-03-26 13:39 Addendum Peter Freyd
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