algebra for an endofunctor

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
#### Algebra
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[[!include higher algebra - contents]]
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# Contents
* table of contents
{: toc}

## Idea

An _algebra over an endofunctor_ is like an [[algebra over a monad]], but without a notion of [[associativity]] (given that a plain [[endofunctor]] is not equipped with a multiplication-operation that would make it a [[monad]]).

## Definition

+-- {: .num_defn}
###### Definition

For a [[category]] $C$ and [[endofunctor]] $F$, an __algebra__ (or __module__) of $F$ is an [[object]] $X$ in $C$ and a [[morphism]] $\alpha\colon F(X) \to X$. ($X$ is called the __carrier__ of the algebra)

A [[homomorphism]] between two algebras $(X, \alpha)$ and $(Y, \beta)$ of $F$ is a morphism $m\colon X \to Y$ in $C$ such that the following [[commuting diagram|square commutes]]:

$$
  \array{
    F(X)
    &
    \overset{F(m)}{\longrightarrow}
    &
    F(Y)
    \\
    \mathllap{{}^{\alpha}}\big\downarrow
    &&
    \big\downarrow\mathrlap{{}^{\beta}}
    \\
    X
    &
    \underset{m}{\longrightarrow} & Y
  }
  \,.
$$

[[composition|Composition]] of such homomorphisms of algebras is given by composition of the [[underlying]] morphisms in $C$.  This yields the [[category]] of $F$-algebras, which comes with a [[forgetful functor]] to $C$.

=--

+-- {: .num_remark}
###### Remark

The dual concept is a [[coalgebra for an endofunctor]]. Both algebras and coalgebras for endofunctors on $C$ are special cases of [[algebra for a C-C bimodule|algebras for bimodules]].
=--

If $F$ is a [[pointed endofunctor]] with point $\eta : Id \to F$, then by an **algebra** for $F$ one usually means a *pointed algebra*, i.e. one such that $\alpha \circ \eta_X = id_X$.

## Properties

### Relation to algebras over a monad

To a [[category theory|category theorist]], [[algebras over a monad]] may be more familiar than algebras over just an endofunctor.  In fact, when $C$ and $F$ are well-behaved, then algebras over an endofunctor $F$ are equivalent to algebras over a certain monad, the [[algebraically-free monad]] generated by $F$ ([Pirog](#Maciej), [Gambino-Hyland 04, section 6](#GambinoHyland04)).

This is analogous to the relationship between an [[action]] $M \times B \to B$ of a [[monoid]] $M$ and a [[binary function]] $A \times B \to B$ (an action of a [[set]]): such a function is the same thing as an action of the [[free monoid]] $A^*$ on $B$.

Returning to the endofunctor case, the general statement is:

+-- {: .num_prop}
###### Proposition
The [[category]] of algebras of the endofunctor $F\colon \mathcal{C} \to \mathcal{C}$ is [[equivalence of categories|equivalent]] to the category of [[algebra over a monad|algebras]] of the [[algebraically-free monad]] on $F$, should such exist.
=--

Actually, this proposition is merely a definition of the term "algebraically-free monad".  If $F$ has an algebraically-free monad, denoted say $F^*$, then in particular the forgetful functor $F Alg \to C$ has a [[left adjoint]], and $F^*$ is the monad on $C$ generated by this [[adjunction]].  Conversely, if such a left adjoint exists, then the monad it generates is algebraically-free on $F$; for the straightforward proof, see for instance ([Pirog](#Maciej)). An explicit construction of the algebraically free monad in terms of [inductive types](#initalg) is given below.

Algebraically-free monads exist in particular when $C$ is a [[locally presentable category]] and $F$ is an [[accessible functor]]; see [[transfinite construction of free algebras]].

+-- {: .num_remark}
###### Remark

It turns out that an algebraically-free monad on $F$ is also [[free object|free]] in the sense that it receives a universal arrow from $F$ relative to the [[forgetful functor]] from [[monads]] to [[endofunctors]].  The converse, however, is not necessarily true: a free monad in this sense need not be algebraically-free.  It is true when $C$ is [[complete category|complete]], however.

=--

Entirely analogous facts are true for pointed algebras over pointed endofunctors.

## Relationship to inductive types {#initalg}

The [[initial algebra of an endofunctor]] provides [[categorical semantics]] for [[inductive types]].

The construction of an algebraically free monad may be cast in the language of such initial algebras. Suppose $C$ is a category with coproducts and $F: C \to C$ is an endofunctor. Let $F$-$alg$ be the category of $F$-algebras, and let $U: F\text{-}alg \to C$ be the usual forgetful functor. A [[left adjoint]] to $U$ then takes an object $d$ of $C$ to the initial algebra $\Phi(d)$ of the endofunctor $c \mapsto d + F(c)$, provided this initial algebra exists. For, by the usual comma category description (see for example [[adjoint functor theorem]]), $\Phi(d)$ is the [[initial object]] of the category $(d \downarrow U)$. However, an object of $(d \downarrow U)$ is a triple $(c, \alpha: F(c) \to c, \beta: d \to c)$, equivalently a pair $(c, \gamma: d + F(c) \to c)$, equivalently an algebra of $c \mapsto d + F(c)$. Hence an initial object of $(d \downarrow U)$ is an initial algebra of an endofunctor.

The [[monad]] structure of the algebraically free monad $F^\ast = U\Phi$ may be straightforwardly extracted from this initial algebra description. This is made explicit in [Pirog](#Maciej). For example, to describe the multiplication $\mu: F^\ast F^\ast \to F^\ast$, let $d$ be an object; then $F^\ast d$ has an algebra structure $[i, \theta]: d + F(F^\ast d) \to F^\ast d$. It therefore also has a structure of algebra over the endofunctor $c \mapsto F^\ast d + F(c)$, namely $[1, \theta]: F^\ast d + F(F^\ast d) \to F^\ast d$. But since $F^\ast F^\ast d$ is the initial algebra for the monad $c \mapsto F^\ast d + F(c)$, we obtain a unique algebra map $F^\ast F^\ast d \to F^\ast d$. This is the component $\mu_d$ of the monad multiplication.

## Related concepts

* [[free monad]]

* [[endofunctor]], [[pointed endofunctor]]

* [[algebra over a monad]], [[algebra over a profunctor]], [[coalgebra over an endofunctor]]

* [[algebraically compact category]]

## References

A textbook account of the basic theory is in [chapter 10](http://www.andrew.cmu.edu/course/80-413-713/notes/chap10.pdf) of

* [[Steve Awodey]], _Category theory_ lecture notes (2011) ([web](http://www.andrew.cmu.edu/course/80-413-713/))

The relation to [[free monads]] is discussed in

* [Maciej Pirog](http://maciejcs.wordpress.com/), _[Free monads and their algebras](http://maciejcs.wordpress.com/2012/04/17/free-monads-and-their-algebras/)_
 {#Maciej}

* [[Nicola Gambino]], [[Martin Hyland]], _Wellfounded trees and dependent polynomial functors_. In Types for proofs and programs, volume 3085 of Lecture Notes in Comput. Sci., pages 210–225. Springer-Verlag, Berlin, 2004 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.4534))
 {#GambinoHyland04}

[[!redirects algebra of a functor]]
[[!redirects algebra of an endofunctor]]
[[!redirects algebra for a functor]]
[[!redirects algebra for an endofunctor]]
[[!redirects algebra over a functor]]
[[!redirects algebra over an endofunctor]]
[[!redirects algebras of a functor]]
[[!redirects algebras of an endofunctor]]
[[!redirects algebras for a functor]]
[[!redirects algebras for an endofunctor]]
[[!redirects algebras over a functor]]
[[!redirects algebras over an endofunctor]]
[[!redirects algebras of functors]]
[[!redirects algebras of endofunctors]]
[[!redirects algebras for functors]]
[[!redirects algebras for endofunctors]]
[[!redirects algebras over functors]]
[[!redirects algebras over endofunctors]]
[[!redirects F-algebra]]

[[!redirects algebra of a pointed endofunctor]]
[[!redirects algebra for a pointed endofunctor]]
[[!redirects algebra over a pointed endofunctor]]
[[!redirects algebras of a pointed endofunctor]]
[[!redirects algebras for a pointed endofunctor]]
[[!redirects algebras over a pointed endofunctor]]
[[!redirects algebras of pointed endofunctors]]
[[!redirects algebras for pointed endofunctors]]
[[!redirects algebras over pointed endofunctors]]


algebra for an endomorphism

[[!redirects algebra over an endomorphism]]
## Definition

If $K$ is a [[bicategory]] and $f \colon a \to a$ is an [[endomorphism]] in $K$, then a (**left**) $f$-**algebra** or $f$-**module** is given by a 1-cell $x \colon b \to a$ together with a 2-cell $\lambda \colon f x \Rightarrow x$.

One can also define right modules/algebras, comodules/coalgebras and bimodules as for [[module over a monad|monads]].

## Examples

If $K$ is $Cat$, an [[algebra for an endofunctor]] $F \colon C \to C$ is the same thing as an $F$-algebra $A \colon \ast \to C$ in the sense above.

Every [[module over a monad]] $(t, \eta, \mu)$ is an algebra over the underlying endomorphism $t$.

An [[algebra for a profunctor]] (q.v.) $H \colon C ⇸ C$ on $X \colon D \to C$ is essentially the same as a $H$-coalgebra $C(1,X) \Rightarrow H \circ C(1,X)$ in $Prof$, the bicategory of categories and [[profunctors]].


algebra modality

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## Idea

The notion of algebra modality is used to define [[differential categories|codifferential categories]]. However, it could be use in other types of [[doctrines|categorical doctrines]].

## Definition

\begin{definition}
An [[algebra modality]] in a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, I)$ is given by a [[monad]] $(S,m,u)$ and two [[natural transformations]] $\eta:I \rightarrow S(A)$ and $\nabla:S(A) \otimes S(A) \rightarrow S(A)$ such that for every $A \in \mathcal{C}$, $(S(A), \nabla, \eta)$ is a [[commutative monoid]] in $(\mathcal{C},\otimes,I)$ and this diagram commutes:
\begin{tikzcd}
S(S(A)) \otimes S(S(A)) \arrow[dd, "m \otimes m"'] \arrow[rr, "\nabla"] &  & S(S(A)) \arrow[dd, "m"] \\
                                                                            &  &                           \\
S(A) \otimes S(A) \arrow[rr, "\nabla"']                                   &  & S(A)
\end{tikzcd}
\end{definition}

## Properties

\begin{proposition}
If $R$ is a commutative rig, then the symmetric algebra defines an algebra modality in $Mod_{R}$.

* $Sym(A)$ is the [[symmetric algebra]] of the module $A$.
* We have $\nabla_{A}:Sym(A) \otimes Sym(A) \rightarrow Sym(A)$.
* We have $\eta_{A}:A \rightarrow Sym(A)$.
* The unit $A \rightarrow Sym(A)$ of the monad is just the injection $x \mapsto x$.
* The multiplication $Sym(Sym(A)) \rightarrow Sym(A)$ of the monad is given on pure tensors by $(x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)}) \boxtimes_{s} ... \boxtimes_{s}  (x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}) \mapsto x_{1}^{(1)} \otimes_{s} ... \otimes_{s} x_{n_{1}}^{(1)} \otimes ... \otimes x_{1}^{(p)} \otimes_{s} ... \otimes_{s} x_{n_{p}}^{(p)}$.  It is a kind of composition of polynomials.

\end{proposition}

\begin{proposition}
[[rigs|Commutative rigs]] are exactly the [[algebra over a monad|algebras]] (over a monad) over the algebra modality $Sym$ in the category of commutative monoids.
\end{proposition}

## References

Algebra modalities are part of the definition of a codifferential category ie. a category $\mathcal{C}$ such that $\mathcal{C}^{op}$ is a differential category.

