nLab simplicial local system

and

rational homotopy theory

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

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 $\left(V,d\right)$, 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)

${d}_{i}:{F}_{\sigma }\to {F}_{{d}_{i}\sigma }\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}{s}_{i}:{F}_{\sigma }\to {F}_{{s}_{i}\sigma }$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

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

where $Y:\Delta \to \left[{\Delta }^{\mathrm{op}},\mathrm{Set}\right]$ is the Yoneda embedding for which $Y\left({\Delta }^{n}\right)$ is the standard simplicial $n$-simplex, so that $c:Y\left({\Delta }^{n}\right)\to K$ is an $n$-simplex $c\in {K}_{n}$ of the simplicial set $n$.

A simplicial local system is then just a functor

$F:\left(Y,{\mathrm{const}}_{K}\right)\to 𝒞$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

$\left({\phi }^{*}F{\right)}_{\sigma }={F}_{\phi \sigma };\phantom{\rule{1em}{0ex}}{d}_{i}={d}_{i};\phantom{\rule{1em}{0ex}}{s}_{i}={s}_{i}.$(\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 $F\left(K\right)$ as follows : an element $\Phi$ of ${F}^{p}\left(K\right)$ is a function which assigns to each simplex $\sigma$ of $K$ an element ${\Phi }_{\sigma }\in {F}_{\sigma }^{p}$ such that for all $\sigma$

${\Phi }_{{d}_{i}\sigma }={d}_{i}\left({\Phi }_{\sigma }\right)\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}{\Phi }_{{s}_{i}\sigma }={s}_{i}\left({\Phi }_{\sigma }\right).$\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 $F\left(K\right)$ 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 $F\left(K\right)$ 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 $F\left(\phi \right):\left({\phi }^{*}F\right)\left(L\right)\to F\left(K\right)$ given by

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

If $\phi$ is an inclusion of $L$ into $K$, then we denote $\left({\phi }^{*}F\right)\left(L\right)$ simply by $F\left(L\right)$ and call the morphism $F\left(K\right)\to F\left(L\right)$ 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 $\left\{{M}_{n}\right\}$ 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 $\Phi \in {F}^{p}\left(L\right)$ 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 $\Delta \left[n\right]$ to $K$ that sends the unique non-degenerate $n$-simplex of the standard $n$-simplex $\Delta \left[n\right]$ to the element $\sigma$. In particular, if $F$ is a local system over $K$, then we can form ${\sigma }^{*}F$ over $\Delta \left[n\right]$. We will say that $F$ is extendable if for each $\sigma$ the restriction

${\sigma }^{*}\left(F\right)\left(\Delta \left[n\right]\right)\to {\sigma }^{*}\left(F\right)\left(\partial \Delta \left[n\right]\right)$\sigma^*(F)(\Delta[n]) \to \sigma^*(F)(\partial\Delta[n])

is surjective, where $\partial \Delta \left[n\right]$ 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 $F\left(K\right)\to F\left(L\right)$ 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 $F\left(K\right)\to F\left(L\right)$ by $F\left(K,L\right)$ and call it the space of relative global sections. (A description of $F\left(K,L\right)$ 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 $\mathrm{dim}\sigma$.

A local system $F$ over $K$ is a local system of coefficients if for each $\sigma$ and each $i$,

${d}_{i}:{F}_{\sigma }\to {F}_{{d}_{i}\sigma }\phantom{\rule{1em}{0ex}}\mathrm{and}{s}_{i}:{F}_{\sigma }\to {F}_{{s}_{i}\sigma }$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 $i$

${d}_{i}^{*}:H\left({F}_{\sigma }\right)\to H\left({F}_{{d}_{i}\sigma }\right)\phantom{\rule{1em}{0ex}}\mathrm{and}{s}_{i}^{*}:H\left({F}_{\sigma }\right)\to H\left({F}_{{s}_{i}\sigma }\right)$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

${\phi }_{\sigma }:{F}_{\sigma }\to {G}_{\sigma },\phantom{\rule{1em}{0ex}}\sigma \in K,$\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,

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

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

is an isomorphism.

Revised on May 29, 2011 06:34:50 by Tim Porter (95.147.238.52)