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A module $A$ over a commutative ring $k$ with two homomorphisms $\phi$ and $\epsilon$ such that the diagrams
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{{TEX|done}}{{MSC|16T15}}
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A module $A$ over a commutative ring $k$ with two homomorphisms, "comultiplication" $\phi : A \to A \otimes_k A$ and "counit" $\epsilon : A \to k$ such that the diagrams
 
$$
 
$$
\begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ \phi\,\downarrow & \  & \downarrow\,1 \otimes \phi \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array}
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\begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ {}^\phi\downarrow & \  & \downarrow{}^{1 \otimes \phi} \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array}
 
$$
 
$$
 
and
 
and
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c0226706.png" /></td> </tr></table>
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\begin{array}{ccccc} A \otimes A & \stackrel{\phi}{\leftarrow} & A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ & & & & \\ & \searrow{}^{\epsilon\otimes1}\ & \Vert & {}^{1\otimes\epsilon}\swarrow & \\ & & & & \\ & & A & & \end{array}
 
+
$$
are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c0226707.png" />.
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are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring $k$.
  
 
Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. [[Hopf algebra|Hopf algebra]]).
 
Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. [[Hopf algebra|Hopf algebra]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
Given a co-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c0226708.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c0226709.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267010.png" /> be the module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267011.png" />-module homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267014.png" /> define the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267015.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267017.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267018.png" />. For any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267019.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267020.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267021.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267022.png" />. Then the multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267023.png" /> can also be seen as the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267024.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267025.png" /> is a unit element for this multiplication making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267026.png" /> an associative algebra with unit, the dual algebra. In general the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267027.png" /> is not an isomorphism and there is no natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267028.png" />-module homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267029.png" />. Thus there is no equally natural construction associating a co-algebra to an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267030.png" />, even when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267031.png" /> is a field. In that case there does however exist an [[Adjoint functor|adjoint functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267032.png" /> to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267033.png" /> which associates to a co-algebra its dual algebra, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267034.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267038.png" /> denote, respectively, the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267039.png" />-algebras and the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267040.png" />-co-algebras, [[#References|[a2]]]; cf. also [[Hopf algebra|Hopf algebra]]. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267041.png" /> is free of finite rank over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267042.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267043.png" /> is an isomorphism and the dual co-algebra can be defined.
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Given a co-algebra $A$ over $k$, let $A^*$ be the module of $k$-module homomorphisms from $A$ to $k$. For $f,g \in A^*$ define the product $fg : A \to k$ by the formula $fg : a \mapsto (f\otimes g)(\phi(a))$, where $k \otimes_k k$ is identified with $k$. For any two $k$-modules $M,N$ define $\rho : M^* \otimes N^* \to (M \otimes N)^*$ by $\rho(f\otimes g)(m\otimes n) = f(m)g(n)$. Then the multiplication on $A^*$ can also be seen as the composite $A^* \otimes A^* \to (A\otimes A)^* \stackrel{\phi^*}{\to} A^*$. The element $\epsilon : A \to k$ is a unit element for this multiplication making $A^*$ an associative algebra with unit, the dual algebra. In general the mapping $\rho$ is not an isomorphism and there is no natural $k$-module homomorphism $M^* \otimes N^* \to (M \otimes N)^*$. Thus there is no equally natural construction associating a co-algebra to an algebra over $k$, even when $k$ is a field. In that case there does however exist an [[adjoint functor]] $A \mapsto A^0$ to the functor $C \to C^*$ which associates to a co-algebra its dual algebra, i.e. $\textsf{Alg}_k (A,C^*) \sim \textsf{Coalg}_k(A^0,C)$ for $A \in \textsf{Alg}_k$, $C \in \textsf{Coalg}_k$, where $\textsf{Alg}_k$ and $\textsf{Coalg}_k$ denote, respectively, the category of $k$-algebras and the category of $k$-co-algebras, [[#References|[a2]]]; cf. also [[Hopf algebra]]. But if $B$ is free of finite rank over $k$ then $\rho : B^* \otimes B^* \to (B \otimes B)^*$ is an isomorphism and the dual co-algebra can be defined.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267044.png" /> be the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267045.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267046.png" /> and define
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Let $S$ be the set $s_0, s_1, \ldots$. Let $kS = \oplus_i ks_i$ and define
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267047.png" /></td> </tr></table>
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\phi(s_n) = \sum_{i-0}^n s_i \otimes s_{n-i}\,,\ \ \ \epsilon(s_i) = 1 \ .
 +
$$
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267048.png" /> is a co-algebra.
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Then $kS$ is a co-algebra.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267050.png" /> are two co-algebras, then a morphism of co-algebras is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267051.png" />-module morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267054.png" />. A co-ideal of a co-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267055.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267056.png" />-submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267059.png" />.
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If $(A,\phi)$ and $(B,\psi)$ are two co-algebras, then a morphism of co-algebras is a $k$-module morphism $\alpha:A \to B$ such that $\psi \circ \alpha = (\alpha\otimes\alpha)\circ\phi$ and $\epsilon_B \circ \alpha = \alpha \circ \epsilon_A$. A co-ideal of a co-algebra $A$ is a $k$-submodule $V$ such that $\phi(V) \subset V \otimes A + A \otimes V$ and $\epsilon(V) = 0$.
  