Differential categories were introduced in:

*  {#BluteCocketSeely06} [[R. F. Blute]], [[J. R. B. Cockett]], and [[R. A. G. Seely]]: _Differential categories_. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) ([doi:10.1017/S0960129506005676](https://doi.org/10.1017/S0960129506005676))

and revised in:

* [[Richard Blute]], [[Robin Cockett]], [[Jean-Simon Lemay]] and [[Robert Seely]], _Differential categories revisited,_ Appl. Categ. Struct. 28, 171-235 (2020). ([arXiv:1806.04804](https://arxiv.org/abs/1806.04804), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y))


algebra of an endofunctor > history

< [[algebra of an endofunctor]]
[[!redirects algebra of an endofunctor -- history]]


algebra of functions

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## Definition

### Algebra of function on a set

For $R$ a [[ring]] and $S$ a [[set]], the set of [[functions]] $S \to R$ (to the underlying set of $R$) is itself naturally an [[associative algebra]] over $R$, where addition and multiplication is given pointwise in $S$ by addition and multiplication in $R$: for $f_1, f_2 \colon S \to R$ their sum is the function

$$
  (f_1 + f_2) \colon s \mapsto f_1(s) + f_2(s) \in R
  \,,
$$

their product is the function

$$
  (f_1 \cdot f_2) \colon s \mapsto f_1(s) f_2(s) \in R
$$

and the [[ring]] inclusion $R \to [S,R]$ is given by sending $r \in R$ to the constant function with value $r$.

### Algebra of functions on an $\infty$-stack

More generally, in the context of [[(∞,1)-topos theory]] and [[higher algebra]], there is a notion of [[function algebras on ∞-stacks]].

## Properties

### Relation to free modules

If $S$ is a [[finite set]] or else if one restricts to functions that are non-vanishing only for finitely many elements in $S$, then the algebra of functions with values in $R$ also forms the [[free module]] over $R$ generated by $S$.

### Hadamard product

If $S$ is a [[set]] and $R$ is a [[commutative ring]], then the pointwise multiplication on the function algebra $S \to R$ is the [[Hadamard product]] on the function algebra.

### Duality between algebra and geometry

Sending [[spaces]] to their suitable algebras of functions constitutes a basic [[duality]] operation that relates [[geometry]] and [[algebra]]. For more on this see at _[[Isbell duality]]_.

## Related concepts

* [[pullback of functions]]
* [[composition ring]]

[[!redirects algebras of functions]]
[[!redirects function algebra]]
[[!redirects function algebras]]

[[!redirects function ring]]
[[!redirects function rings]]

[[!redirects ring of functions]]
[[!redirects rings of functions]]


algebra over a Lawvere theory

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## Definition

A [[Lawvere theory]] is encoded in its [[syntactic category]] $T$, a category with finite products such that all objects are finite products of a given object.

An **algebra over a Lawvere theory $T$**, or **$T$-algebra** for short, is a [[model]] for this [[algebraic theory]]: it is a product-preserving [[functor]]

$$
  A : T \to Set
  \,.
$$

The [[category]] of $T$-algebras is the [[full subcategory]] of the [[functor category]] on the product-preserving functors

$$
  T Alg := [T,Set]_\times \subset [T,Set]
  \,.
$$

For more discussion, properties and examples see for the moment [[Lawvere theory]].

## Properties

+-- {: .un_prop}
###### Proposition

The category $T Alg$ has all [[limit]]s and these are computed objectwise, hence the embedding $T Alg \to [T,Set]$ preserves these limits.

=--

+-- {: .un_prop}
###### Proposition

$T Alg$ is a [[reflective subcategory]] of $[T, Set]$:

$$
  T Alg \stackrel{\leftarrow}{\hookrightarrow} [T,Set]
  \,.
$$

=--

+-- {: .proof}
###### Proof

With the above this follows using the [[adjoint functor theorem]].

=--

+-- {: .un_corollary}
###### Corollary

The category $T Alg$ has all [[colimit]]s.

=--

for more see [[Lawvere theory]] for the moment

## Examples

* [[group]]s

* [[ring]]s

* $k$-[[associative algebra]]s

* [[smooth algebra]]s

## Related concepts

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* **algebra over an algebraic theory**

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

[[!redirects algebras over a Lawvere theory]]

[[!redirects algebra over an algebraic theory]]
[[!redirects algebras over an algebraic theory]]

[[!redirects algebra for an algebraic theory]]
[[!redirects algebras for an algebraic theory]]

[[!redirects T-algebra]]


algebra over a monad

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## Idea

There are different, related ways in which one could view the notion of *algebra over a [[monad]]*:

* If one views monads as a generalization of [[algebraic theories]] (see [[algebraic theory#RelationToMonads|algebraic theory -  relation to monads]]), an algebra over a monad is the corresponding generalization of an [[algebra over a Lawvere theory|algebra over a theory]]. In particular, if one views a monad as a way of prescribing particular [[operations]], an algebra is a context where those specified formal expressions can be evaluated to an actual result.
* From the 2-dimensional point of view, an algebra over a monad is a special case of a [[module over a monad]] for the bicategory [[Cat]], where the arrow is from the [[terminal category]].
* If one views monads as a way to model computational effects (see [[monad in computer science]]), an algebra is a context in which the extra effects can be reincorporated into the main data.

Algebras over a monad are usually objects equipped with extra [[structure]], not just [[properties]]. (They can also be seen as [[algebra over an endofunctor|algebras over the underlying endofunctor]], satisfying extra compatibility [[properties]].)

The corresponding [[formal duality|dual]] notion is that of *[[coalgebra over a comonad]]*.

## Definition
 {#Definition}

Let $(T,\eta, \mu)$ be a [[monad]] on a [[category]] $\mathcal{C}$.

### Algebras
 {#Algebras}

\begin{definition}
An **algebra over $T$** (or just **$T$-algebra** or **$T$-module**) consists of:

1. an [[object]] $A$ of $\mathcal{C}$,

1. a [[morphism]] $a \colon T (A) \longrightarrow A$ of $\mathcal{C}$,

such that the following [[commuting diagram|diagrams commute]] in $\mathcal{C}$ (cf. the definition of *[[module object]]* [here](module+object#MonoidsInMonoidalCategory)):

\[
  \label{UnitProperty}
  \text{Unit Property:}
\]
\begin{tikzcd}
  A
  \ar{r}{\eta}
  \ar[swap]{dr}{\mathrm{id}}
  &
  T(A)
  \ar{d}{a}
  \\
  &
  A
\end{tikzcd}

\linebreak

\[
  \label{ActionProperty}
  \text{Action Property:}
\]
\begin{tikzcd}
  T\big(T(A)\big)
  \ar{r}{T(a)}
  \ar{d}{\mu}
  &
  TA
  \ar{d}{a}
  \\
  TA
  \ar{r}{a}
  &
  A
\end{tikzcd}

\end{definition}

\begin{remark}
The diagram (eq:UnitProperty) is also sometimes called the _unit triangle_, and the diagram (eq:ActionProperty) is also called the _multiplication square_ or _algebra square_.
\end{remark}

### Homomorphisms

Let $(A,a)$ and $(B,b)$ be $T$-algebra. A **[[homomorphism]] of $T$-algebras** is a [[morphism]] $f \colon A \to B$ of $\mathcal{C}$ which makes the following [[commuting diagram|diagram commute]].

\begin{tikzcd}
  TA
  \ar{r}{Tf}
  \ar{d}{a}
  &
  TB
  \ar{d}{b}
  \\
  A
  \ar{r}{f}
  &
  B
\end{tikzcd}

The [[category]] formed by $T$-algebras and their homomorphisms is known as the **[[Eilenberg-Moore category]]** of $T$ and often denoted by $\mathcal{C}^T$.

### Free algebras
 {#FreeAlgebras}

Given a [[monad]] $(T,\mu,\eta)$ on a [[category]] $\mathcal{C}$, then for every [[object]] $X$ of $\mathcal{C}$, the object $T X$ is canonically equipped with a $T$-algebra [[structure]], given by the multiplication map $\mu$ of the monad. The relevant diagrams commute by the monad axioms.

$T$-Algebras of this sort are called **[[free construction|free]] $T$-algebras**.

Given any [[morphism]] $\phi \colon X \to Y$ of $\mathcal{C}$, the [[morphism]] $T \phi \colon T X \longrightarrow T Y$ is evidently a [[homomorphism]] of $T$-algebras, by [[natural transformation|naturality]] of $\mu$. But not every homomorphism of $T$-algebras between the free $T$-algebras $T X$ and $T Y$ arises this way, in general.

However, for any morphism of the form
$$
  f \colon X \longrightarrow T Y
$$
in $\mathcal{C}$ (called a $T$-*[[Kleisli morphism]]*), the induced morphism
\[
  \label{AlgebraMapInducedByKleisliMorphism}
  \mu_{Y} \circ T f
  \;\colon\;
  T X
    \xrightarrow{\;\; T f  \;\;}
  T T Y
    \xrightarrow{\;\; \mu_Y \;\;}
  TY
\]

*is* a [[homomorphism]] of $T$-algebras between these free $T$-algebras, as one verifies again using the [[natural transformation|naturality]] of $\mu$. Now, all homomorphisms of $T$-algebras between the free algebras $T X$ and $T Y$ *do* arise this way.

Moreover, given in addition a morphism in $\mathcal{C}$ of the form  $g \colon Y \longrightarrow T Z$,
then, under this association, the [[composition]] of the corresponding $T$-algebra morphisms (eq:AlgebraMapInducedByKleisliMorphism) of $f$ and $g$
equals the $T$-algebra homomorphism corresponding to their *[[Kleisli composition|Kleisli composite]]*, defined by
$$
  g \circ_T f
    \;\coloneqq\;
  X
    \xrightarrow{ f }
  T Y
    \xrightarrow{ T g }
  T T Z
    \xrightarrow{ \mu_Z }
  T Z
  \,.
$$

The [[category]] of [[Kleisli morphisms]] equipped with this [[Kleisli composition]] is called the *[[Kleisli category]]* and is [[equivalence of categories|equivalent]] to the [[full subcategory]] of $T$-algebras on the free $T$-algebras (see [there](Kleisli+category#InTermsOfKleisliMorphisms) for more).

### Tensor product

In the case of a [[commutative monad]] $T$ , one can define a [[tensor product of algebras over a commutative monad|tensor product of monad algebras]], see there for more.

## Examples

Many monads are named after their (free) algebras:

* The algebras of the _free monoid monad_ on [[Set]] are [[monoids]], and the morphisms of algebras the monoid homomorphisms.

* The algebras of the _free commutative monoid monad_ on [[Set]] are [[commutative monoids]], and their morphisms the monoid homomorphisms between them.

* The algebras of the _free group monad_ on [[Set]] are groups, and their morphisms are the group homomorphisms.
* ...and so on.

In these cases, the notion of _free group_, _free monoid_, et cetera coincide with the notion of free algebra given above.

* Given a [[monoid]] or [[group]] $M$, the algebras of the $M$-[[action monad]] on [[Set]] are the $M$-sets, i.e. sets equipped with an [[action]] of $M$. The morphisms are the [[equivariant maps]].
* The example above generalizes to [[action monads]] given by [[monoid objects]] in a general [[monoidal category]]. Famous examples of this construction in mathematics are smooth actions of [[Lie groups]] on [[manifolds]] and actions of [[rings]] on their [[modules]].

