Re: Composing modifications

Re: Composing modifications

[Ronnie Brown]

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.
>

...


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Re: Composing modifications

[JeanBenabou]

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
>

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Composing modifications

[David Leduc]

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


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Re: Composing modifications

[Tom Leinster]

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


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Re: Composing modifications

[David Leduc]

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
>


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Re: Composing modifications

[Ronnie Brown]

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

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Re: Composing modifications

[John Baez]

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


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Re: Composing modifications

[David Leduc]

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


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Re: Composing modifications

[Richard Garner]

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


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Re: Composing modifications

[Robert Seely]

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
>>
>
>
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