Difference between revisions of "Eilenberg-Moore algebra"
m (AUTOMATIC EDIT (latexlist): Replaced 10 formulas out of 12 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.) |
m (fix tex) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 2: | Line 2: | ||
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist | the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist | ||
was used. | was used. | ||
| − | If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. | + | If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. |
Out of 12 formulas, 10 were replaced by TEX code.--> | Out of 12 formulas, 10 were replaced by TEX code.--> | ||
| − | {{TEX| | + | {{TEX|done}} |
''Moore–Eilenberg algebra'' | ''Moore–Eilenberg algebra'' | ||
| − | Given a monad (or [[ | + | Given a monad (or [[triple]]) $T$ in a [[Category|category]] $\mathcal{C}$, a $T$-algebra is a pair $( A , \alpha )$, $\alpha : T A \rightarrow A$, $A \in \mathcal{C}$, such that the diagram |
| − | + | \begin{equation} | |
| − | + | \begin{array}{crccc} | |
| − | + | A & \stackrel{\eta_A}{\rightarrow} & T(A) & & T(T(A)) \\ | |
| + | & {}_{\mathrm{id}_A}\nwarrow & \downarrow{}_\alpha & \stackrel{\mu_A}{\leftarrow} & \downarrow{}_{T(\alpha)} \\ | ||
| + | & & A & & T(A) | ||
| + | \end{array} | ||
| + | \end{equation} | ||
commutes. Such a $T$-algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras $\mathcal{C} ^ { T }$ to $\mathcal{C}$ has a left adjoint, exhibiting the monad $T$ as coming from a pair of adjoint functors (the Eilenberg–Moore construction). | commutes. Such a $T$-algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras $\mathcal{C} ^ { T }$ to $\mathcal{C}$ has a left adjoint, exhibiting the monad $T$ as coming from a pair of adjoint functors (the Eilenberg–Moore construction). | ||
| − | See also [[ | + | See also [[Adjoint functor]]. |
====References==== | ====References==== | ||
| − | <table><tr><td valign="top">[a1]</td> <td valign="top"> F. Borceux, "Handbook of categorical algebra: Categories and structures" , '''2''' , Cambridge Univ. Press (1994) pp. Chap. 4</td></tr></table> | + | <table> |
| + | <tr><td valign="top">[a1]</td> <td valign="top"> F. Borceux, "Handbook of categorical algebra: Categories and structures" , '''2''' , Cambridge Univ. Press (1994) pp. Chap. 4</td></tr> | ||
| + | </table> | ||
Latest revision as of 09:49, 19 July 2020
Moore–Eilenberg algebra
Given a monad (or triple) $T$ in a category $\mathcal{C}$, a $T$-algebra is a pair $( A , \alpha )$, $\alpha : T A \rightarrow A$, $A \in \mathcal{C}$, such that the diagram \begin{equation} \begin{array}{crccc} A & \stackrel{\eta_A}{\rightarrow} & T(A) & & T(T(A)) \\ & {}_{\mathrm{id}_A}\nwarrow & \downarrow{}_\alpha & \stackrel{\mu_A}{\leftarrow} & \downarrow{}_{T(\alpha)} \\ & & A & & T(A) \end{array} \end{equation} commutes. Such a $T$-algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras $\mathcal{C} ^ { T }$ to $\mathcal{C}$ has a left adjoint, exhibiting the monad $T$ as coming from a pair of adjoint functors (the Eilenberg–Moore construction).
See also Adjoint functor.
References
| [a1] | F. Borceux, "Handbook of categorical algebra: Categories and structures" , 2 , Cambridge Univ. Press (1994) pp. Chap. 4 |
Eilenberg-Moore algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-Moore_algebra&oldid=49964