* Re: Composing modifications
@ 2010-03-07 22:23 Ronnie Brown
2010-03-08 3:46 ` JeanBenabou
0 siblings, 1 reply; 10+ messages in thread
From: Ronnie Brown @ 2010-03-07 22:23 UTC (permalink / raw)
To: categories
Dear All,
(this was sent earlier but had some html which may have got it rejected).
Thanks very much for the info.
After I sent off my email I did remember (just about) papers of Tom
Fiore and of Grandis/Pare which were relevant and was collecting
together references, but fortunately you beat me to it, and with any
interesting points.
It may be convenient to have the definition of multiple composition in
KX, the cubical singular complex of X, on the record
as follows:
--------------------------------------------------------
Let $(m) = (m_1, \ldots , m_n)$ be an $n$-tuple of positive integers
and
$$\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0,
m_n]$$ be the map $(x_1 , \ldots , x_n) \mapsto (m_1 x_1, \ldots ,
m_n x_n).$ Then a {\it subdivision of type $(m)$} \index{singular
$n$-cubes!subdivision of type $(m)$} of a map $\alpha : I^n
\rightarrow X$ is a factorisation $\alpha = \alpha' \circ
\phi_{(m)}$; its {\it parts} are the cubes $\alpha_{(r)}$ where $(r)
= (r_1, \ldots , r_n)$ is an $n$-tuple of integers with $1 \leqslant
r_i \leqslant m_i$, $i = 1, \ldots , n,$ and where $\alpha_{(r)} :
I^n \rightarrow X$ is given by
$$(x_1, \ldots , x_n) \mapsto
\alpha'(x_1 + r_1 - 1, \ldots , x_n + r_n - 1).$$
We then say that $\alpha$ is the {\it composite}of the
cubes $\alpha_{(r)}$ and write $\alpha = [\alpha_{(r)}]$. The {\it
domain} of $\alpha_{(r)}$ is then the set $\{(x_1,\ldots,x_n) \in
I^n : r_i-1 \leqslant x_i \leqslant r_i, 1 \leqslant i \leqslant
n\}$.
------------------------------------------------------------------------------------
(I have also put on the arXiv an exposition of Moore Hyperrectangles,
which is the start of another approach. One problem with this is that a
homotopy is more general than a Moore hyperrectangle as defined there,
since a homotopy is a path in a space of such hyperrectangles.)
These compositions satisfy an interchange law, but without associativity
(unlike the Moore approach) we do not get of course any of what one
might call `partitioning' results. One should get these `up to homotopy'
but this seems difficult to formulate.
In our work we also found the necessity of `connections', \Gamma^\pm _i,
which have many advantages including bringing the cubical theory a
little nearer to the simplicial: a cube \Gamma^\pm_i x has two adjacent
faces the same. In the strict theory, or at least for groupoids, the
compositions, at least of two cubes, are recoverable from the
connections, up to homotopy.
Then of course one needs the more general n-fold objects. These occur
topologically from (n+1)-ads
X_*= (X;A_1, ....,A_n) where the A_i are subspaces of X. Let \Phi=
\Phi(X_*) consist of maps of an n-cube into X which take the faces in
direction i into A_i. Then \Phi has n compositions, making it a `weak
n-fold category' . (The amazing fact (Loday) is that \p_1(\Phi, *),
where * is the constant map, gets the structure of cat^n-group = strict
(!!) n-fold groupoid in groups, and these model weak pointed homotopy
(n+1)-types. )
So I am just suggesting that cubical approaches have reached into
methods not easily approachable by simplicial or globular methods and
perhaps more can be made of this.
Will more high powered approaches using operads help in all this?
I should also mention Richard Steiner's paper
Thin fillers in the cubical nerves of omega-categories
<http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/journaldoc.html?cn=Theory_Appl_Categ>
<http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html?pg1=ISSI&s1=240201>,
Theory App Categories (2006) 144--173 as suggesting the possibility of
using some kind of thin structure to define weak cubical categories.
My motivation all along was in terms of `algebraic inverses to
subdivision' .
