Subject: categories: Re: co-

From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Sat, 4 Jul 1998 18:40:15 +0100 (BST)

P.S. -- I don't agree with John that "all categorical 'co-'s" 
are of the same kind as "cosine" or "colatitude" in referring to
something complementary (although I suppose that "cohomology"
might be). I can't see any sense in which the opposite of a
category can be regarded as complementary to it.

Peter Johnstone

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From: Michael Barr <[email-masked]>
Date: Mon, 6 Jul 1998 12:02:12 -0400 (EDT)

Well, after reading the various replies, I concede that I was very likely
wrong about cohomolohy (which is generally accepted as the first co-)
being short for contra-homology.  Therefore, complementary homology seems
the best.  Then people started thinking of it as covariant homology
anyway.  Then the idea took hold that co-meant opposite.  Strange, since
now it has essentially reversed meaning.  That's how the language
developes.  It is through such twists and turns that the same
Indo-european root come to mean black in English and white in French (and
other romance languages).  

A few other comments.  The x- and y-coordinates are (or used to be) called
ordinate and abscissa (or vice versa).  For what it's worth.  I know what
efficient means, but I have no idea what it has to do with coefficient.
It stands to reason that Birkhoff & Mac Lane is the opposite of Mac Lane &
Birkhoff.  I hadn't noticed that, but I sure had noticed that I had no
simple way of recalling which were left and which were right cosets.  And
would products be left limits or right limits and why?

Michael

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From: Vaughan Pratt <[email-masked]>
Date: Mon, 06 Jul 1998 14:50:29 -0700

From: "Dr. P.T. Johnstone" <[email-masked]>
>I can't see any sense in which the opposite of a

Peter's book "Stone spaces" (CUP 1982) studies many categories that
obey the rule that the signature of the opposite of C is the complement
of the signature of C, with respect to both existence and direction
of sup's and inf's of cardinalities 0, finite nonzero, and infinite.
Namely, if sup's of a given arity exist in C then inf's of that arity
do not exist in C^op, and conversely (at least in the examples).

The extremal pair of opposite categories in this regard are CABA,
complete atomic Boolean algebras, which has all inf's and sup's, dual to
Set having none (but for this purpose we could as well take StoneDLat,
dual to Poset).  "In the middle" is CSLat, complete semilattices,
having sup's of all cardinalities and no inf's, dual to itself via a
duality that interchanges sup's and inf's (not to be confused with the
equivalence that performs this interchange).

Top and bottom as zeroary sup and inf can be moved independently from
CABA to Set to give pointed and bipointed sets dual to CABA's lacking
one or both constants 0 and 1.

Likewise binary (and hence ternary etc.) sup and/or inf can be moved from
CABA to Set.  Moving all finitary sup's for example turns Set into SLat
(semilattices, specifically of the meet kind) and CABA into StoneSLat,
aka algebraic lattices.

By viewing Stone topology as the manifestation of infinite sup's and
inf's, the duality of Stone (Stone spaces) and Bool (Boolean algebras)
can be understood as the migration of the infinitary sup's and inf's
of CABA to Set, turning CABA into Bool and Set into Stone.  From this
viewpoint the ability to present limits in Top follows by duality from
the optional removal in CABA of infinitary inf's to yield Top^op, while
Frm (frames) as Loc^op (locales) tightens this up by removing "optional".

Chu(Set,2) provides a uniform setting for this rule.  All the above
categories C have a "common" full concrete embedding in Chu(Set,2)
in which the points remain unchanged (concreteness) while the states
or opens are taken to be the morphisms of C to the realization of the
schizophrenic object 2 as an object of C.  In this setting the above
tendency (but not the converse part) can be made a theorem about Chu
spaces over 2 as follows, at least for the finitary operations.

Theorem 0: If a space A has constant 0 then its dual A\perp does not
have constant 1 (and dually with 0 and 1 interchanged).

Proof.  A row of all 0's precludes a column of all 1's (the impossibility
of having an irresistible force and an immovable object).

Theorem 2: If A has all binary joins then the only binary meets existing
in A\perp are of comparable elements.

Proof.  Suppose x,y are incomparable columns of A (points of A\perp)
whose meet x&y is a column of A.  Then there exist rows a,b of A such that

	     x  y  x&y
	a    0  1   0
	b    1  0   0

is part of A's matrix.  Now form the join of a and b:

       a|b   1  1   0

This contradicts x&y being the meet of x and y.  QED

(Apropos of the comparability issue, note that the category of finite
chains with bottom, with morphisms all monotone bottom-preserving
functions, is self-dual.)

I don't know how to state the corresponding theorem if any for infinitary
sup's and inf's.  And I have no idea how to generalize all this to
Chu(Set,3) and beyond.

I became aware of this complementarity principle for opposite categories
around 1990, and contemplating it led my student Vineet Gupta and myself
to (ordinary) Chu spaces in 1992, at which point we learned that they
were not new (but not well known).

Vaughan Pratt

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From: Zhaohua Luo <[email-masked]>
Date: Tue, 07 Jul 1998 15:27:14 -0400

A category is called " left " if it has a strict initial object and any map
with an initial domain is regular (in the general sense). The opposite of a
left category is called a " right " category. The key fact here is that a left
category is never right unless it is trivial (i.e. any map is an isomorphism).
Since may natural categories are either left or right, this enable us to
introduce the "relative version" of a categorical concept with a dual. The
following short note

On Left and Right Categories

is available at

www.azd.com/lrcat.html

which is inspired by Vaughan Pratt comments:

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From: Vaughan Pratt <[email-masked]>
Date: Wed, 08 Jul 1998 22:25:50 -0700

From: Zhaohua Luo <[email-masked]>
>left category is called a " right " category.

This division of categories into left, right, and other caters for only
a very special case of the general phenomenon "signature of opposite is
complement of signature", namely where *all* the algebra is on the right,
call this C\op.  The more general situation permits the algebra to be
split between C and C\op, e.g. CSLat which puts the sups (say) in C and
sends the infs off to C\op, which turn into sups as they "pass through
the mirror."  Many other partitions of algebra are possible that similarly
do not fit this left-right classification and so fall under "other".

Vaughan Pratt