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\section*{differential graded coalgebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differentialgraded_objects}{}\paragraph*{{Differential-graded objects}}\label{differentialgraded_objects}

[[!include differential graded objects - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{abstract_definition}{Abstract definition}\dotfill \pageref*{abstract_definition} \linebreak
\noindent\hyperlink{detailed_component_definition}{Detailed component definition}\dotfill \pageref*{detailed_component_definition} \linebreak
\noindent\hyperlink{precoalgebra}{Pre-coalgebra}\dotfill \pageref*{precoalgebra} \linebreak
\noindent\hyperlink{coaugmentations}{Coaugmentations}\dotfill \pageref*{coaugmentations} \linebreak
\noindent\hyperlink{the_commutation_morphism}{The commutation morphism}\dotfill \pageref*{the_commutation_morphism} \linebreak
\noindent\hyperlink{tensor_product_of_pregcs}{Tensor product of pre-gcs.}\dotfill \pageref*{tensor_product_of_pregcs} \linebreak
\noindent\hyperlink{coderivations_of_pregraded_coalgebras}{Coderivations of pre-graded coalgebras}\dotfill \pageref*{coderivations_of_pregraded_coalgebras} \linebreak
\noindent\hyperlink{differential_graded_coalgebras}{Differential graded coalgebras}\dotfill \pageref*{differential_graded_coalgebras} \linebreak
\noindent\hyperlink{cocommutative_predgcs}{Cocommutative pre-dgcs}\dotfill \pageref*{cocommutative_predgcs} \linebreak
\noindent\hyperlink{cdgc}{CDGC}\dotfill \pageref*{cdgc} \linebreak
\noindent\hyperlink{connected__cdgcs}{$n$-connected $\eta$ cdgcs}\dotfill \pageref*{connected__cdgcs} \linebreak
\noindent\hyperlink{homalgebras_and_duals}{Hom-algebras and duals}\dotfill \pageref*{homalgebras_and_duals} \linebreak
\noindent\hyperlink{coalgebra_filtrations}{Coalgebra filtrations}\dotfill \pageref*{coalgebra_filtrations} \linebreak
\noindent\hyperlink{filtrations_of_the_primitives}{Filtrations of the primitives}\dotfill \pageref*{filtrations_of_the_primitives} \linebreak
\noindent\hyperlink{the_pregc_structure_on_}{The pre-gc structure on $T(V)$}\dotfill \pageref*{the_pregc_structure_on_} \linebreak
\noindent\hyperlink{shuffles}{Shuffles}\dotfill \pageref*{shuffles} \linebreak
\noindent\hyperlink{the_precgc_structure_on_}{The pre-cgc structure on $\bigwedge V$}\dotfill \pageref*{the_precgc_structure_on_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{differential_graded_algebras}{Differential graded algebras}\dotfill \pageref*{differential_graded_algebras} \linebreak
\noindent\hyperlink{semifree_dgcoalgebras_and_algebras}{Semifree dg-coalgebras and $L_\infty$-algebras}\dotfill \pageref*{semifree_dgcoalgebras_and_algebras} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsFilteredColimits}{As filtered colimits of finite-dimensional pieces}\dotfill \pageref*{AsFilteredColimits} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{abstract_definition}{}\subsubsection*{{Abstract definition}}\label{abstract_definition}

A \emph{dg-coalgebra} is a [[comonoid]] in the [[category]] of [[chain complex]]es.

Equivalently, this is a graded [[coalgebra]] $C$ equipped with a [[coderivation]]

\begin{displaymath}
D : C \to C
\end{displaymath}
that is of degree -1 and squares to 0,

\begin{displaymath}
D^2 = 0 
  \,.
\end{displaymath}
\hypertarget{detailed_component_definition}{}\subsubsection*{{Detailed component definition}}\label{detailed_component_definition}

\hypertarget{precoalgebra}{}\paragraph*{{Pre-coalgebra}}\label{precoalgebra}

A \emph{pre-graded coalgebra} (pre-gc), $(C,\Delta, \varepsilon)$, is a [[graded vector space|pre-gvs]] $C$ together with linear maps of degree 0

\begin{displaymath}
\Delta: C\to C\otimes C,    \varepsilon : C\to k,
\end{displaymath}
such that the obvious (usual) diagrams commute. (When there is no ambiguity, we may write $C$ instead of $(C,\Delta, \varepsilon)$.)

