nLab simplicial local system

Simplicial local system

Definition

A simplicial local system on a simplicial set $X$ with coefficients in a category $C$ (like abelian groups, for example) is a functor

$$\pi_{\le1}(X)\to C,$$

where $\pi_{\le1}(X)$ is (one of the models for) the simplicial fundamental groupoid of $X$.

References

• NB. There is an entry at local systems together with a blog link to David Speyer: Three ways of looking at a local system

  • Introduction and connection to cohomology theories

Here we will concentrate on the combinatorial and simplicial version of local systems.

Local Systems in a simplicial context

By the category of $n$-graded spaces, we mean the category whose objects are the $n$-graded vector spaces

$$V = \sum_{p_1,\ldots,p_n\geq0}V^{p_1,\ldots,p_n}$$

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:

1. a family of objects $F_\sigma =\sum_{p\geq 0} F^p_\sigma$ in $\mathcal{C}$ indexed by the simplices $\sigma$ of $K$;

2. a family of morphisms (called the face and degeneracy operators)

$$d_i :F_\sigma \to F_{d_i\sigma} \quad and\quad s_i : F_\sigma \to F_{s_i\sigma}$$

satisfying the simplicial identities.

Remarks

• 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

$$K : \Delta^{op} \to Set$$

on the simplex category, its category of cells is the comma category

$$(Y, const_K) = \left\{ \array{ Y(\Delta^n) &&\stackrel{}{\to}&& Y(\Delta^{n'}) \\ & {}_c\searrow && \swarrow_{c'} \\ && K } \right\}$$

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

$$F : (Y,const_K) \to \mathcal{C}$$

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.

Back to discussion

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

$$(\varphi^*F)_\sigma = F_{\varphi\sigma} ; \quad d_i = d_i ; \quad s_i = s_i.$$

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 $\sigma$

$$\Phi_{d_i\sigma} = d_i(\Phi_\sigma) \quad and \quad \Phi_{s_i\sigma} = s_i(\Phi_\sigma).$$

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

$$(F(\varphi)\Phi)_\sigma = \Phi_{\varphi\sigma}.$$

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.

Lemma

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

$$\sigma^*(F)(\Delta[n]) \to \sigma^*(F)(\partial\Delta[n])$$

is surjective, where $\partial\Delta[n]$ is the boundary of the $n$-simplex.

Proposition

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.

Proposition

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 $i$,

$$d_i : F_\sigma \to F_{d_i\sigma} \quad and s_i : F_\sigma \to F_{s_i \sigma}$$

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 $i$

$$d_i^* : H(F_\sigma) \to H(F_{d_i\sigma}) \quad and s_i^* : H(F_\sigma) \to H(F_{s_i\sigma})$$

are isomorphisms, in other words if the corresponding cohomology is a local system of coefficients.

Theorem

Let $F$ and $G$ be extendable local systems of differential coefficients over $K$. Assume we are given morphisms

$$\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,$$

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

$$\varphi^* : H(F(K))\to H(G(K))$$

is an isomorphism.

See also

• local system

• twisted cohomology

References

• S. Halperin, Lectures on minimal models, Mémoires de la S. M. F. 2e série, tome 9-10 (1983), p. 1-261 (eduml, pdf)