nLab differential graded coalgebra

Contents

Context

Differential-graded objects

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Definition

Abstract definition

A dg-coalgebra is a comonoid in the category of chain complexes.

Equivalently, this is a graded coalgebra CC equipped with a coderivation

D:CC D : C \to C

that is of degree -1 and squares to 0,

D 2=0. D^2 = 0 \,.

Detailed component definition

Pre-coalgebra

A pre-graded coalgebra (pre-gc), (C,Δ,ε)(C,\Delta, \varepsilon), is a pre-gvs CC together with linear maps of degree 0

Δ:CCC,ε:Ck,\Delta: C\to C\otimes C, \varepsilon : C\to k,

such that the obvious (usual) diagrams commute. (When there is no ambiguity, we may write CC instead of (C,Δ,ε)(C,\Delta, \varepsilon).)

The field kk is a coalgebra for the canonical isomorphism kkkk\to k\otimes k with ε=id k\varepsilon = id_k.

A morphism ψ:CC\psi : C \to C' of pre-gcs is a linear mapping of degree zero such that

(ψψ)Δ=Δψandε=εψ.(\psi \otimes \psi)\circ \Delta = \Delta' \circ \psi and \varepsilon = \varepsilon' \circ \psi.

The linear counit mapping ε:Ck\varepsilon :C \to k is always a morphism of pre-gcs.

Coaugmentations

A coaugmentation of a pre-gc is a morphism η:kC\eta : k \to C. We will write 11 for η(1)\eta(1).

The cokernel C¯\bar{C} of η\eta can be identified with KerεKer \varepsilon and so can be considered as a subspace of CC.

The reduced diagonal Δ¯:C¯C¯C¯\bar{\Delta} : \bar{C} \to \bar{C}\otimes \bar{C}, induced by Δ\Delta is defined by Δx=1x+x1+Δ¯x\Delta x = 1\otimes x + x\otimes 1 + \bar{\Delta }x. The vector space of primitives of CC, denoted P(C)P(C), is the kernel of the reduced diagonal.

A morphism of coaugmented pre-gcs, ψ:(C,η)(C,η)\psi : (C,\eta)\to (C',\eta'), is a morphism of the pre-gcs which satisfies η=ψη\eta' = \psi \circ \eta. It preserves primitives.

The commutation morphism

Let VV and VV' be two pre-gvs. The commutation morphism

τ:VVVV\tau : V\otimes V' \to V'\otimes V

is defined by τ(vv)=(1) |v||v|vv\tau( v\otimes v') = (-1)^{|v||v'|} v'\otimes v, on homogeneous elements.

Tensor product of pre-gcs.

Let (C,Δ,ε)(C,\Delta, \varepsilon) and (C,Δ,ε)(C',\Delta', \varepsilon') be two pre-graded coalgebras. The mappings

CCΔΔCCCCCτC(CC)(CC)C\otimes C'\stackrel{\Delta\otimes \Delta'}{\to}C\otimes C\otimes C' \otimes C'\stackrel{C\otimes \tau \otimes C}{\to}(C\otimes C')\otimes (C\otimes C')

and

CCεεkkkC\otimes C'\stackrel{\varepsilon\otimes \varepsilon'}{\to}k \otimes k\stackrel{\cong}{\to}k

give CCC\otimes C' a pre-gc structure.

If η\eta and η\eta' are coaugmentations of CC and CC' respectively, then ηη\eta\otimes\eta' defines a coaugmentation of CCC\otimes C'.

Coderivations of pre-graded coalgebras

Tim: These are called derivations by some sources, but I think that they are the coderivations of other workers. (to be checked)

If CC is a pre-gc, a coderivation of degree pp\in \mathbb{Z}, is a linear map θHom p(C,C)\theta \in Hom_p(C,C) such that

Δθ=(θid C+τ(θid C)τ)Δ,andεθ=0.\Delta \circ \theta = (\theta \otimes id_C + \tau \circ(\theta \otimes id_C)\circ \tau)\circ \Delta, and \varepsilon\circ \theta = 0.

A coderivation θ\theta of a coaugmented pre-gc (C,η)(C,\eta) is a coderivation of CC such that θη=0\theta\circ \eta = 0.

Differential graded coalgebras

A differential \partial on a (coaugmented) pre-gc, CC, is a coderivation of degree -1 such that =0\partial\circ\partial = 0.

The pair, (C,)(C, \partial) is called a differential (coaugmented) pre-graded coalgebra (pre-dgc). Its homology H(C,)H(C,\partial) will be a pre-gc.

