composition algebra

> This entry is about algebras exhibiting "composition of sums of squares". For the un-related notion of [[rings]] exhibiting a structure akin to [[composition]] of [[endofunctions]] see at *[[composition ring]]*.

***

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

#Contents#
* table of contents
{:toc}

## Definition

Let $k$ be a [[field]] with [[characteristic]] $char(k) \neq 2$. A **unital composition algebra** over $k$ consists of a [[finite-dimensional vector space|finite-dimensional]] [[vector space]] $V$ together with a

* a nondegenerate symmetric [[bilinear form]]

  $$
    \langle - , - \rangle: V \otimes V \to k
    \,,
  $$

* a multiplication map, i.e., a [[bilinear map]]

  $$
    - \cdot - \;\colon\; V \otimes V \to V
    \,,
  $$

* a [[unit]] $e \in V$ for the multiplication, i.e., so that $e \cdot v = v = v \cdot e$,

such that:

$$
  \langle u\cdot v ,\, u \cdot v \rangle
  \;=\;
  \langle u,u \rangle
  \langle v,v \rangle
$$

The bilinear form defines a *[[norm]]* $N(u) = \langle u,u\rangle$.

Beware that there are *no* assumptions on the multiplication such as [[associativity]], [[commutativity]], etc.

Examples of composition algebras include the [[real numbers]], the [[complex numbers]], the [[quaternions]], the [[octonions]], and the [[matrix algebra|algebra of $2 \times 2$ matrices]] over a [[field]].

Since $char(k) \neq 2$, we can recover the [[bilinear form]] from the [[norm]] by the formula

$$\langle u, v \rangle = \frac{N(u + v) - N(u) - N(v)}{2}$$

Since the bilinear form is nondegenerate, we may infer $u = v$ whenever

$$\langle u, w \rangle = \langle v, w \rangle$$

and this will be frequently used in the sequel.

Also since the form is nondegenerate, there exists $v \in V$ such that $N(v) \neq 0$. From $N(v) = N(e v) = N(e)N(v)$, it follows that $N(e) = 1$.

### Para- and non-unital composition algebras

Frequently, one refers to *unital* composition algebras simply as composition algebras. However, this unital property can be weakened or removed altogether. A weakening of this condition leads to **para-unital** composition algebras. This requires the existence of an involution

$$
x\mapsto\bar{x}
$$

such that there exists a distinguished element **1**, called the **para-unit**, such that

$$
x\cdot \mathbf{1} = \mathbf{1}\cdot x = \bar{x}
$$

meaning **1** acts as a unit up to an involution.

On the other hand, removing the existence of a unit altogether leads to the notion of **non-unital** composition algebras.

These two generalizations of composition algebras are intimately related to a generalization of the [[Hurwitz theorem]] (see [generalized Hurwitz theorem](https://ncatlab.org/nlab/show/Hurwitz+theorem#generalized_hurwitz_theorem) for the statement).

## Properties

### Basic identities

The arrangements of the proofs below are based in part on the treatments by Conway and Smith, and by Springer and Veldkamp (see references below).

\begin{proposition} (Scaling)
$\langle u v, u w \rangle = N(u)\langle v, w \rangle$ and $\langle u w, v w \rangle = \langle u, v \rangle N(w)$
\end{proposition}

+-- {: .proof}
######Proof
The left sides, and therefore the right sides of the equations below are equal:

$$N(u(v + w)) = N(u)N(v + w) = N(u)(N(v) + 2\langle v, w \rangle + N(w))$$

$$N(u v + u w) = N(u v) + 2\langle u v, u w \rangle + N(u w) = N(u)N(v) + 2\langle u v, u w \rangle + N(u)N(w)$$

and the result follows by cancellation and division by $2$.
=--

\begin{proposition} (Exchange)
$\langle u v, w x \rangle = 2\langle u, w \rangle \langle v, x \rangle - \langle u x, w v \rangle$
\end{proposition}