Ronnie
Richard Garner wrote:
>
> Dear Ronnie,
>
> There are a number of "cubical" notions of bicategory or higher in the
> literature with which I am sure you are familiar; the earliest being
> Dominic Verity's double bicategories (which are really a particular
> kind of triple category), and later the weak (or pseudo) double
> categories studied in a series of papers by Bob Paré and Marco
> Grandis. However, none of these quite seem to fit the spirit of what
> you are asking for, and certainly do not describe the singular cubical
> complex of a space.
>
...
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-07 22:23 Composing modifications Ronnie Brown
@ 2010-03-08 3:46 ` JeanBenabou
0 siblings, 0 replies; 10+ messages in thread
From: JeanBenabou @ 2010-03-08 3:46 UTC (permalink / raw)
To: Ronnie Brown
Dear Ronnie,
I usually enjoy your remarks and I would probably have enjoyed your
last ones, except that I cannot read it. Has the sorry time finally
arrived when one can no longer work in Mathematics unless he knows
TeX, LaTeX, and/or other sophisticated word processings? Should I,
and many others it would be too long to name, stop doing mathematics?
I'd appreciate answers from all colleagues
> Dear All,
>
> (this was sent earlier but had some html which may have got it
> rejected).
> Thanks very much for the info.
>
> After I sent off my email I did remember (just about) papers of Tom
> Fiore and of Grandis/Pare which were relevant and was collecting
> together references, but fortunately you beat me to it, and with
> any interesting points.
>
> It may be convenient to have the definition of multiple composition
> in KX, the cubical singular complex of X, on the record
> as follows:
> --------------------------------------------------------
> Let $(m) = (m_1, \ldots , m_n)$ be an $n$-tuple of positive integers
> and
> $$\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0,
> m_n]$$ be the map $(x_1 , \ldots , x_n) \mapsto (m_1 x_1, \ldots ,
> m_n x_n).$ Then a {\it subdivision of type $(m)$} \index{singular
> $n$-cubes!subdivision of type $(m)$} of a map $\alpha : I^n
> \rightarrow X$ is a factorisation $\alpha = \alpha' \circ
> \phi_{(m)}$; its {\it parts} are the cubes $\alpha_{(r)}$ where $(r)
> = (r_1, \ldots , r_n)$ is an $n$-tuple of integers with $1 \leqslant
> r_i \leqslant m_i$, $i = 1, \ldots , n,$ and where $\alpha_{(r)} :
> I^n \rightarrow X$ is given by
> $$(x_1, \ldots , x_n) \mapsto
> \alpha'(x_1 + r_1 - 1, \ldots , x_n + r_n - 1).$$
>
> We then say that $\alpha$ is the {\it composite}of the
> cubes $\alpha_{(r)}$ and write $\alpha = [\alpha_{(r)}]$. The {\it
> domain} of $\alpha_{(r)}$ is then the set $\{(x_1,\ldots,x_n) \in
> I^n : r_i-1 \leqslant x_i \leqslant r_i, 1 \leqslant i \leqslant
> n\}$.
> ----------------------------------------------------------------------
> --------------
> (I have also put on the arXiv an exposition of Moore
> Hyperrectangles, which is the start of another approach. One
> problem with this is that a homotopy is more general than a Moore
> hyperrectangle as defined there, since a homotopy is a path in a
> space of such hyperrectangles.)
>
> These compositions satisfy an interchange law, but without
> associativity (unlike the Moore approach) we do not get of course
> any of what one might call `partitioning' results. One should get
> these `up to homotopy' but this seems difficult to formulate.
>
> In our work we also found the necessity of `connections', \Gamma^
> \pm _i, which have many advantages including bringing the cubical
> theory a little nearer to the simplicial: a cube \Gamma^\pm_i x
> has two adjacent faces the same. In the strict theory, or at least
> for groupoids, the compositions, at least of two cubes, are
> recoverable from the connections, up to homotopy.
>
> Then of course one needs the more general n-fold objects. These
> occur topologically from (n+1)-ads
> X_*= (X;A_1, ....,A_n) where the A_i are subspaces of X. Let \Phi=
> \Phi(X_*) consist of maps of an n-cube into X which take the faces
> in direction i into A_i. Then \Phi has n compositions, making it a
> `weak n-fold category' . (The amazing fact (Loday) is that \p_1
> (\Phi, *), where * is the constant map, gets the structure of cat^n-
> group = strict (!!) n-fold groupoid in groups, and these model weak
> pointed homotopy (n+1)-types. )
>
> So I am just suggesting that cubical approaches have reached into
> methods not easily approachable by simplicial or globular methods
> and perhaps more can be made of this.