If (C,)(C,\partial) and (C,)(C',\partial') are two pre-dgcs, then their tensor product (C,)(C,)(C,\partial)\otimes(C',\partial') is a pre-gdgc with the structures defined earlier.

A morphism of (coaugmented) pre-dgcs is a morphism both of (coaugmented) pre-gcs and of pre-dgvs. We denote the resulting categories by preDGCpre DGC (resp. preηDGCpre \eta DGC).

Cocommutative pre-dgcs

A pre-gc CC is cocommutative if τΔ=Δ\tau\circ\Delta = \Delta, similarly for a pre-dgc. The subcategories of cocommutative d.g. coalgebras will be denoted preCDGCpre CDGC (resp. preηCDGCpre \eta CDGC).

CDGC

A cocommutative differential graded coalgebra is a pre-cdgc on a graded vector space of lower grading (so C p=0C_p = 0 for p<0p \lt 0). This gives categories CDGCCDGC (resp. ηCDGC\eta CDGC).

nn-connected η\eta cdgcs

A coaugmented cdgc (C,)(C, \partial) is nn-connected if C¯ p=0\bar{C}_p = 0 for pnp\leq n.

This gives a category CDGC nCDGC_n. Any connected (i.e. 00-connected) cdgc is canonically coaugmented with C¯\bar{C} coinciding with C +C_+.

Hom-algebras and duals

Let (C,)(C,\partial) be a pre-cdgc and (A,d)(A,d) a pre-cdga. The pre-dgvs (Hom(C,A),D)(Hom(C,A),D) is a pre-cdga for the usual differential and the multiplication f.g=μ(fg)Δf.g = \mu\circ (f\otimes g)\circ \Delta,

CΔCC(fg)AAμA, C\stackrel{\Delta}{\to}C\otimes C\stackrel{(f\otimes g)}{\to}A\otimes A\stackrel{\mu}{\to} A,

for f,gHom(C,A)f,g \in Hom(C,A).

In particular #(C,)=(Hom(C,k),D)\#(C,\partial)= (Hom(C,k),D) defines a functor from preCDGCpre CDGC to preCDGApre CDGA, which commutes with homology and is such that #CDGC nCDGA n\# CDGC_n \subseteq CDGA^n. Conversely, if (A,d)(A,d) is a pre-cdga of finite type, #(A,d)\#(A,d) is a pre-cdgc.

Coalgebra filtrations

Let (C,)(C,\partial) be a pre-dgc. A coalgebra filtration (resp. differential coalgebra filratation) of (C,)(C,\partial) is a family of subspaces F pCF_p C, pp\in \mathbb{Z} such that

F pCF p+1C,ΔF pC kF kCF pkC,(resp.andF pCF pC).F_p C\subseteq F_{p+1} C, \quad \Delta F_p C \subseteq \sum_k F_k C\otimes F_{p-k} C, \quad (resp.\quad and \quad \partial F_p C\subseteq F_p C).

Filtrations of the primitives

Let (C,η)(C,\eta) be a coaugmented pre-gc, C¯\bar{C} the cokernel of η\eta, Δ¯\bar{\Delta}, the reduced diagonal.

The iteration of Δ¯\bar{\Delta} is defined by

Δ¯ 1=Δ¯;Δ¯ p=(Δ¯C¯C¯)Δ¯ (p1).\bar{\Delta}^1 = \bar{\Delta}; \quad \bar{\Delta}^p = (\bar{\Delta}\otimes \bar{C} \otimes \ldots \bar{C}) \otimes \bar{\Delta}^{(p-1)}.

The (increasing) filtration of the primitives is F pC=KerΔ¯ pF_p C = Ker\bar{\Delta}^p, p1p\geq 1. It is a graded coalgebra filtration.

If (C,,η)(C,\partial, \eta) is a coaugmented pre-dgc, each F pCF_p C is stable under the differential and, in particular, F 1=P(C)F_1 = P(C). PP thus defines a functor from preηCDGCpre \eta CDGC to preDGVSpre DGVS.

Let μ\mu be the comultiplication of the pre-ga #C\# C, the dual of CC. Elementary results on duality show, for finite type: Imμ¯ pIm\bar{\mu}^p is the orthogonal complement of KerΔ¯ pKer\bar{\Delta}^p, so, in particular, Q(#C)=#P(C)Q(\# C) =\# P(C).

