differential graded coalgebra


Differential-graded objects

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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


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.


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')


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).


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).


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.


As filtered colimits of finite-dimensional pieces


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.


The model structure on dg-coalgebras is due to

Revised on March 8, 2017 12:20:42 by Urs Schreiber (