nLab
simplicial local system

Simplicial local systems

• 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

$$V=\sum _{p_1,\dots ,p_n\ge 0}{V}^{p_1,\dots ,p_n}$$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 $𝒞$ 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 $𝒞$ is:

1. a family of objects $F_{\sigma }={\sum }_{p\ge 0}{F}_{\sigma }^p$ in $𝒞$ indexed by the simplices $\sigma$ of $K$;

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

$$d_i:F_{\sigma }\to F_{d_i\sigma }\mathrm{and}{s}_i:F_{\sigma }\to F_{s_i\sigma }$$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

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

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

$$(Y,{\mathrm{const}}_K)=\left\{\begin{array}{ccccc}Y({\Delta }^n)& & \stackrel{}{\to }& & Y({\Delta }^{n\prime })\\ & {}_c\searrow & & {\swarrow }_{c\prime }\\ & & K\end{array}\right\}$$(Y, const_K) = \left\{ \array{ Y(\Delta^n) &&\stackrel{}{\to}&& Y(\Delta^{n'}) \\ & {}_c\searrow && \swarrow_{c'} \\ && K } \right\}

where $Y:\Delta \to [{\Delta }^{\mathrm{op}},\mathrm{Set}]$ 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

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

from that category to $𝒞$.

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 $\phi :L\to K$ be a simplicial map and $F$ a local system over $K$. The pullback of $F$ to $L$ (or along $\phi$) is the local system ${\phi }^{*}F$ over $L$ given by

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

If $\phi$ is an inclusion of a simplicial subset then we may say that ${\phi }^{*}F$ is the restriction of $F$ to $L$.

Now let $F$ be a local system on $K$ with values in $𝒞$. 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 ${\Phi }_{\sigma }\in F_{\sigma }^p$ such that for all $\sigma$

$${\Phi }_{d_i\sigma }=d_i({\Phi }_{\sigma })\mathrm{and}{\Phi }_{s_i\sigma }=s_i({\Phi }_{\sigma }).$$\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 $𝒞$ (for instance the tensor unit if $𝒞$ 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 $𝒞$ 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 $𝒞$, 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 $\phi :L\to K$ is a simplicial map, it determines a morphism $F(\phi ):({\phi }^{*}F)(L)\to F(K)$ given by

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

If $\phi$ is an inclusion of $L$ into $K$, then we denote $({\phi }^{*}F)(L)$ 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 ($n\ge 0$) 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 ${\Phi }_{\sigma }\in F_{\sigma }^p$ ( $\sigma \in M_n$), $n\ge 0$, 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 $n\ge 0$). 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

$${\sigma }^{*}(F)(\Delta [n])\to {\sigma }^{*}(F)(\partial \Delta [n])$$\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 $\phi :L\to K$ is a simplicial map and $F$ is an extendable system over $K$, then ${\phi }^{*}F$ 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 $F_0\in 𝒞$, 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 $F_n\in 𝒞$ ($n\ge 0$), $F_{\sigma }=F_n$, 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 $i$,

$$d_i:F_{\sigma }\to F_{d_i\sigma }\mathrm{and}{s}_i:F_{\sigma }\to F_{s_i\sigma }$$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 $𝒞$ is one of the categories with differentials above, and for each $\sigma$, and $i$

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

$${\phi }_{\sigma }:F_{\sigma }\to G_{\sigma },\sigma \in K,$$\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,

compatible with the face and degeneracy operators. Then a morphism $\phi :F(K)\to G(K)$ is given by $(\phi \Phi )\sigma ={\phi }_{\sigma }({\Phi }_{\sigma })$, and

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

is an isomorphism.