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October 6, 2006

On n-Transport: Universal Transition

Posted by Urs Schreiber

Further chasing the nn-transport # Illuminati:


To any morphism p:P 2(Y)P 2(X)p : P_2(Y) \to P_2(X) of domains of 2-transport, we may associate a universal pp-local transition. The associated object turns out to be the category of 2-paths in the 2-groupoid Y Y^\bullet. The associated factorization morphism turns out to be the descending trivialized transport.

A remark related to our discussion # of reconstructing # 2-transport from local data.

Let me put it this way:

An orbifold is conveniently thought of # as a groupoid, where isomorphisms connect points that are to be identified under the local group action.

A path in an orbifold is hence a path in a groupoid: a free combination of ordinary paths with morphisms of the groupoid.

This idea plays a prominent role in the field of orbifold string topology #.

Before learning about it’s use there, I had grown fond of the concept of paths in categories - in the sense of free combinations of ordinary paths in the space of objects combined with morphisms in the category - as a convenient means to express nn-transport on XX in terms of local data: here it is the 2-groupoid

(1)Y =Y [3]Y [2]Y Y^\bullet = Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y

associated to the morphism of domains

(2)p:YX p : Y \to X

inside of which we wish to consider 2-paths.

I made some remarks on how to conceive this and the construction used in orbifold string topology in a decent categorical language in these old notes:

\;\; Paths in Categories

If you don’t know what I have in mind when forming the category of nn-paths inside a smooth nn-category - how one freely generates new morphisms by combining paths with existing morphisms and dividing out certain relations - you can find the relevant diagrams in these notes.

But at the time of writing these notes I didn’t realize the nice universal structure behind this concept. This I now try to discuss in

\;\; Universal Transition of Transport

I think one obtains the 2-category

(3)P 2(Y ) P_2(Y^\bullet)

of 2-paths inside Y Y^\bullet as the universal (P 2(Y)pP 2(X)P_2(Y) \stackrel{p}{\to} P_2(X))-local transition.

(I am freely making use of the terminology of TraTriTra.)

In other words, every pp-local transition of 2-transport uniquely factors through P 2(Y )P_2(Y^\bullet) in a suitable sense.

As a nice corollary of this, we find that the factorization morphism

(4)(tra y,g,f):P 2(Y )T (\mathrm{tra}_y,g,f) : P_2(Y^\bullet) \to T

is the “descending” local trivialized morphism.

To appreciate this, consider the example of pp-local ii-trivializations, with ii being the obvious injection

(5)i:Σ(U(1)1)Σ(1DVect ). i : \Sigma(U(1)\to 1) \to \Sigma(1D\mathrm{Vect}_\mathbb{C}) \,.

Such a pp-local ii-trivialization is a line bundle gerbe #.

The factorization morphism (tra y,g,f):P 2(Y )T (\mathrm{tra}_y,g,f) : P_2(Y^\bullet) \to T associated to this line bundle gerbe is in general not ii-trivial - reflecting the fact that a line bundle gerbe is a collection of transition bundles instead of transition functions.

But for sufficiently well-behaved pp (for instance for any good covering of XX) (tra y,g,f)(\mathrm{tra}_y,g,f) is in fact ii-trivializable. Performing this trivialization of the factorization morphism of the original trivialization amounts to passing from the bundle gerbe to its Deligne 3-cocycle #.

Posted at October 6, 2006 4:19 PM UTC

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6 Comments & 10 Trackbacks

Re: On n-Transport: Universal Transition

In “Paths in Categories” you use the term `Moore path’ without definition - I had a search in the literature and the best I could come up with was some double category stuff from a while back. I know what you’re aiming for, but just for conventions, could you put something into the pdf notes?

Posted by: David Roberts on October 9, 2006 6:22 AM | Permalink | Reply to this

Moore paths

In “Paths in Categories” you use the term ‘Moore path’ without definition

Actually, off the top of my head, I can’t give you a reference. I think I originally learned about Moore paths from John Baez.

It’s a simple idea: you make composition of paths associative not by dividing out by reparameterization, but by gluing the parameter ranges.

So a smooth Moore path in XX is an interval

(1)[0,a] [0,a]

together with a smooth map

(2)γ:[0,a]X \gamma : [0,a] \to X

(such that all derivatives of γ\gamma in a neighbourhood of 0 and of aa, respectively, vanish).