>
> Will more high powered approaches using operads help in all this?
>
> I should also mention Richard Steiner's paper
> Thin fillers in the cubical nerves of omega-categories <http://0-
> ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/
> journaldoc.html?cn=Theory_Appl_Categ> <http://0-ams.mpim-
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html?
> pg1=ISSI&s1=240201>, Theory App Categories (2006) 144--173 as
> suggesting the possibility of using some kind of thin structure to
> define weak cubical categories.
>
> My motivation all along was in terms of `algebraic inverses to
> subdivision' .
>
> Ronnie
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-04 7:24 ` Ronnie Brown
2010-03-05 0:25 ` John Baez
2010-03-05 0:43 ` David Leduc
@ 2010-03-05 15:59 ` Richard Garner
2 siblings, 0 replies; 10+ messages in thread
From: Richard Garner @ 2010-03-05 15:59 UTC (permalink / raw)
To: Ronnie Brown
Dear Ronnie,
There are a number of "cubical" notions of bicategory or
higher in the literature with which I am sure you are
familiar; the earliest being Dominic Verity's double
bicategories (which are really a particular kind of triple
category), and later the weak (or pseudo) double categories
studied in a series of papers by Bob Paré and Marco Grandis.
However, none of these quite seem to fit the spirit of what
you are asking for, and certainly do not describe the
singular cubical complex of a space.
One approach which is indicated in the literature is based on
Michael Batanin's theory of higher operads. In particular,
Tom Leinster, in the refined presentation of this theory
given in his book, discusses the possibility of using it to
give a notion of weak cubical omega-category, though he does
not go into any details. The basic idea is straightforward.
On the category of cubical sets (without diagonals or
symmetries) one has the monad T for strict cubical
omega-categories; and one aims to define the monad for weak
cubical omega-categories as a suitable deformation T' of this
monad. In the language of homotopy theory, one would like T'
to be a cofibrant replacement for T; this then ensures both
that T' looks sufficiently like T, and also that its algebras
are homotopy-invariant in a suitable sense.
To make this more formal, one defines a cubical operad to be
a monad S on the category of cubical sets together with a
cartesian monad morphism S => T. The existence of such a
monad morphism---which I think will be unique if it
exists---places some strong restrictions on the shapes of the
operations out of which the monad S is built: a typical such
operation will take an n_1 x n_2 x ... x n_k grid of
k-dimensional hypercubes, and stick them together in some way
to yield a single k-dimensional hypercube. This is precisely
what one wants for a notion of (weak) cubical category.
To ensure that the monad S looks sufficiently like the monad
T, one needs the notion of a contraction (= acyclic
fibration). In fact, we only need to know how to say that the
monad morphism S => T has a contraction, and this is quite
simple. Suppose we are given a n_1 x ... x n_k grid of
k-dimensional hypercubes as before. Suppose we are also given
operations s_1, ..., s_2k in S which indicate how to compose
up the (k-1)-dimensional grids of hypercubes obtained as the
faces of the original grid. Then we require that there should
be given an operation of S which acts on n_1 x ... x n_k
grids, and whose actions on each of the (k-1)-dimensional
faces are given by s_1, ..., s_2k.
There now arises a category whose objects are cubical operads
S equipped with a contraction on S => T, and whose morphisms
are monad maps over T which commute to the contractions in
the obvious sense. This category is locally presentable, and
hence has an initial object; which we declare to be the monad
T' for weak cubical omega categories. One may in fact show
that T' is a cofibrant replacement for T in a suitable model
structure on cubical operads.
One may of course specialise this construction to low
dimensions, and I have just sat down to work it out in the
case where n=2. Rather surprisingly, it seems that there
might not be a finite axiomatisation of the resultant
structure; at least, one does not immediately leap off the
page which is the usual situation. The best I have been able
to do is the following (which is not exactly the result of
applying the above construction but is a perfectly good
surrogate).
