NB. There is an entry at local systems together with a blog link to David Speyer: Three ways of looking at a local system
Here we will concentrate on the combinatorial and simplicial version of local systems.
By the category of -graded spaces, we mean the category whose objects are the -graded vector spaces
and whose morphisms are the linear maps, homogeneous of multidegree zero.
The category of -graded differential vector spaces has for objects pairs , where is an -graded vector space, is a linear map of total degree 1, and . The morphisms are the linear maps, homogeneous of multidegree zero, which commute with .
We will denote by one of the following categories:
-graded vector spaces.
The category of -graded algebras,
The subcategory of commutative -graded algebras,
-graded differential vector spaces,
The subcategory of -graded differential algebras,
The subcategory of commutative -graded differential algebras.
Urs: How does the -grading affect the nature of the following definition? It seems that chain homotopies are not used in the following, just the 1-categorical structure?
In the ‘differential’ examples, the differential will usually be denoted . Almost always we will be restricting ourselves to the case . Extensions of any results or definitions to the general case are usually routine.
Let be a simplicial set. A local system on with values in is:
a family of objects in indexed by the simplices of ;
a family of morphisms (called the face and degeneracy operators)
satisfying the simplicial identities.
Here we will often just refer to ‘local system’ rather than the fuller ‘simplicial local system’, if no confusion will be likely to result.
There is an obvious way of assigning a small category to a simplicial set in which the simplices are the objects and the face and degeneracy maps generate the morphisms:
A simplicial local system is then just a functor
from that category to .
Urs: Here it says “a local system”. I suppose “simplicial local system” is meant? We should have a discussion about how this notion of simplicial local system relates to the functors from fundamental groupoids discussed at local system.
Tim: That has been amended! Halperin just calls them ‘local systems’, so in the notes that were the basis for this so did I. I copied and pasted from them, so this slip may occur elsewhere as well.
Let be a simplicial map and a local system over . The pullback of to (or along ) is the local system over given by
If is an inclusion of a simplicial subset then we may say that is the restriction of to .
Now let be a local system on with values in . Define a graded space as follows : an element of is a function which assigns to each simplex of an element such that for all
Urs: Do I understand correctly that when the simplicial local system is expressed as a functor, then is the space of natural transformations from the simplicial local system constant on the generator (if any) of (for instance the tensor unit if is graded vector spaces).
For ordinary local systems this gives the flat sections.
Tim: I’m not sure.
The linear structure is the obvious one, defined ‘componentwise’ and if is one of the algebra (resp. differential) variants of the generic receiving category then the multiplication (resp. the differential) is defined componentwise as well. In this way becomes an object of , called the object of global sections of .
Tim: This construction also has (I think) a neat categorical description, that will be worth investigating. It would seem to be the analogue of the Grothendieck construction / homotopy colimit (at least partially) in this context. (enlightenment sought!!!)
If is a simplicial map, it determines a morphism given by
If is an inclusion of into , then we denote simply by and call the morphism restriction.
Now suppose is a local system over . Assume are subsets () such that This family generates a subsimplicial set and if then .
Tim: Aha! All will be revealed in the next entry ‘Differential forms on a simplicial set’ … when I get to putting it in! There is some more to go here as well, describing special properties, but it was getting late last night.
Suppose ( ), , satisfy and (this is with , and ). Then there is a unique element extending .
The proof is by induction and can be found in Halperin’s notes if required.
For any simplicial set , any -simplex determines a unique simplicial map, which we will also write as from to that sends the unique non-degenerate -simplex of the standard -simplex to the element . In particular, if is a local system over , then we can form over . We will say that is extendable if for each the restriction
is surjective, where is the boundary of the -simplex.
Suppose is a simplicial map and is an extendable system over , then is an extendable local system over .
The proof is easy.
Suppose that is a subsimplicial set and is an extendable local system over . Then the restriction morphism is surjective.
The proof is again by induction up the skeleta of and , for details see Halperin, p.XII 10.
If is an extendable local system over and , we denote the kernel of by and call it the space of relative global sections. (A description of is given in detail in Halperin, p.XII-12.)
It may be useful to have some more of the terminology of local systems available. A local system over is constant if for some , each and each and is the identity map on . We say is constant by dimension if for some sequence (), , for and , depend only on .
A local system over is a local system of coefficients if for each and each ,
are isomorphisms. Finally is a local system of differential coefficients if is one of the categories with differentials above, and for each , and
are isomorphisms, in other words if the corresponding cohomology is a local system of coefficients.
Let and be extendable local systems of differential coefficients over . Assume we are given morphisms
compatible with the face and degeneracy operators. Then a morphism is given by , and
is an isomorphism.