Composition of ([0,a],γ)([0,a],\gamma) with ([0,a],γ)([0,a'],\gamma'), if composable, is then

(3)([0,a+a],σ{γ(σ) (σa) γ(σa) (σa)). \left([0,a+a'], \sigma \mapsto \left\lbrace \array{ \gamma(\sigma) & (\sigma \leq a) \\ \gamma'(\sigma-a) & (\sigma \geq a) } \right. \right) \,.

So Moore paths in XX form a category. But not a groupoid.

Posted by: urs on October 9, 2006 10:22 AM | Permalink | Reply to this

Re: Moore paths

Since there is more than one Moore relevant
to this cafe’, the Moore of Moore paths
is J C Moore

will try to locate the reference

Posted by: jim stasheff on October 13, 2006 3:48 PM | Permalink | Reply to this

Re: Moore paths and others

In addition to path composition end to end,
there is composition when the latter half
of the first is the reverse of the first half of the second which then get cancelled
This is origianlly due to Lashof but reinvented by Witten
It is associative

Japanese physicists (Hata, Itoh, Kugo, Kunitomo, Ogawa) went one step further -
allowing cancellation of any terminal portion of the first with the reverse
intial protion of the second
but !! this is only homotopy associative
but !!! satisfying a strict pentagon

Then there are homotopy relations:
1. as in the fundamental group(oid)
2. thin homotopy
3. directed thin homotopy
whihc produces a category NOT a groupoid


Posted by: jim stasheff on October 13, 2006 4:05 PM | Permalink | Reply to this

Re: Moore paths and others

[…] reinvented by Witten […]

…in his string field theory paper. The diagram is for instance on p. 43 of hep-th/0311017 .

One interesting aspect of this is that it leads to a kind of matrix multiplication for “string fields”.

Let P 1 LW(X)P_1^{LW}(X) the Lashof-Witten category of paths in XX. Write each morphism in terms of its two parts

(1) xγz = x(γ 1,γ 2)z = xγ 1yγ 2z. \begin{aligned} & x \stackrel{\gamma}{\to} z \\ =& x \stackrel{(\gamma_1,\gamma_2)}{\to} z \\ =& x \stackrel{\gamma_1}{\to} y \stackrel{\gamma_2}{\to} z \,. \end{aligned}

Now let CC be some monoidal category and let Σ(C)\Sigma(C) be its suspension, i.e., Hom Σ(C)(,)=C\mathrm{Hom}_{\Sigma(C)}(\bullet,\bullet) = C.

Then what is a lax functor

(2)tra:P 1 LW(X)Σ(C) \mathrm{tra} : P_1^{LW}(X) \to \Sigma(C)


Among other things, it’s an assignment

(3)(γ 1,γ 2)V γ 1,γ 2 (\gamma_1,\gamma_2) \mapsto V_{\gamma_1,\gamma_2}

of objects of CC to pairs of paths in XX, together with a morphism (the compositor)

(4)μ γ 1,γ 2,γ 3:V γ 1,γ 2V γ¯ 2,γ 3V γ 1,γ 3. \mu_{\gamma_1,\gamma_2,\gamma_3} \;:\; V_{\gamma_1,\gamma_2} \otimes V_{\bar \gamma_2,\gamma_3} \to V_{\gamma_1,\gamma_3} \,.

Suppose that CC is in addition abelian. Then we may “integrate out” the path in the middle, to obtain something like

(5) γ 2imμ γ 1,γ 2,γ 3, \oplus_{\gamma_2} \mathrm{im} \mu_{\gamma_1,\gamma_2,\gamma_3} \,,

which looks a little like a product of matrices, whose indices are not just numbers but paths (half-strings).

The **-product in string field theory is commonly interpreted in terms of such a generalized matrix product, for instance in section 5.1.1 of hep-th/0106010.

Posted by: urs on October 13, 2006 4:39 PM | Permalink | Reply to this

Re: Moore paths and others

Since the Witten reference has been given
here is the earlier one by Lashof

MR0082099 (18,497f) Lashof, R. Classification of fibre bundles by the loop space of the base. Ann. of Math. (2) 64 (1956), 436–446.


Posted by: jim stasheff on October 23, 2006 3:35 PM | Permalink | Reply to this
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