Let (C,η)(C,\eta) be a coaugmented pre-gc and F pCF_p C the filtration of its primitives. CC is conilpotent if C= kF kCC = \bigcup_k F_k C. A connected coalgebra is conilpotent and conilpotency is preserved by tensor product.

The pre-gc structure on T(V)T(V)

We will denote by T(V)T'(V), the gvs T(V)T(V), together with the coalgebra structure in which the reduced diagonal is given by

Δ¯(v 1v n)= p=1 n1(v 1v p)(v p+1v n).\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n).

The counit and the coaugmentation are the natural mappings T(V)kT(V)\to k and kT(V)k\to T(V), respectively.

The coalgebra T(V)T' (V) is non-commutative if dimV>1dim V\gt 1 and has VV as its vector space of primitives.

If CC is a conilpotent pre-gc, then any morphism f:CVf : C\to V of pre-gvs for which f(1)=0f(1) = 0, admits a unique lifting to a pre-gc morphism f^:CT(V)\hat{f}:C \to T'(V).

Shuffles

A (p,q)(p,q)-shuffle σ\sigma is a permutation of {1,,p+q}\{1, \ldots, p+q\} such that

σ(i)<σ(j)if1i<jporp+1i<jp+q.\sigma(i) \lt \sigma(j) \quad if \quad 1\leq i\lt j\leq p \quad or \quad p+1 \leq i \lt j\leq p+q.

The pre-cgc structure on V\bigwedge V

We will denote V\bigwedge' V, the gvs V\bigwedge V together with the coalgebra structure in which the reduced diagonal is given by

Δ¯(v 1v n)= p=1 n1 σε(σ)(v σ(1)v σ(p))(v σ(p+1)v σ(n)),\bar{\Delta}(v_1\wedge \ldots \wedge v_n) = \sum_{p=1}^{n-1} \sum_\sigma\varepsilon(\sigma)(v_{\sigma(1)}\wedge \ldots \wedge v_{\sigma(p)})\otimes(v_{\sigma(p+1)}\wedge \ldots \wedge v_{\sigma(n)}),

in which the second sum is over all (p,np)(p,n-p)-shuffles and ε(σ)\varepsilon(\sigma) is the Koszul sign of σ\sigma.

The counit and coaugmentation are the natural mappings Vk\bigwedge V \to k, and kVk\to \bigwedge V respectively.

If CC is a conilpotent pre-cgc, any pre-gvs morphism f:CVf:C \to V for which f(1)=0f(1) = 0 admits a unique lifting to a pre-cgc morphism f^:CV\hat{f} : C \to \bigwedge' V.

There is an injective homomorphism of pre-gcs

χ:VT(V)\chi : \bigwedge{\!}' V \to T'(V)

given by

χ(x 1x n)= νε(ν)x ν(1)x ν(n),\chi(x_1\wedge \ldots x_n) = \sum_\nu \varepsilon(\nu)x_{\nu(1)}\otimes \ldots \otimes x_{\nu(n)},

where the sum is over all permutations and ε(σ)\varepsilon(\sigma) is the corresponding Koszul sign. It has, as image, all the symmetric tensors (in the graded sense).

On V\bigwedge' V and T(V)T'(V), the filtration of the primitives comes from a gradation

F pV= pV= kp kV;F_p\bigwedge{\!}' V = \bigwedge^{\leq p}V = \bigoplus_{k\leq p}\bigwedge^k V;
F pT(V)=T p(V)= kpT k(V).F_p T'(V) = T^{\leq p}(V) = \bigoplus_{k\leq p} T^k(V).

Differential graded algebras

Dually, a dg-algebra is a monoid in chain complexes.

Semifree dg-coalgebras and L L_\infty-algebras

The notion of dg-coalgebra whose underlying coalgebra is cofree is related by duality to that of semifree dga.

Semicofree dg-coalgebras concentrated in negative degree and with differential of degree -1 are the same as L-∞-algebras

Model category structure

There is a model structure on dg-coalgebras.

Properties

As filtered colimits of finite-dimensional pieces

Proposition

Every dg-coalgebra is the filtered colimit of its finite-dimensional sub-dg-coalgebras.

This is due to (Getzler-Goerss 99), in generalization to the analogous fact for plain coalgebras, see at coalgebra – As filtered colimits.

See also at L-infinity algebra the section Ind-Conilpotency. This plays a role for instance for constructing model structures for L-infinity algebras, see there.

References

The model structure on dg-coalgebras is due to

Last revised on March 8, 2017 at 17:20:42. See the history of this page for a list of all contributions to it.