Definition. A cubical bicategory is given by sets of objects,
of vertical arrows, of horizontal arrows and of squares,
satisfying the obvious source and target criteria, together
with operations of identity and binary composition for
vertical and horizontal arrows, satisfying no laws at all;
and finally, for every n x m grid of squares (where possibly
n or m are zero), and every way of composing up the
horizontal and vertical boundaries using the nullary and
binary compositions, a composite square with those
boundaries. The coherence axioms which this structure must
satisfy say that any two ways of composing up a diagram of
squares must give the same answer.
I would be very interested to know if anyone can extract from
this definition a finite collection of composition operations
on squares, and a finite collection of equations between
them, which together generate all the others. The key
obstable seems to be problem that identity 1-cells are not
strict in either direction.
Richard
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-04 7:24 ` Ronnie Brown
2010-03-05 0:25 ` John Baez
@ 2010-03-05 0:43 ` David Leduc
2010-03-05 15:59 ` Richard Garner
2 siblings, 0 replies; 10+ messages in thread
From: David Leduc @ 2010-03-05 0:43 UTC (permalink / raw)
To: Ronnie Brown
Dear Ronnie,
> Has anyone formulated a cubical (square?) version of bicategory or higher?
What is that? I know the Barendregt's lambda cube. Is it related?
David
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-04 7:24 ` Ronnie Brown
@ 2010-03-05 0:25 ` John Baez
2010-03-05 0:43 ` David Leduc
2010-03-05 15:59 ` Richard Garner
2 siblings, 0 replies; 10+ messages in thread
From: John Baez @ 2010-03-05 0:25 UTC (permalink / raw)
To: categories
Ronnie Brown wrote;
Has anyone formulated a cubical (square?) version of bicategory or higher?
>
Dominic Verity invented "double bicategories", which are the fully weakened
version of double categories. Unfortunately, his thesis can only be
obtained by special shipment from Australia. Someday someone will scan it
and make it more widely available!
Luckily, you can see the definition in this paper:
Jeffrey Morton
Double bicategories and double cospans
http://arxiv.org/abs/math/0611930
For higher dimensions, try:
Marco Grandis
HIGHER COSPANS AND WEAK CUBICAL CATEGORIES
(COSPANS IN ALGEBRAIC TOPOLOGY, I)
http://tac.mta.ca/tac/volumes/18/12/18-12abs.html
and also parts II and III of this series.
Best,
jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-03 13:04 ` David Leduc
2010-03-04 7:24 ` Ronnie Brown
@ 2010-03-04 21:25 ` Robert Seely
1 sibling, 0 replies; 10+ messages in thread
From: Robert Seely @ 2010-03-04 21:25 UTC (permalink / raw)
To: David Leduc
There are very good reasons to want a "tricategory of bicategories" in
the lax sense, it's just unfortunate that such doesn't exist. This
prompted my collegues and me to consider a slightly modified notion
(pun intended), which you can find in two papers (both TAC 2003):
Modules (Cockett-Koslowski-Seely-Wood) (TAC 11(2003)17, pp 375-396.)
Morphisms and modules for poly-bicategories (Cockett-Koslowski-Seely)
(TAC 11(2003)2, pp 15-74.)
http://www.tac.mta.ca/tac/index.html#vol11 or from my webpage
http://www.math.mcgill.ca/rags/
The key idea is to consider modules between morphisms (lax functors)
of bicategories, and what we call modulations between modules, instead
of lax natural transformations and modifications. This all works very
well in the multi-bicategory setting (or even in the poly-bicategory
setting), but under certain well-known completeness conditions, the
"multi-" ("poly-") structure is representable (i.e. we have ordinary
bicategorical structure).
The Cockett-Koslowski-Seely-Wood paper is probably easier going for
most readers here; the Cockett-Koslowski-Seely paper gives a more
general setting, and is perhaps more detailed (well, it is 3x as
long). (It also has prettier pictures.)
-= rags =-
On Wed, 3 Mar 2010, David Leduc wrote:
> Dear Nick and Tom,
>
> Thank you very much for your replies. It is very helpful.
>
> I had in mind to form a tricategory of bicategories, therefore I guess
> I was talking of what Tom calls strong transformations. They are also
> called weak??? I am a bit confused by the lax, weak, pseudo, strict
> and so on terminology in higher category theory. Nick, could you
> confirm that you were talking of strong transformations in your mail?