The field $k$ is a coalgebra for the canonical isomorphism $k\to k\otimes k$ with $\varepsilon = id_k$.

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

\begin{displaymath}
(\psi \otimes \psi)\circ \Delta = \Delta' \circ \psi   and    \varepsilon = \varepsilon' \circ \psi.
\end{displaymath}
The linear counit mapping $\varepsilon :C \to k$ is always a morphism of pre-gcs.

\hypertarget{coaugmentations}{}\paragraph*{{Coaugmentations}}\label{coaugmentations}

A \emph{coaugmentation} of a pre-gc is a morphism $\eta : k \to C$. We will write $1$ for $\eta(1)$.

The cokernel $\bar{C}$ of $\eta$ can be identified with $Ker \varepsilon$ and so can be considered as a subspace of $C$.

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

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

\hypertarget{the_commutation_morphism}{}\paragraph*{{The commutation morphism}}\label{the_commutation_morphism}

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

\begin{displaymath}
\tau : V\otimes V' \to V'\otimes V
\end{displaymath}
is defined by $\tau( v\otimes v') = (-1)^{|v||v'|} v'\otimes v$, on homogeneous elements.

\hypertarget{tensor_product_of_pregcs}{}\paragraph*{{Tensor product of pre-gcs.}}\label{tensor_product_of_pregcs}

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

\begin{displaymath}
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')
\end{displaymath}
and

\begin{displaymath}
C\otimes C'\stackrel{\varepsilon\otimes \varepsilon'}{\to}k \otimes k\stackrel{\cong}{\to}k
\end{displaymath}
give $C\otimes C'$ a pre-gc structure.

If $\eta$ and $\eta'$ are coaugmentations of $C$ and $C'$ respectively, then $\eta\otimes\eta'$ defines a coaugmentation of $C\otimes C'$.

\hypertarget{coderivations_of_pregraded_coalgebras}{}\paragraph*{{Coderivations of pre-graded coalgebras}}\label{coderivations_of_pregraded_coalgebras}

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

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

\begin{displaymath}
\Delta \circ \theta = (\theta \otimes id_C + \tau \circ(\theta \otimes id_C)\circ \tau)\circ \Delta,  and  \varepsilon\circ \theta = 0.
\end{displaymath}
A coderivation $\theta$ of a coaugmented pre-gc $(C,\eta)$ is a coderivation of $C$ such that $\theta\circ \eta = 0$.

\hypertarget{differential_graded_coalgebras}{}\paragraph*{{Differential graded coalgebras}}\label{differential_graded_coalgebras}

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

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

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

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

\hypertarget{cocommutative_predgcs}{}\paragraph*{{Cocommutative pre-dgcs}}\label{cocommutative_predgcs}

A pre-gc $C$ is \emph{cocommutative} if $\tau\circ\Delta = \Delta$, similarly for a pre-dgc. The subcategories of cocommutative d.g. coalgebras will be denoted $pre CDGC$ (resp. $pre \eta CDGC$).

\hypertarget{cdgc}{}\paragraph*{{CDGC}}\label{cdgc}

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

\hypertarget{connected__cdgcs}{}\paragraph*{{$n$-connected $\eta$ cdgcs}}\label{connected__cdgcs}

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

This gives a category $CDGC_n$. Any connected (i.e. $0$-connected) cdgc is canonically coaugmented with $\bar{C}$ coinciding with $C_+$.

\hypertarget{homalgebras_and_duals}{}\paragraph*{{Hom-algebras and duals}}\label{homalgebras_and_duals}

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

\begin{displaymath}
C\stackrel{\Delta}{\to}C\otimes C\stackrel{(f\otimes g)}{\to}A\otimes A\stackrel{\mu}{\to} A,
\end{displaymath}
for $f,g \in Hom(C,A)$.

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

\hypertarget{coalgebra_filtrations}{}\paragraph*{{Coalgebra filtrations}}\label{coalgebra_filtrations}

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

\begin{displaymath}
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).
\end{displaymath}
\hypertarget{filtrations_of_the_primitives}{}\paragraph*{{Filtrations of the primitives}}\label{filtrations_of_the_primitives}

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

The iteration of $\bar{\Delta}$ is defined by

\begin{displaymath}
\bar{\Delta}^1 = \bar{\Delta}; \quad \bar{\Delta}^p = (\bar{\Delta}\otimes \bar{C} \otimes \ldots \bar{C}) \otimes \bar{\Delta}^{(p-1)}.
\end{displaymath}
The (increasing) filtration of the primitives is $F_p C = Ker\bar{\Delta}^p$, $p\geq 1$. It is a graded coalgebra filtration.