* The algebras of the [[maybe monad]] $(-)_*\colon Set \to Set$, which adds a disjoint point, are the pointed sets.
* The algebras of the [[power set]] monad are the sup-[[semilattices]].
* The algebras of the [[distribution monad]] are [[convex spaces]], and more generally algebras of [[probability monads]] correspond to generalized [[convex spaces]] or [[conical spaces]] (see [[probability monad#algebras_expectation_values|probability monad - algebras]]).

## Generalizations

An algebra over a monad is a special case of a [[module over a monad]] in a [[bicategory]]. See there for more information.

The Eilenberg-Moore and Kleisli categories are also special cases of more general 2-dimensional [[universal constructions]], namely the [[Eilenberg-Moore object]] and the [[Kleisli object]]. See those pages for more information.

## Related concepts

* **algebra over a monad**, [[module over a monad]], [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[algebra over a profunctor]]

  [[∞-algebra over an (∞,1)-monad]]

  * [[model structure on algebras over a monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

* [[Eilenberg-Moore category]], [[Kleisli category]], [[Eilenberg-Moore object]], [[Kleisli object]]

## References

### General

> See the *[References](monad#References)* at *[[monad]]*, such as:

* [[Saunders MacLane]], §VI.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

An introduction to the basic ideas, which gives some intuition for newcomers, can be found in

* [[Paolo Perrone]], Chapter 5 of: _Notes on Category Theory with examples from basic mathematics_,  [[arXiv:1912.10642](http://arxiv.org/abs/1912.10642)]

### Kleisli/extension system-style
 {#ReferencesKleisliStyle}

For monads presented in "[[extension system]]"/"[[Kleisli triple]]"-form (the way traditionally used for [[monads in computer science]] -- i.e. in terms of a "bind"-operation taking [[Kleisli maps]] to actual [[morphisms]], not explicitly referring to the monad product) there is the corresponding "Kleisli-triple style" or "Mendler style" [[Uustalu (2021), p. 4](#Uustalu21Lecture2)] for presenting the algebra/module-structures for these monads:

* {#MarmolejoWood10} [[F. Marmolejo]], [[Richard J. Wood]], Def. 3.1 in: *Monads as extension systems -- no iteration is necessary* [[TAC]] **24** 4 (2010) 84-113 [[tac:24-04](http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html)]

* [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], Def. 2.11 in: *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), doi:10.2168/LMCS-11(1:3)2015]

  > (stated in the generality of [[relative monads]])

* {#Uustalu21Lecture2} [[Tarmo Uustalu]], p. 4 of: *Monads and Interaction Lecture 2*  lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021) [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs2.pdf), [[Uustalu-Monads2.pdf:file]]]

See also

* [[R. F. C. Walters]], Chapter I of: *A categorical approach to universal algebra*, Ph.D. Thesis (1970) [[anu:1885/133321](https://openresearch-repository.anu.edu.au/handle/1885/133321)]

[[!redirects algebra for a monad]]
[[!redirects algebra over a monad]]
[[!redirects algebras of a monad]]
[[!redirects algebras for a monad]]
[[!redirects algebras over a monad]]
[[!redirects algebras for monads]]
[[!redirects algebras over monads]]

[[!redirects Eilenberg-Moore algebra]]
[[!redirects Eilenberg-Moore algebras]]


algebra over an operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

An [[operad]] is a structure whose elements are _formal operations_, closed under the operation of plugging some formal operations into others. An **algebra over an operad** is a structure in which the formal operations are interpreted as actual operations on an object, via a suitable [[action]].

Accordingly, there is a notion of [[module over an algebra over an operad]].

## Definition

Let $M$ be a [[closed monoidal category|closed symmetric monoidal category]] with monoidal unit $I$, and let $X$ be any object. There is a canonical or tautological [[operad]] $Op(X)$ whose $n^{th}$ component is the internal hom $M(X^{\otimes n}, X)$; the operad identity is the map

$$1_X: I \to M(X, X)$$

and the operad multiplication is given by the composite

$$\array{
M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1}, X) \otimes \ldots \otimes M(X^{\otimes n_k}, X) & \stackrel{1 \otimes func_\otimes}{\to} & M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1 + \ldots + n_k}, X^{\otimes k}) \\
 & \stackrel{comp}{\to} & M(X^{\otimes n_1 + \ldots + n_k}, X)
}$$

Let $O$ be any operad in $M$. An **algebra over** $O$ is an object $X$ equipped with an operad map $\xi: O \to Op(X)$. Alternatively, the data of an $O$-algebra is given by a sequence of maps

$$O(k) \otimes X^{\otimes k} \to X$$

which specifies an action of $O$ via finitary operations on $X$, with compatibility conditions between the operad multiplication and the structure of plugging in $k$ finitary operations on $X$ into a $k$-ary operation (and compatibility with actions by permutations).

An _algebra over an [[operad]]_ can equivalently be defined as a [[category over an operad]] which has a single [[object]].

If $M$ is cocomplete, then an operad in $M$ may be defined as a monoid in the symmetric monoidal category $(M^{\mathbb{P}^{op}}, \circ)$ of permutation representations in $M$, aka [[species]] in $M$, with respect to the substitution product $\circ$. There is an [[actegory]] structure
$M^{\mathbb{P}^{op}} \times M \to M$ which arises by restriction of the monoidal product $\circ$ if we consider $M$ as fully embedded in $M^{\mathbb{P}^{op}}$:

$$i: M \to M^{\mathbb{P}^{op}}: X \mapsto (n \mapsto \delta_{n 0} \cdot X)$$

(interpret $X$ as concentrated in the 0-ary or "constants" component), so that an operad $O$ induces a [[monad]] $\hat{O}$ on $M$ via the actegory structure. As a functor, the monad may be defined by a [[coend]] formula

$$\hat{O}(X) = \int^{k \in \mathbb{P}} O(k) \otimes X^{\otimes k}$$

An $O$-algebra is the same thing as an algebra over the monad $\hat{O}$.

**Remark** If $C$ is the [[symmetric monoidal category|symmetric monoidal]] [[enriched category|enriching category]], $O$ the $C$-enriched operad in question, and $A \in Obj(C)$ is the single [[hom-object]] of the [[category over an operad|O-category]] with single object, it makes sense to write $\mathbf{B}A$ for that $O$-category. Compare the discussion at [[monoid]] and [[group]], which are special cases of this.

## Examples

### Over single-coloured operads

* an [[associative algebra]] is an algebra over the [[associative operad]].

  * an [[A-infinity algebra]] is an algebra over a cofibrant [[resolution]] of $Assoc$.

* a [[commutative algebra]] is an algebra over the [[commutative operad]].

  * an [[E-infinity algebra]] is an algebra over a cofibrant [[resolution]] of the commutative operad

* etc.

### Over coloured operads

* There is a [[coloured operad]] $Mod_P$ whose algebras are pairs consisting of a $P$-algebra $A$ and a [[module]] over $A$;

* For a single-coloured operad $P$ there is a coloured operad $P^1$ whose algebras are triples consisting of two $P$ algebras and a [[morphism]] $A_1 \to A_2$ between them.

* Let $C$ be a set. There is a $C$-coloured operad whose algebras are $V$-[[enriched categories]] with $C$ as their set of objects.

## Literature

### Related $n$Lab entries

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* **algebra over an operad**

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

### Generalizations

* S. N. Tronin, _Algebras over multicategories_, Russ Math. (2016) 60: 52. [doi](http://dx.doi.org/10.3103/S1066369X16020092); Rus. original: С. Н. Тронин, _Об алгебрах над мультикатегориями_, Изв. вузов. Матем., 2016, № 2,  62–74

[[!redirects algebra over operad]]
[[!redirects algebra for an operad]]
[[!redirects algebra of an operad]]

[[!redirects algebras over an operad]]
[[!redirects algebras over operads]]

[[!redirects algebras over operad]]
[[!redirects algebras for an operad]]
[[!redirects algebras of an operad]]


algebra spectrum

+-- {: .rightHandSide}
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _algebra spectrum_ or _[[A-∞ algebra]]-spectrum_ over a [[ring spectrum]] is the analog in the [[higher algebra]] of [[stable homotopy theory]] of an [[associative algebra]] over a [[ring]] on ordinary [[algebra]].

## Definition

Abstractly, an _$A_\infty$-algebra spectrum_ over an [[E-∞ ring spectrum]] $R$ is [[algebra in an (∞,1)-category]] in the [[stable (∞,1)-category]] of $R$-[[module spectra]].

Concretely this [[(∞,1)-category]] is [[presentable (∞,1)-category|presented]] by the [[model structure on monoids]] in the [[monoidal model category|monoidal]] $R$-modules in the [[model structure on symmetric spectra]].

(...)

## Properties

### Stable monoidal Dold-Kan correspondence

Let $R := H \mathbb{Z}$ be the [[Eilenberg-MacLane spectrum]] for the [[integer]]s.

+-- {: .num_prop}
###### Proposition

There is a zig-zag of [[monoidal Quillen adjunction|lax monoidal]] [[Quillen equivalences]]

$$
  H \mathbb{Z} Mod
    \stackrel{\overset{Z}{\longrightarrow}}{\underset{U}{\leftarrow}}
  Sp^\Sigma(sAb)
    \stackrel{\overset{L}{\leftarrow}}{\underset{\phi^* N}{\longrightarrow}}
  Sp^\Sigma(Ch_+)
   \stackrel{\overset{D}{\longrightarrow}}{\underset{R}{\leftarrow}}
  Ch_\bullet
  \,,
$$

between [[monoidal model categories]] satisfying the [[monoid axiom in a monoidal model category]]:

* the model structure for $H \mathbb{Z}$-[[module spectra]];

* the [[model structure on symmetric spectrum objects]] in [[simplicial abelian group]]s and in [[chain complex]]es;

* and the [[model structure on chain complexes]] (unbounded).

This induces a [[Quillen equivalence]] between the corresponding [[model structures on monoids]] in these [[monoidal category|monoidal categories]], which on the left is the model structure on $H \mathbb{Z}$-algebra spectra and on the right the [[model structure on dg-algebras]]:

$$
  H \mathbb{Z} Alg \simeq dgAlg_\mathbb{Z}
  \,.
$$

=--

This is due to ([Shipley](#Shipley)). The corresponding [[equivalence of (∞,1)-categories]] for $R$ a commutative rings with the intrinsically defined [[(∞,1)-category]] of [[En-algebra|E1-algebra]] objects on the left appears as ([Lurie, prop. 7.1.4.6](#Lurie)).

+-- {: .num_remark}
###### Remark

This is a stable version of the [[monoidal Dold-Kan correspondence]]. See there for more details.

=--

## Related concepts

* [[ring spectrum]], [[module spectrum]]

* [[symmetric algebra spectrum]]

* [[A-∞ algebra]], [[algebra in an (∞,1)-category]]

## References

An account in terms of [[(∞,1)-category theory]] is in section 7.1.4 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

The equivalence of $H \mathbb{Z}$-algebra spectra with [[dg-algebra]]s is due to

* [[Brooke Shipley]], _$H \mathbb{Z}$-algebra spectra are differential graded algebras_ , Amer. Jour. of Math. 129 (2007) 351-379. ([arXiv:math/0209215](http://arxiv.org/abs/math/0209215))
 {#Shipley}

Eilenberg-MacLane spectra $H R$ for $R$ itself a [[dg-algebra]] are discussed in

* [[Daniel Dugger]], [[Brooke Shipley]], _Topological equivalences for differential graded algebras_ ([arXiv:math/0604259](http://arxiv.org/abs/math/0604259))

See also the references at [[stable homotopy theory]].

