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 $n$-graded spaces, we mean the category whose objects are the $n$-graded vector spaces
and whose morphisms are the linear maps, homogeneous of multidegree zero.
The category of $n$-graded differential vector spaces has for objects pairs $(V,d)$, where $V$ is an $n$-graded vector space, $d$ is a linear map of total degree 1, and $d^2 = 0$. The morphisms are the linear maps, homogeneous of multidegree zero, which commute with $d$.
We will denote by $\mathcal{C}$ one of the following categories:
$n$-graded vector spaces.
The category of $n$-graded algebras,
The subcategory of commutative $n$-graded algebras,
$n$-graded differential vector spaces,
The subcategory of $n$-graded differential algebras,
The subcategory of commutative $n$-graded differential algebras.
Urs: How does the $n$-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 $d$. Almost always we will be restricting ourselves to the case $n = 1$. Extensions of any results or definitions to the general case are usually routine.
Let $K$ be a simplicial set. A local system $F$ on $K$ with values in $\mathcal{C}$ is:
a family of objects $F_\sigma =\sum_{p\geq 0} F^p_\sigma$ in $\mathcal{C}$ indexed by the simplices $\sigma$ of $K$;
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:
regarding the simplicial set as a functor
on the simplex category, its category of cells is the comma category
where $Y : \Delta \to [\Delta^{op}, Set]$ is the Yoneda embedding for which $Y(\Delta^n)$ is the standard simplicial $n$-simplex, so that $c : Y(\Delta^n) \to K$ is an $n$-simplex $c \in K_n$ of the simplicial set $n$.
A simplicial local system is then just a functor
from that category to $\mathcal{C}$.
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 $\varphi : L \to K$ be a simplicial map and $F$ a local system over $K$. The pullback of $F$ to $L$ (or along $\varphi$) is the local system $\varphi^*F$ over $L$ given by
If $\varphi$ is an inclusion of a simplicial subset then we may say that $\varphi^*F$ is the restriction of $F$ to $L$.
Now let $F$ be a local system on $K$ with values in $\mathcal{C}$. Define a graded space $F(K)$ as follows : an element $\Phi$ of $F^p(K)$ is a function which assigns to each simplex $\sigma$ of $K$ an element $\Phi_\sigma \in F^p_\sigma$ such that for all $\sigma$
Urs: Do I understand correctly that when the simplicial local system is expressed as a functor, then $F(K)$ is the space of natural transformations from the simplicial local system constant on the generator (if any) of $\mathcal{C}$ (for instance the tensor unit if $\mathcal{C}$ 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 $\mathcal{C}$ 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 $F(K)$ becomes an object of $\mathcal{C}$, called the object of global sections of $F$.
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 $\varphi : L \to K$ is a simplicial map, it determines a morphism $F(\varphi) : (\varphi^*F)(L)\to F(K)$ given by
If $\varphi$ is an inclusion of $L$ into $K$, then we denote $(\varphi^*F)(L)$ simply by $F(L)$ and call the morphism $F(K)\to F(L)$ restriction.
Now suppose $F$ is a local system over $K$. Assume $M_n \subset K_n$ are subsets ($n \geq 0$) such that $d_i : M_n \to M_{n-1}$ This family $\{M_n\}$ generates a subsimplicial set $L\subset K$ and if $s_i\sigma \in M_{n+1}$ then $\sigma = d_i s_i\sigma \in M_n$.
Urs: So what are simplicial local systems used for? Is there a good motivating example? Relating it to the other definition of local system, maybe?
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 $\Phi_\sigma \in F^p_\sigma$ ( $\sigma \in M_n$), $n \geq 0$, satisfy $\Phi_{d_i\sigma} = d_i\Phi_\sigma$ and $\Phi_{s_i\sigma} = s_i\Phi_\sigma$ (this is with $s_i\sigma\in M_n$, and $n\geq 0$). Then there is a unique element $\Phi\in F^p(L)$ extending $\Phi_\sigma$.
The proof is by induction and can be found in Halperin’s notes if required.
For any simplicial set $K$, any $n$-simplex $\sigma \in K_n$ determines a unique simplicial map, which we will also write as $\sigma$ from $\Delta[n]$ to $K$ that sends the unique non-degenerate $n$-simplex of the standard $n$-simplex $\Delta[n]$ to the element $\sigma$. In particular, if $F$ is a local system over $K$, then we can form $\sigma^*F$ over $\Delta[n]$. We will say that $F$ is extendable if for each $\sigma$ the restriction
is surjective, where $\partial\Delta[n]$ is the boundary of the $n$-simplex.
Suppose $\varphi :L \to K$ is a simplicial map and $F$ is an extendable system over $K$, then $\varphi^*F$ is an extendable local system over $L$.
The proof is easy.
Suppose that $L\subset K$ is a subsimplicial set and $F$ is an extendable local system over $K$. Then the restriction morphism $F(K)\to F(L)$ is surjective.
The proof is again by induction up the skeleta of $K$ and $L$, for details see Halperin, p.XII 10.
If $F$ is an extendable local system over $K$ and $L\subset K$, we denote the kernel of $F(K)\to F(L)$ by $F(K,L)$ and call it the space of relative global sections. (A description of $F(K,L)$ 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 $F$ over $K$ is constant if for some $F_0 \in \mathcal{C}$, each $F_\sigma = F_0$ and each $d_i$ and $s_j$ is the identity map on $F_0$. We say $F$ is constant by dimension if for some sequence $F_n\in \mathcal{C}$ ($n \geq 0$), $F_\sigma = F_n$, for $\sigma \in K_n$ and $d_i$, $s_j$ depend only on $\dim \sigma$.
A local system $F$ over $K$ is a local system of coefficients if for each $\sigma$ and each $i$,
are isomorphisms. Finally $F$ is a local system of differential coefficients if $\mathcal{C}$ is one of the categories with differentials above, and for each $\sigma$, and $i$
are isomorphisms, in other words if the corresponding cohomology is a local system of coefficients.
Let $F$ and $G$ be extendable local systems of differential coefficients over $K$. Assume we are given morphisms
compatible with the face and degeneracy operators. Then a morphism $\varphi : F(K)\to G(K)$ is given by $(\varphi\Phi)\sigma = \varphi_\sigma (\Phi_\sigma)$, and
is an isomorphism.