> Now I am not sure anymore what are natural transformations in category
> theory. They are strict transformations, right?
>
> Unfortunately, I do not have a copy of the paper "Coherence for
> tricategories" by Gordon, Power and Street. I guess such reference
> would help me a lot with such questions.
>
> I have another question. For strong and strict (and maybe lax?)
> transformations, we have the interchange law relating vertical and
> horizontal composition. What is the equivalent of interchange law for
> compositions of modifications?
>
> Thank you,
>
> David
>
>
> On 3/3/10, Tom Leinster <tl@maths.gla.ac.uk> wrote:
>> Dear David,
>>
>>> I am reading Basic Bicategories by Tom Leinster, and I have basic
>>> questions about modifications.
>>>
>>> 1) Suppose that n, n', m and m' are transformations such that m * n
>>> and m' * n' are well defined, where * denotes horizontal (=
>>> Godement) composition of transformations.
>>> From given modifications a:m-->m' and b:n-->n' is there a way to
>>> derive a modification from m * n to m' * n' ?
>>
>> The first thing to be careful about is horizontal composition of
>> transformations.
>>
>> In that paper, "transformation" was used to mean what might more
>> systematically be called "lax transformation". The paper also refers to
>> "strong" transformations (Gray's terminology?, also called pseudo or
>> weak), and strict transformations. For horizontal composition of
>> transformations, the situation is this:
>>
>> i. Lax: can't be done
>> ii. Strong: can be done, after making a fairly harmless non-canonical
>> choice of "left" or "right"
>> iii. Strict: can be done, canonically.
>>
>> So in order for your question to make sense, I think you need to assume
>> that the transformations are strong, at least. And in that case, yes,
>> there is a canonical way to horizontally compose modifications in the way
>> that you describe.
>>
>>> 2) There are two ways to compose transformations: vertical and
>>> horizontal. What are the ways to compose modifications?
>>
>> Provided that you're using strong or strict transformations (so that
>> horizontal composition makes sense), there are three ways. You could call
>> them vertical, horizontal and... transversal?
>>
>> But it's probably better to adopt a more systematic terminology and talk
>> about "i-composition" for i = 0, 1, 2. Here i is the dimension of the
>> cell that your two composable things have in common. For example, suppose
>> that we were talking about composing 2-cells x and y inside a 2-category.
>> Then:
>>
>> * vertical composition would be "1-composition", because you can do it
>> when the 1-dimensional domain dom(x) of x is equal to the 1-dimensional
>> codomain cod(y) of y
>>
>> * horizontal composition would be "0-composition", because you can do it
>> when the 0-dimensional domain dom(dom(x)) of x is equal to the
>> 0-dimensional codomain cod(cod(y)) of y.
>>
>> Best wishes,
>> Tom
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
--
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-03-03 13:04 ` David Leduc
@ 2010-03-04 7:24 ` Ronnie Brown
2010-03-05 0:25 ` John Baez
` (2 more replies)
2010-03-04 21:25 ` Robert Seely
1 sibling, 3 replies; 10+ messages in thread
From: Ronnie Brown @ 2010-03-04 7:24 UTC (permalink / raw)
To: David Leduc
Dear All,
Has anyone formulated a cubical (square?) version of bicategory or higher?
The model is surely the singular cubical complex KX of a topological
space. This has a great deal of structure of multiple compositions and
tensor products which we have fairly easily written down and exploited
in papers with Philip Higgins. By working with filtered spaces and
taking certain homotopy classes we also get (non trivially!) associated
strict structures.
There is also a notion of fibrant (=Kan) cubical set, and of fibration,
which seems not available globularly (?). So one formulation of a weak
cubical omega-category is to say it comes with a cubical fibration to a
strict cubical omega-category. (Such exists for RX_*, the cubical
singular complex of a filtered space.
(with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'',
{\em J. Pure Appl. Algebra} 22 (1981) 11-41.
) This is a definition with one example and no theorems (as yet)!
By contrast, there is a singular globular complex GX of a space, see for
example
`A new higher homotopy groupoid: the fundamental globular
$\omega$-groupoid of a filtered space', Homotopy, Homology and
Applications, 10 (2008), No. 1, pp.327-343.
but I think nobody has written down an axiomatisation.
