Re: Composing modifications
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/ ]
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/ ]
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/ ]
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/ ]
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/ ]
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/ ]
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/ ]
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/ ]
participants (6)
-
David Leduc -
JeanBenabou -
John Baez -
Richard Garner -
Robert Seely -
Ronnie Brown