[[!redirects algebra spectra]]


algebrad

The notion of **algebrad** is due to

* [[Nikolai Durov]], _Classifying vectoids and generalisations of operads_, [arxiv/1105.3114](http://arxiv.org/abs/1105.3114), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273

based on the earlier conference talk:

* _Classifying vectoids and generalizations of operads_ , The International Conference "Contemporary Mathematics", June 12, 2009, [link](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=34084)

A **vectoid** is a finitely complete and [[cocomplete category]] $C$ with a [[small set]] of [[generators]], where all [[epimorphisms]] are universally [[effective epimorphism|effective]] and where the following "completeness/totality" axiom holds: every [[functor]]  $F : C^{op} \to Set$ commuting with all [[colimit]]s is [[representable functor|representable]].

The concept of vectoid simultaneously generalizes  [[topoi]] and [[abelian categories]] of $O$-[[module]]s for [[ringed topos|ringed topoi]]: intuitively it is roughly to the category of $O$-modules for a ringed topos $(X,O)$ what a [[generalized ring]] (algebraic monad in Set) is to a ring.
There are monoidal, symmetric monoidal and usual variant of vectoids; for monoidal versions one needs to impose a cocontinuity of the tensor product in each argument,
Vectoids are organized in a 2-category $Vectoid$ of vectoids. The name vectoid because of some analogies of that 2-category with the category of vector spaces, including a universal property of an external tensor product between vectoids which is similar to the universal property of the tensor product for vector spaces; where (bi)cocontinuous functors are analogous to (bi)cocontinuous maps. A monad in the 2-category of vectoids, that is a monoid with respect to the composition product, is called an **algebrad**.

A main source of algebrads are classifying vectoids; classifying vectoid  come from the problem of representing $Cat$-valued 2-presheaves on $Vectoid$. One looks at algebrads whose underlying 1-cell is an endomorphism of a classifying vectoid. Examples of the algebrads of that kind are symmetric [[operad]]s (which come from the classifier of objects), algebraic [[monad]]s (from the classifier of (cocommutative) [[coalgebra]]s) and a new type from the classifier of algebras. While the moduli spaces of algebras are hard to construct, the classifying vectoids are in these examples constructed with relatively little pain.

Abstract (in Russian, from [link](http://www.pdmi.ras.ru/EIMI/2009/cm/abs.html)):

Н. В. Дуров Классифицирующие вектоиды и обобщения операд

Первая треть доклада посвящена изложению основ теории вектоидов, представляющих собой совместное обобщение понятия топоса, окольцованного топологического пространства и обобщенного кольца. Далее будет обсуждаться задача построения "классифицирующего вектоида" или "пространства модулей" для различных алгебраических структур, и будет приведена простая комбинаторная конструкция таких вектоидов в простейших случаях (классификатор объектов, алгебр и коалгебр). Оказывается, что моноиды в категории эндоморфизмов классифицирующего вектоида представляют собой естественное обобщение понятие операды, зависящее от выбора классифицирующего вектоида. Например, классические операды получаются из классификатора объектов, а алгебраические монады - из классификатора коалгебр. Случай классификатора алгебр представляется не менее естественным, хотя соответствующее обобщение операды, по всей видимости, является новым. Изложению его основных свойств и будет посвящена заключительная часть доклада.

English: We start with a brief discussion of "vectoids", which are a common generalization of topoi, ringed spaces and generalized rings. After that we list several "classifying vectoid" or "moduli space" construction problems for different algebraic structures, and present a straightforward combinatorial construction of such classifying vectoids for the simplest cases (such as the classification of objects, algebras and coalgebras). Once a classifying vectoid is constructed, one can study monoids with respect to the composition in the category of its endomorphisms; such monoids turn out to be a natural generalization of the notion of an operad. Each choice of a classifying vectoid leads to its own kind of generalization, for example, the classifying vectoid of objects leads to (classical) operads, and that of coalgebras - to algebraic monads. The case of classifying vectoid of algebras seems to be as natural as these two other cases; however, corresponding generalization of operads, which we call "algebrads", appears to be new. Therefore, an elementary description of algebrads will be given.

(ZS: it seems that the time constraint unfortunately caused some cuts in the last part of the plan of the abstract)

(for now this entry also redirects vectoid)

+--{.query}
[[Todd Trimble|Todd]]: "Commutes with colimits" must really mean: $F: A^{op} \to Set$ takes colimits in $A$ to limits in $Set$, and the axiom is that such continuous functors are representable. This reminds me of notions of totality in category theory.

[[Mike Shulman]]: Yes, that exact condition has been studied by category theorists under the name of a "compact" category.  That's a terrible name, of course, so even the odd-sounding (to me) "vectoid" is better.  I think the original reference is Isbell's paper [Small subcategories and completeness](http://www.ams.org/mathscinet-getitem?mr=0224670), and one later one is [Compact and hypercomplete categories](http://www.ams.org/mathscinet-getitem?mr=614376) by Börger, Tholen, Wischnewsky, and Wolff.  The property is implied by totality (= the Yoneda embedding has a left adjoint), and implies hypercompleteness (= admits every limit which it could conceivably admit, subject to local smallness).
=--

## Related entries

* [[2-ring]]

[[!redirects vectoid]]
[[!redirects vectoids]]


algebraic analysis

**Algebraic analysis** is a program introduced by [[Mikio Sato]] from around 1958, based on the idea that the study of differential equations should be done in a coordinate-free manner, and operations should follow general nonsense geometric and algebraic constructions. One of the first steps was the introduction of the concept of [[D-module]], and of a [[holonomic D-module]], [[hyperfunction]]s (as a sheaf theoretic approach to distribution theory), then relying on homological algebra in [[derived categories]] (it seems that Sato introduced them independently from Grothendieck-Verdier, without publication at the time). Study of nonlinear and nonholonomic system was supposed to reduce on study of holonomic systems on more complicated spaces, e.g. on the cartesian square of the original space and so on. This depends on subtle properties of the study of singularities and other aspects and works only in some generality, with ongoing progress. Singularity theory and the lagrangian geometry are very important aspects of the algebraic analysis. Later Sato introduced [[microlocal analysis|microlocalization]] and his program joined young [[Masaki Kashiwara]] around 1968. Numerous connections to mathematical physics (e.g. [[holonomic quantum field]]s, [[integrable systems]]) and [[Hodge theory]] gradually entered into the program.

It seems that the vision of this program fits well with [[nPOV]]. See also [[D-geometry]].

* interview with [[Mikio Sato]] in [Notices AMS](http://www.ams.org/notices/200702/fea-sato-2.pdf)

* [[Masaki Kashiwara]], Takahiro Kawai, Tatsuo Kimura, _Foundations of algebraic analysis_, Transl. from Japanese by Goro Kato. Princeton Mathematical Series __37__, 1986. xii+255 pp.
[MR87m:58156](http://www.ams.org/mathscinet-getitem?mr=855641); [[J.-L. Brylinski]], Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) __18__ (1988), no. 1, 104–108, [doi](http://dx.doi.org/10.1090/S0273-0979-1988-15622-4)

* [[Tadao Oda]], *Introduction to algebraic analysis on complex manifolds*, Algebraic varieties and analytic varieties (Tokyo, 1981), 29–48, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam (1983) $[$[doi:10.2969/aspm/00110029](https://doi.org/10.2969/aspm/00110029), [MR85a:14010](http://www.ams.org/mathscinet-getitem?mr=715644)$]$

A short but quite complete overview of algebraic analysis can be found in:

* [[Pierre Schapira]] Triangulated categories for the analysts. In "Triangulated categories" London Math. Soc. LNS 375 Cambridge University Press, pp 371-389 (2010) [pdf](https://webusers.imj-prg.fr/~pierre.schapira/ResPapers/Tricat.pdf);


algebraic approaches to differential calculus

Derivatives and differentials are usually expressed in terms of limits in the sense of analysis. However it became clear in about the last half century that much of the knowledge on usual [[differential calculus]] can be inferred from using just algebraic properties of differentials and derivatives, most notably the Leibniz rule for differentiating products ($D (f g) = (D f) g + f (D g)$); an alternative [[synthetic differential geometry|synthetic]] formalism also appeared which did not use limiting procedures as well.

A [[derivation]] of an [[associative algebra]] is simply a linear [[endomorphism]] satisfying the Leibniz rule. Then for example the [[tangent vector fields]] on a [[smooth manifold]] are obtained as derivations of the algebra of $C^\infty$-functions on the manifold. The differential of a map is a linearized approximation. This is clear in various non-classical analytic setups, for example for maps between [[Banach spaces]] and for differentiable [[manifolds]].

This linearization idea has been obtained at the level of [[sheaves]] of $\mathcal{O}$-modules by [[Grothendieck]] for algebraic varieties a the view toward the differential calculus for varieties in prime characteristics. It is interesting that he related differential calculus to resolutions of the diagonal, where he considered the sheaves of modules supported on [[infinitesimal neighborhood]]s of diagonal. Indeed, to define a derivative in analysis, one needs to start with consideration of differences of values of a function at points which are close to one to another, $x$ and $x + \Delta x$, and this means that means that the pair $(x, x + \Delta x)$ is close to the diagonal of the cartesian square $X \times X$. In [[numerical analysis]], various approximation schemas for higher order differential operators are involved which obviously live around higher diagonals in $X^n = X \times X \times \ldots \times X$ ($n$ times). One of the products of that thinking is Grothendieck's notion of a [[regular differential operator]]. This led later to the creation of the theory of [[D-modules]] which are sheaves of modules over the [[sheaf]] of rings of regular differential operators over a scheme, or a complex analytic manifold.

We plan in the nLab to cover many aspects of the interaction between geometry and differential calculi of various sorts including [[synthetic differential geometry]] and algebraic counterparts of notions from differential calculi. It is hard to say, however, where [[homological algebra]] belongs: the differential in the sense of homological algebra is rather a notion which can be systematized into the more general subject of [[homotopical algebra]], but in some cases it is related to analogues of [[exterior differentiation]] for the de Rham [[complex]] of [[differential forms]] (say on manifolds). But there is also an analogue of the [[Taylor series]] for functors in some homotopical contexts (say [[Goodwillie calculus]]).

One should also point out that an elaborate schema for
differential calculus in
noncommutative geometry has been proposed by Tsygan
in terms of algebras with higher brackets. In one version,
a differential calculus is given there by a Gerstenhaber
algebra and a Batalin-Vilkovisky module over it.

See [[derivation]], [[regular differential operator]], [[differential form]], [[differential bimodule]], [[universal differential envelope]], [[differential forms in synthetic differential geometry]], [[connection]], [[connection for a differential graded algebra]], [[D-module]], [[Fox derivative]], [[crystal]], [[Fermat theory]]...

[[!redirects algebraic approach to differential calculus]]


algebraic category

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Algebraic categories
* table of contents
{: toc}

## Idea

An algebraic category is a [[concrete category]] which behaves very much like the categories familiar from [[algebra]], such as [[Grp]], [[Ring]], and [[Vect]], but characterised in category-theoretic terms.  But many other categories are also algebraic, most famously [[compact Hausdorff space|CompHausTop]]; one can describe these in purely algebraic terms, but only using infinitary (perhaps even largely many) operations.

There are several definitions of 'algebraic' in the literature.  Here, we will follow AHS (see references) in using a generous interpretation, but other authors follow Johnstone in using 'algebraic' to mean monadic (a stricter requirement), while some authors add finiteness conditions that remove examples such as $Comp Haus Top$.  However, all of these notions are related, and we will discuss them here.

The definitions in AHS also includes a requirement violating the [[principle of equivalence]], which we omit: that of unique strict lifts of isomorphisms, which serves to fix algebraic categories up to [[isomorphism of categories|isomorphism]] (instead of mere [[equivalence of categories|equivalence]]).