A co-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267060.png" /> over a co-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267061.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267062.png" />-module with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267063.png" />-module morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267066.png" /> the canonical isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022670/c02267067.png" />. There are obvious notions of homomorphisms of co-modules, etc.
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A co-module $M$ over a co-algebra $(C,\phi)$ is a $k$-module with a $k$-module morphism $\psi:M \to M\otimes C$ such that $(\psi\otimes1)\circ\psi = (1\otimes\phi)\circ\psi$ and $(1\otimes\epsilon)\circ\psi$ the canonical isomorphism $M \to M \otimes k$. There are obvious notions of homomorphisms of co-modules, etc.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Sweedler,   "Hopf algebras" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Abe,   "Hopf algebras" , Cambridge Univ. Press  (1980)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Sweedler, "Hopf algebras" , Benjamin  (1969) {{ZBL|0194.32901}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Abe, "Hopf algebras" , Cambridge Univ. Press  (1980) {{ISBN|0-521-60489-3}} {{ZBL|0476.16008}}</TD></TR>
 +
</table>

Latest revision as of 08:10, 26 November 2023

2020 Mathematics Subject Classification: Primary: 16T15 [MSN][ZBL]

A module $A$ over a commutative ring $k$ with two homomorphisms, "comultiplication" $\phi : A \to A \otimes_k A$ and "counit" $\epsilon : A \to k$ such that the diagrams $$ \begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ {}^\phi\downarrow & \ & \downarrow{}^{1 \otimes \phi} \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array} $$ and $$ \begin{array}{ccccc} A \otimes A & \stackrel{\phi}{\leftarrow} & A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ & & & & \\ & \searrow{}^{\epsilon\otimes1}\ & \Vert & {}^{1\otimes\epsilon}\swarrow & \\ & & & & \\ & & A & & \end{array} $$ are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring $k$.

Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. Hopf algebra).

References

[1] S. MacLane, "Homology" , Springer (1963)


Comments

Given a co-algebra $A$ over $k$, let $A^*$ be the module of $k$-module homomorphisms from $A$ to $k$. For $f,g \in A^*$ define the product $fg : A \to k$ by the formula $fg : a \mapsto (f\otimes g)(\phi(a))$, where $k \otimes_k k$ is identified with $k$. For any two $k$-modules $M,N$ define $\rho : M^* \otimes N^* \to (M \otimes N)^*$ by $\rho(f\otimes g)(m\otimes n) = f(m)g(n)$. Then the multiplication on $A^*$ can also be seen as the composite $A^* \otimes A^* \to (A\otimes A)^* \stackrel{\phi^*}{\to} A^*$. The element $\epsilon : A \to k$ is a unit element for this multiplication making $A^*$ an associative algebra with unit, the dual algebra. In general the mapping $\rho$ is not an isomorphism and there is no natural $k$-module homomorphism $M^* \otimes N^* \to (M \otimes N)^*$. Thus there is no equally natural construction associating a co-algebra to an algebra over $k$, even when $k$ is a field. In that case there does however exist an adjoint functor $A \mapsto A^0$ to the functor $C \to C^*$ which associates to a co-algebra its dual algebra, i.e. $\textsf{Alg}_k (A,C^*) \sim \textsf{Coalg}_k(A^0,C)$ for $A \in \textsf{Alg}_k$, $C \in \textsf{Coalg}_k$, where $\textsf{Alg}_k$ and $\textsf{Coalg}_k$ denote, respectively, the category of $k$-algebras and the category of $k$-co-algebras, [a2]; cf. also Hopf algebra. But if $B$ is free of finite rank over $k$ then $\rho : B^* \otimes B^* \to (B \otimes B)^*$ is an isomorphism and the dual co-algebra can be defined.

Let $S$ be the set $s_0, s_1, \ldots$. Let $kS = \oplus_i ks_i$ and define $$ \phi(s_n) = \sum_{i-0}^n s_i \otimes s_{n-i}\,,\ \ \ \epsilon(s_i) = 1 \ . $$

Then $kS$ is a co-algebra.

If $(A,\phi)$ and $(B,\psi)$ are two co-algebras, then a morphism of co-algebras is a $k$-module morphism $\alpha:A \to B$ such that $\psi \circ \alpha = (\alpha\otimes\alpha)\circ\phi$ and $\epsilon_B \circ \alpha = \alpha \circ \epsilon_A$. A co-ideal of a co-algebra $A$ is a $k$-submodule $V$ such that $\phi(V) \subset V \otimes A + A \otimes V$ and $\epsilon(V) = 0$.

A co-module $M$ over a co-algebra $(C,\phi)$ is a $k$-module with a $k$-module morphism $\psi:M \to M\otimes C$ such that $(\psi\otimes1)\circ\psi = (1\otimes\phi)\circ\psi$ and $(1\otimes\epsilon)\circ\psi$ the canonical isomorphism $M \to M \otimes k$. There are obvious notions of homomorphisms of co-modules, etc.

References

[a1] M. Sweedler, "Hopf algebras" , Benjamin (1969) Zbl 0194.32901
[a2] E. Abe, "Hopf algebras" , Cambridge Univ. Press (1980) ISBN 0-521-60489-3 Zbl 0476.16008
How to Cite This Entry:
Co-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-algebra&oldid=40931
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article