Multiple compositions are difficult (for me, at any rate) in the
globular (and simplicial!) situation, so I tend to prefer the simple
minded approach. I have spent many happy hours subdividing squares by
horizontal and vertical lines into lots and lots of little squares!
Ronnie
David Leduc wrote:
> Dear Nick and Tom,
>
> Thank you very much for your replies. It is very helpful.
>
> I had in mind to form a tricategory of bicategories, therefore I guess
> I was talking of what Tom calls strong transformations. They are also
> called weak??? I am a bit confused by the lax, weak, pseudo, strict
> and so on terminology in higher category theory. Nick, could you
> confirm that you were talking of strong transformations in your mail?
> Now I am not sure anymore what are natural transformations in category
> theory. They are strict transformations, right?
>
> Unfortunately, I do not have a copy of the paper "Coherence for
> tricategories" by Gordon, Power and Street. I guess such reference
> would help me a lot with such questions.
>
> I have another question. For strong and strict (and maybe lax?)
> transformations, we have the interchange law relating vertical and
> horizontal composition. What is the equivalent of interchange law for
> compositions of modifications?
>
> Thank you,
>
> David
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
[not found] ` <alpine.LRH.2.00.1003030251180.22708@taylor.maths.gla.ac.uk>
@ 2010-03-03 13:04 ` David Leduc
2010-03-04 7:24 ` Ronnie Brown
2010-03-04 21:25 ` Robert Seely
0 siblings, 2 replies; 10+ messages in thread
From: David Leduc @ 2010-03-03 13:04 UTC (permalink / raw)
To: Tom Leinster, Nick.Gurski; +Cc: categories
Dear Nick and Tom,
Thank you very much for your replies. It is very helpful.
I had in mind to form a tricategory of bicategories, therefore I guess
I was talking of what Tom calls strong transformations. They are also
called weak??? I am a bit confused by the lax, weak, pseudo, strict
and so on terminology in higher category theory. Nick, could you
confirm that you were talking of strong transformations in your mail?
Now I am not sure anymore what are natural transformations in category
theory. They are strict transformations, right?
Unfortunately, I do not have a copy of the paper "Coherence for
tricategories" by Gordon, Power and Street. I guess such reference
would help me a lot with such questions.
I have another question. For strong and strict (and maybe lax?)
transformations, we have the interchange law relating vertical and
horizontal composition. What is the equivalent of interchange law for
compositions of modifications?
Thank you,
David
On 3/3/10, Tom Leinster <tl@maths.gla.ac.uk> wrote:
> Dear David,
>
>> I am reading Basic Bicategories by Tom Leinster, and I have basic
>> questions about modifications.
>>
>> 1) Suppose that n, n', m and m' are transformations such that m * n
>> and m' * n' are well defined, where * denotes horizontal (=
>> Godement) composition of transformations.
>> From given modifications a:m-->m' and b:n-->n' is there a way to
>> derive a modification from m * n to m' * n' ?
>
> The first thing to be careful about is horizontal composition of
> transformations.
>
> In that paper, "transformation" was used to mean what might more
> systematically be called "lax transformation". The paper also refers to
> "strong" transformations (Gray's terminology?, also called pseudo or
> weak), and strict transformations. For horizontal composition of
> transformations, the situation is this:
>
> i. Lax: can't be done
> ii. Strong: can be done, after making a fairly harmless non-canonical
> choice of "left" or "right"
> iii. Strict: can be done, canonically.
>
> So in order for your question to make sense, I think you need to assume
> that the transformations are strong, at least. And in that case, yes,
> there is a canonical way to horizontally compose modifications in the way
> that you describe.
>
>> 2) There are two ways to compose transformations: vertical and
>> horizontal. What are the ways to compose modifications?
>
> Provided that you're using strong or strict transformations (so that
> horizontal composition makes sense), there are three ways. You could call
> them vertical, horizontal and... transversal?
>
> But it's probably better to adopt a more systematic terminology and talk
> about "i-composition" for i = 0, 1, 2. Here i is the dimension of the
> cell that your two composable things have in common. For example, suppose
> that we were talking about composing 2-cells x and y inside a 2-category.