## Definitions

Let $A$ be a [[concrete category]]; that is, $A$ is equipped with a [[forgetful functor]] $U\colon A \to Set$ to the [[Set|category of sets]].  For some authors, such a category is called 'concrete' only if $U$ is [[representable functor|representable]], but that follows in all the cases considered below; in particular, if $A$ has free objects (that is, if $U$ has a [[left adjoint]] $F$), then $U$ is representable by $F(1)$, where $1$ is a [[singleton]].

+-- {: .un_defn}
###### Definition (based on AHS 23.38)

The concrete category $A$ is __algebraic__ if the following conditions hold:

*  $A$ has [[free objects]].
*  The category $A$ has all binary [[coequalizers]].
*  The forgetful functor $U$ preserves and reflects [[extremal epimorphisms]].
=--

+-- {: .un_defn}
###### Definition (based on AHS ...)

The concrete category $A$ is __monadic__ if the following conditions hold:

*  $A$ has [[free objects]].
*  The [[adjunction]] $F \dashv U$ is [[monadic adjunction|monadic]].
=--

+-- {: .un_defn}
###### Definition (based on AHS 24.11)

An algebraic (or monadic) category is __bounded__ if the following condition holds:

*  For some [[cardinal number]] $\kappa$ and every $\kappa$-[[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.
=--

+-- {: .un_defn}
###### Definition (based on AHS 24.4)

An algebraic (or monadic) category is __finitary__ if the following condition holds:

*  For every finitely [[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.
=--

Note that this is a weakening of the condition that the forgetful functor $U$ is [[finitary functor|finitary]] (that is, that $U$ preserves directed colimits); every universal cocone in $Set$ is jointly surjective, but not conversely.

## Properties

Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor $U$ preserves [[congruences]].  (AHS 23.41)

A category is algebraic if and only if it is a [[reflective subcategory]] of a monadic category with [[regular epimorphism|regular epic]] reflector; given an algebraic category, this monadic category is the [[Eilenberg–Moore category]] of the monad $U \circ F$.  (AHS 24.3)

Every monadic category is the category of algebras for some [[variety of algebras]], although we must allow potentially a [[proper class]] of infinitary axioms; that is, every monadic category is [[equationally presentable category|equationally presentable]].  Similarly, every algebraic category is the category of algebras for some [[quasivariety of algebras]]; that is, we allow [[conditional statements]] of equations among the axioms.  (AHS 24.11)

As special cases of the last item:

*  A concrete category is bounded monadic if and only if it is equationally presentable (presented by a variety) with a small set of operations (and hence equations).
*  A concrete category is bounded algebraic if and only if it is presented by a quasivariety with a small set of operations.
*  A concrete category is finitary monadic if and only if it is the category of algebras for some finitary variety; that is, we have only a small set of finitary operations.
*  A concrete category is finitary algebraic if and only if it is the category of algebras for some finitary quasivariety.

Also, every algebraic category whose [[forgetful functor]] preserves [[filtered colimits]] is the category of [[models]] for some [[first-order theory]]. The converse is false.

## Examples

The typical categories studied in [[algebra]], such as [[Grp]], [[Ring]], [[Vect]], etc, are all finitary monadic categories.  The [[monad]] $U \circ F$ may be thought of as mapping a set $x$ to the set of words with alphabet taken from $x$ and the connections between letters taken from the appropriate algebraic operations, with two words identified if they can be proved equal by the appropriate algebraic axioms.

The category of cancellative [[monoids]] is finitary algebraic but not monadic. The category [[Field]] of [[fields]] is not even algebraic.

Assuming the [[ultrafilter principle]], the category of [[compact Hausdorff spaces]] is monadic, but not bounded algebraic.  The monad in question takes a set $x$ to the set of [[ultrafilters]] on $x$.  (Without the ultrafilter principle, this monad still exists, but it may be quite small, possibly even the [[identity monad]]; passing to [[locales]] does not help.)

Similarly, the category of [[Stone space]]s is algebraic, but not monadic or bounded algebraic.

## References

The original definitions can be found in

* [[F. William Lawvere]], _Algebraic theories, algebraic categories, and algebraic functors_. 1965 Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), 413–418. North-Holland, Amsterdam.

Our definitions are taken from

*  **AHS**: [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]]; _Abstract and Concrete Categories: [[The Joy of Cats]]_, Sections 23 & 24; [web](http://katmat.math.uni-bremen.de/acc).

Actually, AHS discusses the more general concept of algebraic (etc) *functors*, generalising from $U\colon A \to Set$ to arbitrary functors (not necessarily faithful, not necessarily to $Set$).  We actually take our definitions from AHS\'s characterisation theorems in the case of faithful functors to $Set$.  We probably should discuss the more general concept, perhaps at [[algebraic functor]]; we already have [[monadic functor]].

*  [[Peter Johnstone]]; _[[Stone Spaces]]_, Section 3.8

For Johnstone, a concrete category is 'algebraic' if and only if it is monadic.  However, Johnstone also discusses [[equationally presentable category|equationally presentable categories]].

Another modern reference is

* [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico M. Vitale]],
_Algebraic theories_,
Cambridge Tracts in Mathematics 184 (2011), Cambridge University Press.
doi:10.1017/cbo9780511760754.

[[!redirects algebraic category]]
[[!redirects algebraic categories]]
[[!redirects bounded algebraic category]]
[[!redirects bounded algebraic categories]]
[[!redirects finitary algebraic category]]
[[!redirects finitary algebraic categories]]

[[!redirects monadic category]]
[[!redirects monadic categories]]
[[!redirects bounded monadic category]]
[[!redirects bounded monadic categories]]
[[!redirects finitary monadic category]]
[[!redirects finitary monadic categories]]


algebraic cobordism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Algebraic cobordism_ is the bigraded [[generalized cohomology theory]] represented by the [[motivic Thom spectrum]] $MGL$.  Hence it is the algebraic or [[motives|motivic]] analogue of [[complex cobordism]].  The $(2n,n)$-graded part has a geometric description via [[cobordism]] classes, at least over fields of characteristic zero.

## Definition

Let $S$ be a [[scheme]] and $MGL_S$ the [[motivic Thom spectrum]] over $S$.  **Algebraic cobordism** is the [[generalized motivic cohomology theory]] $MGL_S^{*,*}$ represented by $MGL_S$:

... formula here ...

## Properties

### The (2n,n)-graded part

Let $S = Spec(k)$ where $k$ is a field of characteristic zero.
A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by [[Marc Levine]] and [[Fabien Morel]].  More precisely, [Levine-Morel](#LevineMorel) constructed the universal _oriented cohomology theory_ $\Omega^* : \Sm_k \to CRing^*$.  Here _oriented_ signifies the existence of [[direct image]] or [[Gysin homomorphisms]] for [[proper morphisms of schemes]].  This implies the existence of[[Chern classes]] for [[vector bundles]].

+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
There is a canonical isomorphism of graded rings
  $$ \mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k)) $$
where $\mathbf{L}^*$ denotes the [[Lazard ring]] with an appropriate grading.
=--

+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
Let $i : Z \hookrightarrow X$ be a [[closed immersion]] of smooth $k$-schemes and $j : U \hookrightarrow X$ the complementary [[open immersion]].  There is a canonical exact sequence of graded abelian groups
  $$ \Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0, $$
where $d = \codim(Z, X)$.
=--

+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings
  $$ \Omega^*(k) \longrightarrow MU^{2*}(pt) $$
is invertible.
=--

+-- {: .num_theorem}
###### Theorem
**([Levine 2008](#Levine2008))**.
The canonical homomorphisms of graded rings
  $$ \Omega^*(X) \longrightarrow MGL^{2*,*}(X) $$
are invertible for all $X \in \Sm_k$.
=--

## Related concepts

* [[motivic Thom spectrum]]
* [[complex cobordism]]
* [[motivic homotopy theory]]
* [[motivic cohomology]]
* [[algebraic K-theory]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

There are two notions of "algebraic cobordism", not closely related, one due to [Snaith 77](#Snaith77), and one due to [Levine-Morel 01](#LevineMorel01).

### Snaith's construction
 {#SnaithConstructiom}

* {#Snaith77} [[Victor Snaith]],  _Towards algebraic cobordism_,  Bull. Amer. Math. Soc. 83 (1977), 384-385 ([doi:10.1090/S0002-9904-1977-14281-X](https://doi.org/10.1090/S0002-9904-1977-14281-X))

* {#Snaith79} [[Victor Snaith]], _Algebraic Cobordism and K-theory_, Mem. Amer. Math. Soc. no 221 (1979)

* {#GepnerSnaith08} [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math.  2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817))

The construction in [Snaith 77](#Snaith77), motivated from the [[Conner-Floyd isomorphism]], uses a variant of his general construction ([[Snaith's theorem]]) of a periodic [[multiplicative cohomology theory]] $X(b)^*(-)$ out of a pair consisting of a homotopy commutative [[H-monoid]] $X$ and a class $b\in \pi_n(X)$:

When $X = B S^1$ (the [[classifying space]] of the [[circle group]]) and $b$ is a generator of $\pi_2(BS^1)\cong\mathbb{Z}$ then $X(b)^*(-)$ is isomorphic with 2-periodic [[complex K-theory]].

When $X = B U$ and $b$ a generator of $\pi_2(BU)\cong\mathbb{Z}$ one obtains $MU^*[u_2,u_2^{-1}]$ where [[MU]] is the (topological) [[complex cobordism cohomology]] and $u_2$ is the periodicity element.

Then Snaith introduces a variant of such constructions with a more general ring $A$ replacing the complex numbers; and uses the Quillen's description of [[algebraic K-theory]] of a ring $A$ in terms of the [[classifying space]] $B GL(A)$; this way he obtains an algebraic cobordism theory.

Later, [Gepner-Snaith 08](#GepnerSnaith08) returned to the question of algebraic cobordism this time using the motivic version of algebraic cobordism of Voevodsky, namely the [[motivic spectrum]] $M GL$ representing universal oriented motivic cohomology theory (which is different from Morel-Voevodsky algebraic cobordism), and to the motivic version of [[Conner-Floyd isomorphism]] for which they give a comparably short proof.

### Morel-Levine's construction

* {#LevineMorel01} [[Marc Levine]], [[Fabien Morel]], _Cobordisme algébrique I_, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723--728, 2001 (doi:10.1016/S0764-4442(01)01833-X); _Cobordisme algébrique II_, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815--820, 2001 (doi:10.1016/S0764-4442(01)01832-8).

* [[Marc Levine]], _Algebraic cobordism_, Proceedings of the ICM, Beijing 2002, vol. 2, 57--66, [math.KT/0304206](http://arxiv.org/abs/math/0304206)

* {#LevineMorel} [[Marc Levine]], [[Fabien Morel]], _Algebraic cobordism_, Springer 2007, [pdf](https://www.uni-due.de/~bm0032/publ/AlgCobordBook4.pdf).

* [[Marc Levine]], _A survey of algebraic cobordism_ ([pdf](http://www.uni-due.de/~bm0032/publ/SurveyAlgCobord.pdf))

* [[Marc Levine]], _Three lectures on algebraic cobordism_, University of Western Ontario Mathematics Department, 2005, [Lecture I](http://www.uni-due.de/~bm0032/publ/CobordismLec1LS.pdf), [Lecture II](http://www.uni-due.de/~bm0032/publ/CobordismLec2LS.pdf), [Lecture III](http://www.uni-due.de/~bm0032/publ/CobordismLec4LS.pdf).

