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simplicial local system

Simplicial local systems

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

Local Systems in a simplicial context

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

V= p 1,,p n0V p 1,,p nV = \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 nn-graded differential vector spaces has for objects pairs (V,d)(V,d), where VV is an nn-graded vector space, dd is a linear map of total degree 1, and d 2=0d^2 = 0. The morphisms are the linear maps, homogeneous of multidegree zero, which commute with dd.

We will denote by 𝒞\mathcal{C} one of the following categories:

  • nn-graded vector spaces.

  • The category of nn-graded algebras,

  • The subcategory of commutative nn-graded algebras,

  • nn-graded differential vector spaces,

  • The subcategory of nn-graded differential algebras,

  • The subcategory of commutative nn-graded differential algebras.

Urs: How does the nn-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 dd. Almost always we will be restricting ourselves to the case n=1n = 1. Extensions of any results or definitions to the general case are usually routine.

Let KK be a simplicial set. A local system FF on KK with values in 𝒞\mathcal{C} is:

  1. a family of objects F σ= p0F σ pF_\sigma =\sum_{p\geq 0} F^p_\sigma in 𝒞\mathcal{C} indexed by the simplices σ\sigma of KK;

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

d i:F σF d iσands i:F σF s iσd_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:Δ opSet K : \Delta^{op} \to Set

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

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

where Y:Δ[Δ op,Set]Y : \Delta \to [\Delta^{op}, Set] is the Yoneda embedding for which Y(Δ n)Y(\Delta^n) is the standard simplicial nn-simplex, so that c:Y(Δ n)Kc : Y(\Delta^n) \to K is an nn-simplex cK nc \in K_n of the simplicial set nn.

A simplicial local system is then just a functor

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

from that category to 𝒞\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 φ:LK\varphi : L \to K be a simplicial map and FF a local system over KK. The pullback of FF to LL (or along φ\varphi) is the local system φ *F\varphi^*F over LL given by

(φ *F) σ=F φσ;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 φ\varphi is an inclusion of a simplicial subset then we may say that φ *F\varphi^*F is the restriction of FF to LL.

Now let FF be a local system on KK with values in 𝒞\mathcal{C}. Define a graded space F(K)F(K) as follows : an element Φ\Phi of F p(K)F^p(K) is a function which assigns to each simplex σ\sigma of KK an element Φ σF σ p\Phi_\sigma \in F^p_\sigma such that for all σ\sigma

Φ d iσ=d i(Φ σ)andΦ s iσ=s i(Φ σ).\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)F(K) is the space of natural transformations from the simplicial local system constant on the generator (if any) of 𝒞\mathcal{C} (for instance the tensor unit if 𝒞\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 𝒞\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)F(K) becomes an object of 𝒞\mathcal{C}, called the object of global sections of FF.

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 φ:LK\varphi : L \to K is a simplicial map, it determines a morphism F(φ):(φ *F)(L)F(K)F(\varphi) : (\varphi^*F)(L)\to F(K) given by

(F(φ)Φ) σ=Φ φσ.(F(\varphi)\Phi)_\sigma = \Phi_{\varphi\sigma}.

If φ\varphi is an inclusion of LL into KK, then we denote (φ *F)(L)(\varphi^*F)(L) simply by F(L)F(L) and call the morphism F(K)F(L)F(K)\to F(L) restriction.

Now suppose FF is a local system over KK. Assume M nK nM_n \subset K_n are subsets (n0n \geq 0) such that d i:M nM n1d_i : M_n \to M_{n-1} This family {M n}\{M_n\} generates a subsimplicial set LKL\subset K and if s iσM n+1s_i\sigma \in M_{n+1} then σ=d is iσM n\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 Φ σF σ p\Phi_\sigma \in F^p_\sigma ( σM n\sigma \in M_n), n0n \geq 0, satisfy Φ d iσ=d iΦ σ\Phi_{d_i\sigma} = d_i\Phi_\sigma and Φ s iσ=s iΦ σ\Phi_{s_i\sigma} = s_i\Phi_\sigma (this is with s iσM ns_i\sigma\in M_n, and n0n\geq 0). Then there is a unique element ΦF p(L)\Phi\in F^p(L) extending Φ σ\Phi_\sigma.

The proof is by induction and can be found in Halperin’s notes if required.

For any simplicial set KK, any nn-simplex σK n\sigma \in K_n determines a unique simplicial map, which we will also write as σ\sigma from Δ[n]\Delta[n] to KK that sends the unique non-degenerate nn-simplex of the standard nn-simplex Δ[n]\Delta[n] to the element σ\sigma. In particular, if FF is a local system over KK, then we can form σ *F\sigma^*F over Δ[n]\Delta[n]. We will say that FF is extendable if for each σ\sigma the restriction

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

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

Proposition

Suppose φ:LK\varphi :L \to K is a simplicial map and FF is an extendable system over KK, then φ *F\varphi^*F is an extendable local system over LL.

The proof is easy.

Proposition

Suppose that LKL\subset K is a subsimplicial set and FF is an extendable local system over KK. Then the restriction morphism F(K)F(L)F(K)\to F(L) is surjective.

The proof is again by induction up the skeleta of KK and LL, for details see Halperin, p.XII 10.

If FF is an extendable local system over KK and LKL\subset K, we denote the kernel of F(K)F(L)F(K)\to F(L) by F(K,L)F(K,L) and call it the space of relative global sections. (A description of F(K,L)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 FF over KK is constant if for some F 0𝒞F_0 \in \mathcal{C}, each F σ=F 0F_\sigma = F_0 and each d id_i and s js_j is the identity map on F 0F_0. We say FF is constant by dimension if for some sequence F n𝒞F_n\in \mathcal{C} (n0n \geq 0), F σ=F nF_\sigma = F_n, for σK n\sigma \in K_n and d id_i, s js_j depend only on dimσ\dim \sigma.

A local system FF over KK is a local system of coefficients if for each σ\sigma and each ii,

d i:F σF d iσands i:F σF s iσd_i : F_\sigma \to F_{d_i\sigma} \quad and s_i : F_\sigma \to F_{s_i \sigma}

are isomorphisms. Finally FF is a local system of differential coefficients if 𝒞\mathcal{C} is one of the categories with differentials above, and for each σ\sigma, and ii

d i *:H(F σ)H(F d iσ)ands i *:H(F σ)H(F s iσ)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 FF and GG be extendable local systems of differential coefficients over KK. Assume we are given morphisms

φ σ:F σG σ,σK,\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,

compatible with the face and degeneracy operators. Then a morphism φ:F(K)G(K)\varphi : F(K)\to G(K) is given by (φΦ)σ=φ σ(Φ σ)(\varphi\Phi)\sigma = \varphi_\sigma (\Phi_\sigma), and

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

is an isomorphism.

References

  • S. Halperin, Lectures on minimal models, Mémoires de la S. M. F. 2e série, tome 9-10 (1983), p. 1-261 (eduml, pdf)
Revised on March 21, 2014 14:23:12 by Igor Khavkine (37.117.178.75)