> Then:
>
> * vertical composition would be "1-composition", because you can do it
> when the 1-dimensional domain dom(x) of x is equal to the 1-dimensional
> codomain cod(y) of y
>
> * horizontal composition would be "0-composition", because you can do it
> when the 0-dimensional domain dom(dom(x)) of x is equal to the
> 0-dimensional codomain cod(cod(y)) of y.
>
> Best wishes,
> Tom
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Composing modifications
2010-02-27 14:49 David Leduc
@ 2010-03-03 3:02 ` Tom Leinster
[not found] ` <alpine.LRH.2.00.1003030251180.22708@taylor.maths.gla.ac.uk>
1 sibling, 0 replies; 10+ messages in thread
From: Tom Leinster @ 2010-03-03 3:02 UTC (permalink / raw)
To: David Leduc
Dear David,
> I am reading Basic Bicategories by Tom Leinster, and I have basic
> questions about modifications.
>
> 1) Suppose that n, n', m and m' are transformations such that m * n
> and m' * n' are well defined, where * denotes horizontal (=
> Godement) composition of transformations.
> From given modifications a:m-->m' and b:n-->n' is there a way to
> derive a modification from m * n to m' * n' ?
The first thing to be careful about is horizontal composition of
transformations.
In that paper, "transformation" was used to mean what might more
systematically be called "lax transformation". The paper also refers to
"strong" transformations (Gray's terminology?, also called pseudo or
weak), and strict transformations. For horizontal composition of
transformations, the situation is this:
i. Lax: can't be done
ii. Strong: can be done, after making a fairly harmless non-canonical
choice of "left" or "right"
iii. Strict: can be done, canonically.
So in order for your question to make sense, I think you need to assume
that the transformations are strong, at least. And in that case, yes,
there is a canonical way to horizontally compose modifications in the way
that you describe.
> 2) There are two ways to compose transformations: vertical and
> horizontal. What are the ways to compose modifications?
Provided that you're using strong or strict transformations (so that
horizontal composition makes sense), there are three ways. You could call
them vertical, horizontal and... transversal?
But it's probably better to adopt a more systematic terminology and talk
about "i-composition" for i = 0, 1, 2. Here i is the dimension of the
cell that your two composable things have in common. For example, suppose
that we were talking about composing 2-cells x and y inside a 2-category.
Then:
* vertical composition would be "1-composition", because you can do it
when the 1-dimensional domain dom(x) of x is equal to the 1-dimensional
codomain cod(y) of y
* horizontal composition would be "0-composition", because you can do it
when the 0-dimensional domain dom(dom(x)) of x is equal to the
0-dimensional codomain cod(cod(y)) of y.
Best wishes,
Tom
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
* Composing modifications
@ 2010-02-27 14:49 David Leduc
2010-03-03 3:02 ` Tom Leinster
[not found] ` <alpine.LRH.2.00.1003030251180.22708@taylor.maths.gla.ac.uk>
0 siblings, 2 replies; 10+ messages in thread
From: David Leduc @ 2010-02-27 14:49 UTC (permalink / raw)
To: categories
Dear all,
I am reading Basic Bicategories by Tom Leinster, and I have basic
questions about modifications.
1) Suppose that n, n', m and m' are transformations such that m * n
and m' * n' are well defined, where * denotes horizontal (=
Godement) composition of transformations.
>From given modifications a:m-->m' and b:n-->n' is there a way to
derive a modification from m * n to m' * n' ?
2) There are two ways to compose transformations: vertical and
horizontal. What are the ways to compose modifications?
Thanks for your help,
David
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 10+ messages in thread
end of thread, other threads:[~2010-03-08 3:46 UTC | newest]
Thread overview: 10+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2010-03-07 22:23 Composing modifications Ronnie Brown
2010-03-08 3:46 ` JeanBenabou
-- strict thread matches above, loose matches on Subject: below --
2010-02-27 14:49 David Leduc
2010-03-03 3:02 ` Tom Leinster
[not found] ` <alpine.LRH.2.00.1003030251180.22708@taylor.maths.gla.ac.uk>
2010-03-03 13:04 ` David Leduc
2010-03-04 7:24 ` Ronnie Brown
2010-03-05 0:25 ` John Baez
2010-03-05 0:43 ` David Leduc
2010-03-05 15:59 ` Richard Garner
2010-03-04 21:25 ` Robert Seely
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