* [[Marc Levine]], [[Fabien Morel]], [[Oberwolfach]] Arbeitsgemeinschaft mit aktuellem Thema, April 2005 [report](http://www.ems-ph.org/journals/show_abstract.php?issn=1660-8933&vol=2&iss=2&rank=2), [notes](http://www.ems-ph.org/journals/show_pdf.php?issn=1660-8933&vol=2&iss=2&rank=2)

* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _A universality theorem for Voevodsky's algebraic cobordism spectrum_, Homology, Homotopy and Applications, 2008, 10(2), 211-226 ([arXiv:0709.4116](http://arxiv.org/abs/0709.4116))

* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory_, Inventiones mathematicae, 2009, 175(2), 435-451, [doi:10.1.1.244.7301](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301), [arXiv:0709.4124](http://arxiv.org/abs/0709.4124).

* [[Marc Hoyois]], _From algebraic cobordism to motivic cohomology_, [pdf](http://math.mit.edu/~hoyois/papers/hopkinsmorel.pdf), [arXiv](http://arxiv.org/abs/1210.7182).

* [[Markus Spitzweck]], _Algebraic cobordism in mixed characteristic_ ([arXiv:1404.2542](http://arxiv.org/abs/1404.2542))

* [[Marc Levine]], [[Girja Shanker Tripathi]], _Quotients of MGL, their slices and their geometric parts_, [arXiv:1501.02436](http://arxiv.org/abs/1501.02436).

More chat about the relation to [[motivic homotopy theory]]:

* _Interdependence between A^1-homotopy theory and algebraic cobordism_, [MO/36659](http://mathoverflow.net/questions/36659/interdependence-between-a1-homotopy-theory-and-algebraic-cobordism/36698#36698).

A simpler construction was given in

* M. Levine, R. Pandharipande, _Algebraic cobordism revisited_ ([math.AG/0605196](http://arxiv.org/abs/math/0605196))

A [[Borel-Moore homology]] version of $MGL^{*,*}$ is considered in

* [[Marc Levine]], _Oriented cohomology, Borel-Moore homology and algebraic cobordism_, [arXiv](http://arxiv.org/abs/0807.2257).

The comparison with $MGL^{2*,*}$ is in

* {#Levine2008} [[Marc Levine]], _Comparison of cobordism theories_, Journal of Algebra, 322(9), 3291-3317, 2009, [arXiv](http://arxiv.org/abs/0807.2238).

The construction was extended to [[derived algebraic geometry|derived schemes]] in the paper

* Parker Lowrey, [[Timo Schuerg]].  _Derived algebraic bordism_, 2012, [arXiv:1211.7023](http://arxiv.org/abs/1211.7023).

The close connection of algebraic cobordism with [[K-theory]] is discussed in

* José Luis González, Kalle Karu.  _Universality of K-theory_.  2013.  [arXiv:1301.3815](http://arxiv.org/abs/1301.3815).

An algebraic analogue of h-cobordism:

* Aravind Asok, Fabien Morel, _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_ ([arXiv:0810.0324](http://arxiv.org/abs/0810.0324)).

A construction of algebraic cobordism as a non-[[A1-homotopy theory|$\mathbb{A}^1$-invariant]] [[cohomology theory]] on [[derived schemes]] and the resulting [[Conner-Floyd isomorphism]]:

* Toni Annala, [[Marc Hoyois]], Ryomei Iwasa, _Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory_ ([arXiv:2303.02051](https://arxiv.org/abs/2303.02051)).

[[!redirects algebraic cobordism spectrum]]
[[!redirects algebraic cobordism spectra]]


algebraic correspondences

See [[pure motives]] for now.


algebraic curve

#Contents#
* table of contents
{:toc}

## Idea

An algebraic curve is an [[algebraic variety]] of dimension $1$. Typically
one restricts considerations to either affine or projective algebraic curves. Most often one treats the plane algebraic curves, i.e. curves
with an embedding into $\mathbf{A}^2$ or $\mathbf{P}^2$; they are the locus of solutions of a single algebraic equation.

An _algebraic curve_ over a [[field]] $F$ is the locus of solutions of $(n-1)$-[[polynomials]] in $n$-[[variables]] of [[type]] $F$, provided the
Krull dimension of the ring is $1$.

## Properties

### General

* Every projective algebraic curve is [[birationally equivalent]] to a plane algebraic curve
* [[Mordell conjecture]]: every algebraic curve of genus $g\geq 2$ defined over rationals has at least one point over rationals
* To a nonsingular curve $C$ over the field of complex numbers one associates an [[abelian variety]], namely its [[Jacobian variety]] together with the period map or Abel-Jacobi map $C\to J(C)$.

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[complex curve]], [[Riemann surface]]

* [[arithmetic curve]]

* [[elliptic curve]]

* [[stable curve]],

* [[moduli stack of curves]]

* [[Jacobian variety]],

* [[Mordell conjecture]],

* [[Bezout's theorem]]

## References

* [[William Fulton]], _Algebraic curves_ ([pdf](http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf))

* Wikipedia, _[Algebraic curve](http://en.wikipedia.org/wiki/Algebraic_curve)_

* Patrick J. Morandi, _Error Correcting Codes and Algebraic Curves_ , lecture notes New Mexico State University 2001. ([pdf](http://www.math.nmsu.edu/~pmorandi/math601f01/LectureNotes.pdf))

[[!redirects algebraic curves]]


algebraic cycle

#Contents#
* table of contents
{:toc}

## Definition

Let $X$ be a [[noetherian scheme]].  One defines a **$k$-dimensional algebraic [[cycle]]** as an element of the [[free abelian group]] $Z_k(X)$ generated by the [[closed subspace|closed]] [[integral scheme|integral]] subschemes of [[dimension]] $k$, and dually a $k$-[[codimension|codimensional]] cycle is an element of the [[free group]] $Z^k(X)$ generated by the closed integral subschemes of [[codimension]] $k$ in $X$.  (An important special case is the group $Z^1(X)$ of 1-codimensional cycles, better known as the group of **[[Weil divisors]]**.)  One usually writes a cycle as a formal sum

$$ C = \sum_{Z \subset X} n_Z.[Z] $$

## Direct image and inverse image of cycles

For [[proper morphisms]] $f : X \to Y$, one defines the [[direct image]] of a $k$-cycle by assigning

$$ f_*([Z]) = \deg_{R(f(Z))}(R(Z)) . [f(Z)] $$

when $\dim(f(Z)) = \dim(Z)$ and 0 otherwise.  Here $f(Z)$ is considered as an integral subscheme of $Y$ with the [[reduced scheme|reduced]] subscheme structure induced from $Y$.  $R(Z)$ denotes the field of [[rational functions]] on $Z$ and $\deg$ denotes the degree of the [[field extension]].  One gets homomorphisms $f_* : Z_k(X) \to Z_k(Y)$ for each $k$.

For [[flat morphisms]] of [[relative dimension]] $n$, one defines the inverse image of a $k$-cycle by assigning, for a closed integral subscheme $Z \subset Y$ of dimension $k$,

$$ f^*([Z]) = \sum_{Z_\alpha \subset f^{-1}(Z)} \length_{O_{X,z_\alpha}}(O_{f^{-1}(Z),z_\alpha}) [Z_\alpha] $$

where the sum is taken over the [[irreducible components]] $Z_\alpha$ of $f^{-1}(Z)$, $\length$ denotes length of [[modules]], and $z_\alpha$ are the [[generic points]] of $Z_\alpha$.  Hence one gets homomorphisms $f^* : Z_k(Y) \to Z_{k+n}(X)$.

## Weil divisors and rational functions

A **[[Weil divisor]]** on $X$ is a 1-codimensional cycle.

A [[rational function]] $r \in R(X)$ on an [[integral scheme]] $X$ corresponds via canonical isomorphisms $R(X) \to \Frac(O_{X,x})$, for every $x \in X$, to elements $a_x/b_x \in \Frac(O_{X,x})$, and one defines the **order of vanishing** of $r$ at $x$ as

$$ \ord_x(r) = \length_{O_{X,x}}(O_{X,x}/(a_x)) - \length_{O_{X,x}}(O_{X,x}/(b_x)) $$

where $\length$ denotes length of [[modules]].

Then one defines the **[[Weil divisor]] associated to the rational function $r$** as

$$ div(r) = \sum_{Z \subset X} \ord_z(r).[Z] $$

where the sum goes over closed integral subschemes $Z$ of codimension 1 and with [[generic point]] $z \in Z$.

## Related concepts

* [[Chow groups]]

* [[Cartier divisor]]

* [[Abel-Jacobi map]]

* [[standard conjectures]]

## References

Standard references are

* [[EGA IV]], section 21
* [[Stacks Project]], [02QQ](http://stacks.math.columbia.edu/tag/02QQ)

On the relation with [[Weil cohomology theories]], [[algebraic K-theory]], [[Beilinson-Lichtenbaum conjectures]], and [[motivic cohomology]]:

* [[Marc Levine]], _Algebraic cycle complexes_, talk notes, 2008, [pdf](https://www.uni-due.de/~bm0032/publ/CycleComplexes.pdf).

A relation to [[iterated integrals]] and [[diffeological spaces]] is discussed in

* [[Richard Hain]], _Iterated Integrals and Algebraic Cycles: Examples and Prospects_ ([arXiv:math/0109204](http://arxiv.org/abs/math/0109204))

[[!redirects algebraic cycles]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

* [[geometric definition of higher categories]]

* **algebraic definition of higher categories**

***

# Contents
* automatic table of contents goes here
{:toc}

## Idea

In an _algebraic definition_ of [[(n,r)-categories]] [[composition]] of [[higher morphism]]s is a genuine algebraic operation. As opposed to a _geometric model_ where composites are only guaranteed to exist, in an algebraic model there is prescription for finding these composites.

These choices may be given in an elementary way as binary composition operations as in an ordinary [[category]], or the composition operation may be controled by sophisticated algebraic structures such as [[operad]]s that, while keeping track of all specified composites, may allow large combinations of possible composition operations.

The composition operation in an algebraic model for an $(n,r)$-category are subject to [[associativity]]  [[coherence law]]s. On the geometric side these reflect the fact that the spaces of possible choices of composites are contractible.

Typically an algebraic model for higher categories admits a [[nerve]] operation that turns it into an equivalent geometric model. Conversely, typically one can obtain an algebraic model from a geometric model by making _choices_ of composites.

## Examples

* The series of notions

  * [[category]], [[groupoid]]

  * [[bicategory]]

  * [[tricategory]]

  * [[tetracategory]]

  are algebraic models for [[n-categories]] with $1 \leq n \leq 4$ given in terms of direct explicit operations: no [[operad]]s or other tools are used but all the possible composition operations are defined elementarily and all the relevant [[coherence law]]s are demanded explicitly. This makes these models very concrete and hands-on. But it also has the disadvantage that beginning with tricategories these definitions become quite unwieldy.

* For [[strict n-categories]] all subtleties with [[associator]]s and [[coherence law]]s are absent (by definition) and therefore there are straightforward algebraic models for these See [[strict omega-category]], [[strict omega-groupoid]] and [[n-fold category]]. The drawback is of course that these strict models capture only a very restricted part of higher category theory.

* The [[Michael Batanin|Batanin]]/[[Tom Leinster|Leinster]] approach to higher categories involves algebraic structure imposed all at once (using higher [[operad|operads]]) on a globular set. See [[Batanin omega-category]].

* The [[Todd Trimble|Trimble]]/[[Peter May|May]] approach to higher categories involves algebraic structure imposed in stages by a process of iterative enrichment. See also [[Trimble n-category]].