If $(C,\partial, \eta)$ is a coaugmented pre-dgc, each $F_p C$ is stable under the differential and, in particular, $F_1 = P(C)$. $P$ thus defines a functor from $pre \eta CDGC$ to $pre DGVS$.

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

Let $(C,\eta)$ be a coaugmented pre-gc and $F_p C$ the filtration of its primitives. $C$ is \emph{conilpotent} if $C = \bigcup_k F_k C$. A connected coalgebra is conilpotent and conilpotency is preserved by tensor product.

\hypertarget{the_pregc_structure_on_}{}\paragraph*{{The pre-gc structure on $T(V)$}}\label{the_pregc_structure_on_}

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

\begin{displaymath}
\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).
\end{displaymath}
The counit and the coaugmentation are the natural mappings $T(V)\to k$ and $k\to T(V)$, respectively.

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

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

\hypertarget{shuffles}{}\paragraph*{{Shuffles}}\label{shuffles}

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

\begin{displaymath}
\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.
\end{displaymath}
\hypertarget{the_precgc_structure_on_}{}\paragraph*{{The pre-cgc structure on $\bigwedge V$}}\label{the_precgc_structure_on_}

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

\begin{displaymath}
\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)}),
\end{displaymath}
in which the second sum is over all $(p,n-p)$-shuffles and $\varepsilon(\sigma)$ is the Koszul sign of $\sigma$.

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

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

There is an injective homomorphism of pre-gcs

\begin{displaymath}
\chi : \bigwedge{\!}' V \to T'(V)
\end{displaymath}
given by

\begin{displaymath}
\chi(x_1\wedge \ldots x_n) = \sum_\nu \varepsilon(\nu)x_{\nu(1)}\otimes \ldots \otimes x_{\nu(n)},
\end{displaymath}
where the sum is over all permutations and $\varepsilon(\sigma)$ is the corresponding Koszul sign. It has, as image, all the symmetric tensors (in the graded sense).

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

\begin{displaymath}
F_p\bigwedge{\!}' V = \bigwedge^{\leq p}V = \bigoplus_{k\leq p}\bigwedge^k V;
\end{displaymath}
\begin{displaymath}
F_p T'(V) = T^{\leq p}(V) = \bigoplus_{k\leq p} T^k(V).
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\hypertarget{differential_graded_algebras}{}\subsubsection*{{Differential graded algebras}}\label{differential_graded_algebras}

Dually, a [[dg-algebra]] is a [[monoid]] in [[chain complex]]es.

\hypertarget{semifree_dgcoalgebras_and_algebras}{}\subsubsection*{{Semifree dg-coalgebras and $L_\infty$-algebras}}\label{semifree_dgcoalgebras_and_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]]

\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

There is a [[model structure on dg-coalgebras]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AsFilteredColimits}{}\subsubsection*{{As filtered colimits of finite-dimensional pieces}}\label{AsFilteredColimits}

\begin{prop}
\label{}\hypertarget{}{}
Every dg-coalgebra is the [[filtered colimit]] of its finite-dimensional sub-dg-coalgebras.

\end{prop}
This is due to (\hyperlink{GetzlerGoerss99}{Getzler-Goerss 99}), in generalization to the analogous fact for plain [[coalgebras]], see at \emph{\href{coalgebra#AsFilteredColimits}{coalgebra -- As filtered colimits}}.

See also at \emph{[[L-infinity algebra]]} the section \emph{\href{L-infinity-algebra#IndConilpotency}{Ind-Conilpotency}}. This plays a role for instance for constructing [[model structures for L-infinity algebras]], see \href{model+structure+for+L-infinity+algebras#OnProAlg}{there}.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The [[model structure on dg-coalgebras]] is due to

\begin{itemize}%
\item [[Ezra Getzler]], [[Paul Goerss]], \emph{A model category structure for differential graded coalgebras}, 1999 (\href{http://www.math.northwestern.edu/~pgoerss/papers/model.ps}{ps}, [[GetzlerGoerss99.pdf:file]])

\end{itemize}
[[!redirects differential graded coalgebras]]

[[!redirects dg-coalgebra]] [[!redirects dg-coalgebras]]



\end{document}