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algebraic categories and classes of algebras
0.1 Introduction
Classes of algebras can be categorized at least in two types: either classes of specific algebras, such as: group algebras, Kalgebras, groupoid algebras, logic algebras, and so on, or general ones, such as general classes of: categorical algebras, higher dimensional algebra (HDA), supercategorical algebras, universal algebras, and so on.
0.2 Basic concepts and definitions

Class of algebras
Definition 0.1.
A class of algebras is defined in a precise sense as an algebraic object in the groupoid category.

Monad on a category $\mathcal{C}$, and a Talgebra in $\mathcal{C}$
Definition 0.2.
Let us consider a category $\mathcal{C}$, two functors: $T:\mathcal{C}\to\mathcal{C}$ (called the monad functor) and $T^{2}:\mathcal{C}\to\mathcal{C}=T\circ T$, and two natural transformations: $\eta:1_{\mathcal{C}}\to T$ and $\mu:T^{2}\to T$. The triplet $(\mathcal{C},\eta,\mu)$ is called a monad on the category $\mathcal{C}$. Then, a Talgebra $(Y,h)$ is defined as an object $Y$ of a category $\mathcal{C}$ together with an arrow $h:TY\to Y$ called the structure map in $\mathcal{C}$ such that:
(a) $Th:T^{2}\to TY,$ (b) $h\circ Th=h\circ\mu_{Y},$ where: $\mu_{Y}:T^{2}Y\to TY;$ and
(c) $h\circ\eta_{Y}=1_{Y}.$

Category of EilenbergMoore algebras of a monad $T$ An important definition related to abstract classes of algebras and universal algebras is that of the category of EilenbergMoore algebras of a monad $T$:
Definition 0.3.
The category $\mathcal{C}^{T}$ of $T$algebras and their morphisms is called the EilenbergMoore category or category of EilenbergMoore algebras of the monad T.
0.3 Remarks

a. Algebraic category definition
Remark 0.1.
With the above definition, one can also define a category of classes of algebras and their associated groupoid homomorphisms which is then an algebraic category.
Another example of algebraic category is that of the category of C*algebras.
Generally, a category $\mathcal{A}_{C}$ is called algebraic if it is monadic over the category of sets and settheoretical mappings, $Set$; thus, a functor $G:\mathcal{D}\to\mathcal{C}$ is called monadic if it has a left adjoint $F:\mathcal{C}\to\mathcal{D}$ forming a monadic adjunction $(F,G,\eta,\epsilon)$ with $G$ and $\eta,\epsilon$ being, respectively, the unit and counit; such a monadic adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ is defined by the condition that category $\mathcal{D}$ is equivalent to the to the EilenbergMoore category $\mathcal{C}^{T}$ for the monad
$T=GF.$ 
Remark 0.2.
Although all classes can be regarded as equivalence, weak equivalence, etc., classes of algebras (either specific or general ones), do not define identical, or even isomorphic structures, as the notion of ‘equivalence’ can have more than one meaning even in the algebraic case.
0.4 Note:
See also the entry about abstract and concrete algebras in Expositions.
Mathematics Subject Classification
1800 no label found18E05 no label found08A05 no label found08A70 no label found Forums
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Comments
class $ algebras$???
What exactly is the point of setting off that word in that manner sometimes and sometimes not?
Re: class $ algebras$???
He's trying to emphacize (using italics). But he's doing it the wrong way. He should've written \emph{algebras}.
Please file a correction explaining math mode and text mode italics .
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: class $ algebras$???
That thought did cross my mind. But what do you make of the fact that bci1 felt it necessary to do this in the title also?
He sent me a long and detailed message which says there are broader issues here than a onetime TeX faux pas by a single author. With his permission, I could quote the message, or he could restate it himself to all of us.
Re: class $ algebras$???
It would be best if he replied to things like this as well as corrections in the forums instead of by sending private messages to people; that way the entire community can be involved and no one feels as though they are being personally discriminated against or disrespected. Plus this approach is just more in the spirit of this website, at least in my humble opinion.
Re: class $ algebras$???
Please read the docs entry on Controlling Linking, which has information on how to avoid linking to particular entries, or forcing linking to particular ones:
http://planetmath.org/?op=getobj&from=collab&id=33
Changing the name of your entry to something like $algebra$ or \emph{algebra} will not help neither the linking system nor the search engine to give your entry more precedence to others. Moreover, $algebra$ is plain wrong (since it is a word, not a math expression, and latex interprets each letter of the word as a math symbol), while \emph{algebra} in the title is not coherent with the usual policy for naming entries in the encyclopedia (plus, it probably messes up the automatic linking and search engine).