* These models turn out to be closely related to an original idea by [[Alexandre Grothendieck]] that was resurrected and formalized by [[Georges Maltsiniotis]]: [[Grothendieck-Maltsiniotis ∞-categories]]

* Using the tool of [[model structure on algebraic fibrant objects]] many geometric models for higher categories may be realized equivalently as algebraic models. This is notably true for [[∞-groupoid]]s and [[(∞,1)-categories]]. See [Algebraic  fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories).

## Properties

When an [[(∞,1)-category]] of [[(n,r)-categories]] is presented by a [[model category]], then algebraic models tend to be fibrant objects (while geometric models tend to be cofibrant objects). For instance in all the [[folk model structure]] on algebraic higher categories and groupoids, all objects are fibrant.

In a strict version of the [[homotopy hypothesis]], one may make this 'algebraicity' of the model structure (not to be confused with the notion at [[algebraic model structure]]) a requirement.

[[!redirects algebraic definition of higher categories]]
[[!redirects algebraic definitions of higher categories]]
[[!redirects algebraic definition of higher category]]
[[!redirects algebraic model of higher categories]]
[[!redirects algebraic model of higher category]]


algebraic dynamics

While standard dynamics studies processes in real time, sometimes it also considers discrete dynamical systems. The step of discrete dynamics is in the appropriate category.

In __algebraic dynamics__ one typically studies discrete dynamical systems on algebraic varieties. Such a system is given by a regular endomorphism $D: X\to X$ of a variety $X$.

* L. Szpiro, _Algebraic dynamics_, [pdf](http://wfs.gc.cuny.edu/lszpiro/www/papers/AlgebraicDynamics2.pdf)

The case over [[number field]]s is also called arithmetic dynamics,
see

* wikipedia [arithmetic dynamics](http://en.wikipedia.org/wiki/Arithmetic_dynamics)


algebraic effect

[[!redirects algebraic side effect]]
[[!redirects algebraic side effects]]
[[!redirects algebraic effects]]

#Contents#
* table of contents
{:toc}

## Idea

**Algebraic effects** are computational [[side effect|effects]] that can be represented by an equational theory, or [[algebraic theory]], whose operations produce the effects at hand.

They provide an alternative to using [[monads (in computer science)]] for modelling _effects_ with a [[functional programming language]]. This approach has been realized in the programming language [[Eff]].

[[continuation monad|Continuations]] are not algebraic effects.

## References

* [[Andrej Bauer]], _[Programming with algebraic effects and handlers](http://math.andrej.com/2012/03/08/programming-with-algebraic-effects-and-handlers/)_


algebraic element

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

\tableofcontents

## Definition

Given a [[field]] $K$ and a field extension $K \subseteq F$, an element $\alpha \in F$ is **algebraic** if the subfield $K(\alpha) \subseteq F$ is a finite-degree [[field extension]] of $K$.

## Related concepts

* [[algebraic number]]

* [[algebraic extension]]

* [[transcendental element]]

## References

See also

* Wikipedia, _[Algebraic element](https://en.wikipedia.org/wiki/Algebraic_element)_

[[!redirects algebraic element]]
[[!redirects algebraic elements]]


algebraic equation

#Contents#
* table of contents
{:toc}

## Idea

An [[algebraic equation]] is an [[equation]] of the form

$$
  P(x_1, \cdots, x_n) = 0
$$

where $P(x_1, \cdots, x_n)$ is a [[polynomial]] with [[coefficients]] in some [[field]].

## Related concepts

* [[linear equation]]

* [[Diophantine equation]]

## References

* Wikipedia, _[Algebraic equation](http://en.wikipedia.org/wiki/Algebraic_equation)_

[[!redirects algebraic equations]]


algebraic extension

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[field extension]]

## Related concepts

* [[perfect field]]

* [[transcendental extension]]

## References

* Wikipedia, _[Algebraic extension](http://en.wikipedia.org/wiki/Algebraic_extension)_

[[!redirects algebraic extensions]]


algebraic function

## References

* Wikipedia, _[Algebraic function](http://en.wikipedia.org/wiki/Algebraic_function)_

[[!redirects algebraic functions]]


algebraic fundamental group

#Contents#
* table of contents
{:toc}

## Idea

The _algebraic fundamental group_ is the [[fundamental group]] of a [[scheme]], as defined by [[Grothendieck]] in [[SGA1]]. It is essentially the fundamental group as seen by [[étale homotopy]], the _[[étale fundamental group]]_.

For [[fields]] this is essentially the [[Galois group]]. For [[smooth varieties]] it is the [[Galois group]] of the maximal [[unramified extension]] of the [[function field of a variety|function field]] of the variety (e.g. [MK 09, p. 3](#Kim09)).

In [[arithmetic geometry]] one also speaks of the _arithmetic fundamental group_.

##Definition

 Let $S$ be a connected [[scheme]]. Recall that a finite [[étale cover]] of $S$ is a finite flat
surjection $X\to S$ such that each fibre at a point $s \in S$ is the spectrum of a finite [[étale algebra]] over the local ring at $s$. Fix a [[geometric point]] $\overline{s} : Spec(\Omega) \to \Omega$.

For a finite étale cover, $X\to S$,  we consider the [[geometric fibre]], $X\times_S Spec (\Omega)$, over  $\overline{s}$, and denote by $Fib_\overline{s} (X)$ its
underlying set. This gives a set-valued functor on the category of finite étale covers
of $X$.

The *algebraic fundamental group*, $\pi_1(S,  \overline{s})$ is defined to be the automorphism group
of this functor.

For more on this area, see at _[[étale homotopy]]_.

(This entry is a stub and needs more work, including the linked entries that do not yet exist! Also explanation of $\Omega$. It is adapted from the first reference below.)

##References

* [[Tamás Szamuely ]], [Heidelberg Lectures on Fundamental Groups](http://pagine.dm.unipi.it/tamas/pia.pdf), 2010.

or in a lengthier form:

*   [[Tamás Szamuely ]], Galois Groups and Fundamental Groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, 2009.

An earlier version is to be found [here](http://math.uchicago.edu/~aanders/books/szamuely_-_galois_groups_and_fundamental_groups.pdf).

A paper on a closely related subject is

* Feng-Wen An, _On the arithmetic fundamental groups_ ([arXiv:0910.0605](http://arxiv.org/abs/0910.0605))

See also

* {#Kim08} [[Minhyong Kim]], _Fundamental groups and Diophantine geometry_, 2008 ([pdf](http://www.ucl.ac.uk/~ucahmki/leeds.pdf))

* {#Kim09} [[Minhyong Kim]], _Galois theory and Diophantine geometry_, 2009 ([pdf](http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf))

[[!redirects algebraic fundamental groups]]

[[!redirects arithmetic fundamental group]]
[[!redirects arithmetic fundamental groups]]


algebraic geometry

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#### Algebra
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#Contents#
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## Overview

__Algebraic geometry__ is, in origin, a [[geometry|geometric study]] of solutions of systems of [[polynomial]] equations and generalizations.

The set of zeros of a set of polynomial equations in finitely many variables over a [[field]] is called an __affine [[algebraic variety|variety]]__ and it is equipped with a particular topology called [[Zariski topology]], whose closed sets are subvarieties. The system of polynomial equations defines an [[ideal]] in the ring of polynomials over the [[ground field]]; one of the first insights of algebraic geometry is that the ideal is a more invariant notion than the original set of equations. The [[quotient object|quotient]] of the ring of polynomials by the defining ideal is the ring of coordinate functions of the affine variety; a basic theorem asserts that this ring is [[noetherian ring|Noetherian]] algebra over the ground field. This ring determines the variety up to a natural notion of [[isomorphism]] of varieties. __Projective varieties__ are zeroes of systems of homogeneous polynomial equations in projective $n$-dimensional space; if the coordinates are rescaled by a common nonzero multiple, then by definition they still define the same point in projective space. Zariski open subsets of affine (or projective) varieties are called __quasiprojective varieties__.

It is often useful to take a [[pullback]] along a morphism of fields, usually called [[base change]]. Since [[Alexander Grothendieck|Grothendieck]], one generalizes the coordinate rings of affine varieties to arbitrary commutative unital rings, not necessarily Noetherian nor finitely generated; and interprets the [[opposite category]] of the category of commutative rings as a category of affine [[scheme]]s $\mathrm{Aff}$; affine schemes are traditionally constructed by the affine [[spectrum]] functor $\mathrm{Spec}:\mathrm{CommRing}^{\mathrm{op}}\to\mathrm{lrSp}$ into the category of locally [[ringed space|ringed]] topological spaces. Points of the affine spectrum are the prime ideals of the ring. Assuming the [[axiom of choice]], every affine scheme corresponding to a ring with $0 \neq 1$ has therefore at least one point. The slice category $\mathrm{Aff}/(\mathrm{Spec} F)$ over a spectrum of a fixed field $F$ contains the category of varieties over $F$ as a full subcategory. However the points of an affine variety and of the corresponding affine spectrum do not coincide: only *maximal* ideals are points of usual varieties. Affine schemes have a natural topology, also called a Zariski topology; the ringed spaces locally isomorphic to affine schemes are called __schemes__. Schemes include projective schemes and more generally quasiprojective schemes; if they are relative over a fixed [[ground field]], then they contain the subcategories of projective (resp. quasiprojective) varieties over the same field.

Although [[Grothendieck]] in the late 1950s envisioned many generalizations of [[scheme]] theory, his coauthor Dieudonné wrote later in EGA that algebraic geometry is the study of algebraic and [[formal schemes]], which is clearly a too dogmatic definition. Grothendieck's school studied in addition locally affine spaces in various [[Grothendieck topology|Grothendieck topologies]] on $\mathrm{Aff}$ (including [[algebraic space]]s), [[algebraic stack]]s ([[Deligne-Mumford stack]]s and [[Artin stack]]s), [[ind-object|ind-schemes]] and so on; in SGA the study of ringed spaces is replaced by more general [[ringed site]]s and [[ringed topos|ringed topoi]]. Modern generalizations include derived schemes, almost schemes (with the theory of almost rings of Gabber developing after some ideas of [Faltings](http://de.wikipedia.org/wiki/Gerd_Faltings)), generalized schemes of [[Nikolai Durov]], so-called schemes over the '[[field of one element]]' $F_1$ of various authors, dg-schemes, slightly noncommutative [[D-scheme]]s etc. Many ideas of scheme theory and the spectral theory of rings have influenced parallel developments in the analytic setup (Stein manifolds ([see the English Wikipedia](http://en.wikipedia.org/wiki/Stein_manifold)), rigid [[analytic geometry|analytic spaces]], etc.) and in the noncommutative setup give rise to [[noncommutative algebraic geometry]]. Deligne has also suggested how to do algebraic geometry in an arbitrary symmetric monoidal category.

## Perspective of structured $(\infty,1)$-toposes

Grothendieck took the viewpoint that the schemes, algebraic spaces etc. are sheaves on $Aff$ in some subcanonical Grothendieck topology (functor of points point of view).

Algebraic geometry starts with study of [[space]]s that are locally modeled on ([[object]]s in the [[category]]) Aff = [[CRing]]${}^{op}$ -- main categories being of algebraic [[scheme]]s and of [[algebraic space]]s; one also allows infinitesimal thickenings leading to formal schemes and other ind-objects in schemes. Hakim, [[Deligne]] and [[Ofer Gabber|Gabber]] extend the setup internally to a symmetric monoidal category, where $Aff$ is replaced by the opposite to the category of monoids in that category; [[Nikolai Durov|Durov]] on the other side takes monoids in the category of endofunctors in [[Set]], i.e. [[monad]]s as the opposite to the local objects in a generalized scheme theory.

