Correction

Correction

[Peter May]

Secretarial error: Ieke Moerdijk will also be speaking
at the MacLane Memorial Conference:

correction

[Eduardo J. Dubuc]

Oops, forgot to say that furthermore, it should hold (for such a
category) that when the non vertices of the cospan are equal, the
"projections' in the commutative square can be taken to be equal.

e.d.

On 25/04/13 11:19, Aleks Kissinger wrote:
  > Oops, forgot to send to list.
  >
  > I think its actually a stronger property, but: perhaps cofiltered
category?
  >
  > On 25 April 2013 04:14, David Yetter<dyetter@math.ksu.edu>  wrote:
  >> Is there an existing name in the literature for a category in which
every cospan admits a completion to a commutative square?  (Just that,
no uniqueness, no universal
  >> properties required, just every cospan sits inside at least one
commutative square).  If so, what have such things been called?  If not,
does anyone have a poetic idea for a good name for
  >> such categories?
  >>
  >> Best Thoughts,
  >> David Yetter
  >>

yes !, a cofiltered category is just a connected such category,

Verdier's formulation, see Mac Lane's book if you don't like the SGA4.

e.d.


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correction

[Michael Barr]

Of course, what I called a relation was really a span, but its image is
the relation I had in mind.  It makes me realize that spans are much
easier to think about than relations because there is no existential
quantifier involved.

Michael


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correction

[Peter Freyd]

Vaughan has noticed that I hadn't broken the commutative habit. So let
me start again. He asked if one can determine a minimal equational
theory with the theory of distributive lattices as its unique maximal
consistent extension. Yes, here's an example:

     x meet 1 = x,
     x meet 0 = 0,
     1 join 1 = 1,
     1 join 0 = 1,
     0 join 1 = 1,
     0 join 0 = 0.

(I was missing the penultimate equation.) There is a Klein-group's
worth of variations. One operation simultaneously interchanges meet
and join, 0 and 1. Another operation simultaneously interchanges the
arguments of the operators. I'll hazard that the resulting four
theories are the only ones that do the trick.

correction

[Marta Bunge]

To access the paper "Constructive Theory of Galois Toposes" go to

http://www.math.mcgill.ca/~bunge/ctgt.ps (.pdf)

as the one I gave earlier does not work.
Sorry!
Marta Bunge
_________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

correction

[Michael MAKKAI]

This is a correction to the talk I gave in Toronto on the 24th of
September. In the definition of "principal computad" (on page 8 of the
slides, for those who have copies of the slides) a clause was omitted. The
definition should read as follows (clause (c) was missing):

	The computad A is *principal* if (a) A is finite, (b) there is a
unique indeterminate of maximal dimension in A, say x; and (c) there is no
proper subcomputad of A containing x.

I add that there was another error in the slides, although it was caught
during the talk and corrected on the projected slide (but not on the
copies). On page 11 of the slides, "fibration" should be "trivial
fibration". 

Michael Makkai

correction

[Paul Taylor]

> My motication (in 1987 this was)
This should have been 1984 (and motivation).

I'd also like to add:

I had some difficulty in getting my ideas about stable coequalisers
across (partly, of course, my fault, because I didn't get the rigt
answer, involving stable transitive closures, until 1993).  People
said, "while programs can't possibly be described by finitary first
order theories, because the theory of the natural numbers is an
example".

I knew that, and it wasn't what I meant.  This illustrates a subtlely
in first order categorical logic: that the relevant structure consists
of *less* than all finite (limits and) stable disjoint colimits. The
categorical structure corresponding to first order logic is a (Heyting)
pretopos, which need not have coequalisers of arbitrary parallel pairs.
For example, the category of compact Hausdorff spaces is a pretopos but
does not have all stable coequalisers. (I have a feeling I haven't
got this quite right, and Peter Freyd is going to jump on me. I shouldn't
be so foolish as to answer mathematical questions on the modem from home!)

More recently, Jiri Rosicky and Peter Johnstone have considered the
question of what theories can be expressed with finite sketches.
As Peter Freyd showed in 1972, this includes the natural numbers.
My results and those of Jiri and Peter are more complicated forms of
Peter Freyd's original observation.

Finally, since my more recent work on categorical recursion, I no
longer think that a coequaliser diagram is the best way of presenting
a WHILE program categorically (though the diagram I would use now
expresses the same logical content).

Paul

correction

[categories]

Date: Thu, 15 May 1997 10:12:30 +0200 (MET DST)
From: Jiri Rosicky <rosicky@math.muni.cz>

At the 64th PSSL at Braunschweig, I gave a talk about cartesian closedness
of exact completions with an intention to cover equilogical spaces in the
sense of Dana Scott (see D.S.Scott, A New category? Domains, Spaces and
Equivalence Relations, preprint 1996). Unfortunately, Peter Johnstone
found a flaw in my argument. I would like to announce the following result
which covers equilogical spaces:

Theorem: Let C be an infinitary extensive category. Then its exact
completion ex(C) is cartesian closed iff C is weakly cartesian closed.
Moreover, the embedding C-->ex(C) preserves exponentials.