This is to be compared to and contrasted with for instance [[differential geometry]], that studies [[space]]s locally modeled on ([[object]]s in the [[category]]) [[CartSp]] -- called (smooth) [[manifold]]s.

The general formalization of the notion of [[space]] and hence, by the general lore of [[space and quantity]], that of [[sheaf]], [[stack]], [[∞-stack]] has originally been crucially inspired by [[Grothendieck]]'s work on algebraic geometry and is more recently being greatly revived and further extended by the developments in [[derived algebraic geometry]] by people like [[Bertrand Toen]] and [[Jacob Lurie]].

In particular in [[Structured Spaces]] Lurie presents a general formalism of [[generalized scheme]]s that encompasses the [[space]]s studied in algebraic geometry and [[derived algebraic geometry]] just as well as ordinary [[smooth manifold]]s, [[derived smooth manifold]] and harmonizes with other axiomatizations such as in [[synthetic differential geometry]]: all of these [[space]] object are realized as special cases of [[structured (∞,1)-topos]]es, differing only in the choice of [[geometry (for structured (∞,1)-toposes)]] that they are modeled on.

From this perspective,

* ordinary algebraic geometry is the study of [[structured (∞,1)-topos]]es for the [[geometry (for structured (∞,1)-toposes)|Zariski or etale geometry]] $\mathcal{G}_{Zar}$, $\mathcal{G}_{et}$ on [[CRing]]${}^{op}$. In fact one has as series of geometries for every integer $n\geq 0$, where classical case is at level $0$ and derived at level $\infty$. Cf. 4.2.9 in [[Structured Spaces]]. The fully faithful embedding of schemes into derived schemes does not commute with limits, what is relevant e.g. for intersection theory.

* [[derived algebraic geometry]] is the study of [[structured (∞,1)-topos]]es for the [[geometry (for structured (∞,1)-toposes)|Zariski or etale pre-geometry]] $\mathcal{T}_{Zar}$, $\mathcal{T}_{et}$ on [[CRing]]${}^{op}$.

Despite of this, an axiomatic formulation of algebraic geometry along the lines of [[synthetic differential geometry]], that would de-emphasize the peculiarities of $CRing^{op}$ and emphasize structural aspects such as to facilitate for instance the transportation or interpretation of results in algebraic geometry to other geometries, is currently hardly to be found in the elementary literature. [[SGA]], specially SGA IV was written however to reflect "algebraic" geometry over any topos.

>Maybe we could talk more about [[synthetic differential geometry applied to algebraic geometry]] to unify perspective of algebraic with differential geometry.

## Related pages

* [[synthetic algebraic geometry]]

* [[contributors to algebraic geometry]]

* [[books in algebraic geometry]],

* [pages in the category "algebraic geometry"](http://www.ncatlab.org/nlab/list/algebraic+geometry)

* [[functorial geometry]]

* [[derived algebraic geometry]]

* [[noncommutative geometry]]

## References
 {#References}

Original references: [[EGA]] and [[SGA]].

Textbook accounts:

* {#ShafarevichVol1} [[Igor Shafarevich]], *Basic Algebraic Geometry 1 -- Varieties in Projective Space*, Springer (1977, 1994, 2013) [[pdf](http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf), [doi:10.1007/978-3-642-57908-0](https://link.springer.com/book/10.1007/978-3-642-57908-0)]

* [[Siegfried Bosch]], *Algebraic Geometry and Commutative Algebra*, Universitext, Springer (2017) [[doi:10.1007/978-1-4471-4829-6](https://doi.org/10.1007/978-1-4471-4829-6)]

*  Ulrich Görtz, Torsten Wedhorn, *Algebraic Geometry I: Schemes*, Springer (2020) [[doi:10.1007/978-3-658-30733-2](https://doi.org/10.1007/978-3-658-30733-2)]

* * [[Robin Hartshorne]], Algebraic Geometry, Graduate Texts in Mathematics volume 52, Springer 1977. [ [doi link](https://doi.org/10.1007/978-1-4757-3849-0) ]

Lecture notes:

* [[Ravi Vakil]], _Foundations Of Algebraic Geometry_, Course notes ([web](http://math.stanford.edu/~vakil/216blog/))

See also the references at _[[functorial geometry]]_.

and see

* _[[The Stacks Project]]_

On [[synthetic algebraic geometry]], i.e. via the [[internal logic]] of the [[sheaf topos]] over a [[scheme]] ([[Zariski topos]]/[[étale topos]]):

* {#Blechschmidt15} [[Ingo Blechschmidt]], _Using the internal language of toposes in algebraic geometry_, talk at [Toposes at IHES](https://indico.math.cnrs.fr/event/747/), November 2015 ([pdf](https://github.com/iblech/internal-methods/blob/master/slides-ihes2015.pdf), [recording](https://www.youtube.com/watch?v=7S8--bIKaWQ))

* [[Ingo Blechschmidt]], *Using the internal language of toposes in algebraic geometry*, PhD thesis (2017) [[pdf](https://rawgit.com/iblech/internal-methods/master/notes.pdf), [[Blechschmidt-InternalLanguage.pdf:file]]]

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

* [[Felix Cherubini]], *A Foundation for Synthetic Algebraic Geometry*, talk at *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (Oct 2023) [slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cherubini-2023-10-18-HoTTEST.pdf), video:[YT](https://www.youtube.com/watch?v=lp4kcmQ0ueY)]

For more see:

* [[Ingo Blechschmidt]], _Internal methods_ [[github](https://github.com/iblech/internal-methods)]

category: algebraic geometry

[[!redirects algebraic geometer]]


algebraic group

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## Definition

Given a (typically [[algebraically closed field|algebraically closed]]) field $k$, an __algebraic $k$-group__ is a [[group object]] in the category of $k$-[[algebraic variety|varieties]].

## Linear algebraic groups and abelian varieties

There are two important classes of algebraic groups whose intersection is trivial (the identity group): Linear algebraic groups and abelian varieties.

Any algebraic group contains a unique normal linear algebraic subgroup $H$ such that their quotient $G/H$ is an abelian variety.

### Linear algebraic group

An algebraic $k$-group is [[linear algebraic group|linear]] if it is a [[Zariski topology|Zariski]]-closed subgroup of the [[general linear group]] $GL(n,k)$ for some $n$.

An algebraic group is linear iff it is affine.

An algebraic group scheme is *affine* if the underlying scheme is [[affine scheme|affine]].

The category of affine group schemes is the [[opposite category|opposite]] of the category of commutative [[Hopf algebras]].

### Abelian variety

Another important class are connected algebraic $k$-groups whose underlying variety is [[projective variety|projective]]; these are automatically commutative so they are called *abelian varieties*.  In dimension $1$ these are precisely the [[elliptic curve]]s. If $k$ is a [[perfect field]] and $G$ an algebraic $k$-group, the theorem of Chevalley says that there is a unique linear subgroup $H\subset G$ such that $G/H$ is an abelian variety.

#### Elliptic curve

An abelian variety of dimension $1$ is called an *[[elliptic curve]]*.

## Other prominent classes of algebraic groups

Some of the definitions of the following classes exist more generally for [[group schemes]].

### Jacobian

(...)

### Unipotent algebraic groups

(See also more generally [[unipotent group scheme]].)

+-- {: .num_defn}
###### Definition
An element $x$ of an affine algebraic group is called *unipotent* if its associated right translation operator $r_x$ on the affine [[coordinate ring]] $A[G]$ of $G$ is locally unipotent as an element of the ring of linear endomorphism of $A[G]$ where ''locally unipotent'' means that its restriction to any finite dimensional stable subspace of $A[G]$ is unipotent as a ring object.
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###### Theorem
([[Jordan-Chevalley decomposition]])
Any commutative linear algebraic group over a perfect field is the  product of a unipotent and a [[semisimple object|semisimple algebraic group]].
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## Properties

The group objects in the category of [[algebraic schemes]] and [[formal scheme]]s are called (algebraic) [[group schemes]] and [[formal groups]], respectively.

Among group schemes are 'the infinite-dimensional algebraic groups' of Shafarevich.

Algebraic analogues of [[loop group]]s are in the category of [[ind-scheme]]s. All linear algebraic $k$-groups are affine.

## Examples

The [[affine line]] $\mathbb{A}^1$ comes canonically with the structure of a group under addition: the [[additive group]] $\mathbb{G}_a$.

The affine line without its origin, $\mathbb{A}^1 - \{0\}$ comes canonically with the structure of a group under multiplication: the [[multiplicative group]] $\mathbb{G}_m$.

## Generalizations

* [[group scheme]]

## Related concepts

* [[isogeny]]

* [[form of an algebraic group]]

## References

* [[Gerhard P. Hochschild]], *Basic Theory of Algebraic Groups and Lie Algebras*, Graduate Texts in Mathematics **75**, Springer (1981) [[doi:10.1007/978-1-4613-8114-3_16](https://doi.org/10.1007/978-1-4613-8114-3_16)]

* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970; English edition is _Introduction to algebraic geometry and algebraic groups_, North-Holland, Amsterdam 1980 (North-Holland)

* M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, _Schemas en groupes_, [[SGA3]]

* {#Milne17} [[James Milne]], *Algebraic Groups -- The theory of group schemes of finite type over a field*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

* A. Borel, _Linear algebraic groups_, Springer (2nd edition much expanded)

* W. Waterhouse, _Introduction to affine group schemes_, GTM 66, Springer 1979.

* S. Lang, _Abelian varieties_, Springer 1983.

* D. Mumford, _Abelian varieties_, 1970, 1985.

* J. C. Jantzen, _Representations of algebraic groups_, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

* T. Springer, _Linear algebraic groups_, Progress in Mathematics 9, Birkhäuser Boston (2nd ed. 1998, reprinted 2008)

* Roe Goodman, Nolan R. Wallach,  Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.

* J. Milne, _Algebraic groups, Lie Groups, and their arithmetic subgroups_, [pdf](http://www.jmilne.org/math/CourseNotes/ALA.pdf)

* [[!redirects algebraic groups]]


Algebraic Homotopy

The book

* [[Hans Baues]],

  _Algebraic Homotopy_,

  Cambridge studies in advanced mathematics 15,

  Cambridge University Press, (1989)

  [doi:10.1017/CBO9780511662522](https://doi.org/10.1017/CBO9780511662522)

gives an account of the author's interpretation of  [[Henry Whitehead]]'s [[Algebraic Homotopy Theory]] as described in his ICM talk (1950) and his famous papers, _[[Combinatorial homotopy I]]_, (1949), and _[[Combinatorial homotopy II]]_, again (1949). Although the material contained in the first of these papers became central to the development of homotopy theory (CW complexes etc.) soon after its publication, the second paper, treating harder ideas including those of [[crossed complexes]], was relatively 'unstudied' until much more recently. This book gives one interpretation of the ideas it developed from a modern point of view. That development continued in [[Combinatorial Homotopy and 4-Dimensional Complexes]].

#Contents#

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{:toc}

## Preface

### Introduction

###I Axioms for homotopy theory and examples of cofibration categories

###II  Homotopy theory in a cofibration category

###III The homotopy spectral sequences in a cofibration category

###IV Extensions, coverings and cohomology groups of a category

###V Maps between mapping cones

###VI Homotopy theory of CW-complexes

###VII Homotopy theory of complexes in a cofibration category

### VIII Homotopy theory of Postnikov towers and the Sullivan-de Rham equivalence of rational homotopy categories

###IX Homotopy theory of reduced complexes

###Bibliography

###Index

## References

See also:

*  [[Tim Porter]], _Review of "Algebraic Homotopy'' by H.J.Baues_, in Bull. London Math. Soc. 22 (1990) 196-197.

category: reference