+-- {: .proof}
######Proof
From the scaling identity, we have

$$\langle (u + w)v, (u + w)x \rangle = N(u + w)\langle v, x \rangle = N(u)\langle v, x \rangle + 2\langle u, w \rangle \langle v, x \rangle + N(w)\langle v, x \rangle = \langle u v, u x \rangle + 2\langle u, w \rangle \langle v, x \rangle + \langle w v, w x \rangle$$

but the left-hand side is equal to

$$\langle u v + w v, u x + w x \rangle = \langle u v, u x \rangle + \langle w v, u x \rangle + \langle u v, w x \rangle + \langle w v, w x \rangle$$

and now we equate the right-hand sides and cancel to get the result.
=--

#### Conjugation identities

In any composition algebra, we may define a **conjugation** operator by

$$\bar{v} = 2\langle v, e \rangle e - v$$

Observe that $\bar{v} = v$ just when $v$ is a scalar multiple of the identity. By analogy with the classical case (composition algebras over $\mathbb{R}$), such elements will be called **real**.

The next few propositions develop properties of conjugation.

\begin{proposition} (Adjointness)
$\langle u v, w \rangle = \langle v, \bar{u}w \rangle$ and $\langle u v, w \rangle = \langle u, w\bar{v} \rangle$. $\langle w, u v \rangle = \langle \bar{u} w, v \rangle$ and $\langle w, u v \rangle = \langle w\bar{v}. u \rangle$.
\end{proposition}

+-- {: .proof}
######Proof
Put $x = e$ in the exchange identity to get the first equation in

$$\langle u v, w \rangle = 2\langle u, w \rangle \langle v, e \rangle - \langle u, w v \rangle = \langle u, w(2\langle v, e\rangle e - v)\rangle = \langle u, w\bar{v} \rangle$$

The second adjointness equation is proved similarly; the final two come from symmetry of the form.
=--

\begin{proposition} (Involution)
$v = \bar{\bar{v}}$ for all $v$.
\end{proposition}

+-- {: .proof}
######Proof
For all $u$ we have

$$\langle u, v \rangle = \langle u\bar{v}, e \rangle = \langle u, \bar{\bar{v}} \rangle$$

and the result follows from nondegeneracy.
=--

\begin{proposition} (Unitarity)
$\langle u, v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle$.
\end{proposition}

+-- {: .proof}
######Proof
$\langle u, v \rangle = \langle e, \bar{u}v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle$ where the last equation is symmetry of the bilinear form.
=--

\begin{proposition} (Anti-automorphism)
$\bar{u} \bar{v} = \widebar{v u}$.
\end{proposition}

+-- {: .proof}
######Proof
For all $w$ we have

$$\langle \bar{u}\bar{v}, w \rangle = \langle \bar{v}, u w \rangle = \langle \bar{v}\bar{w}, u \rangle = \langle \bar{w}, v u \rangle = \langle \widebar{v u}, w \rangle$$

using involution and unitarity. The result follows from nondegeneracy of the form.
=--

By the involution and anti-automorphism properties, we see that $\bar{v}v$ is fixed under conjugation: is "real". Better yet,

\begin{proposition} (Reality)
$\bar{u} \cdot (u v) = N(u)v$.
\end{proposition}

+-- {: .proof}
######Proof
For all $w$,

$$\langle \bar{u}\cdot (u v), w \rangle = \langle u v, u w \rangle = N(u)\langle v, w \rangle = \langle N(u)v, w \rangle$$

and the result follows from nondegeneracy.
=--

This last result has several interesting corollaries. Putting $v = e$, we see that

* $N(u) \neq 0$ implies $u$ is invertible, with $u^{-1} = \bar{u}/N(u)$.

* $N(u) = 0$ implies $u$ is a zero divisor, with $\bar{u} u = 0$.

In either case, we have from $\bar{u} = 2\langle u, e \rangle e - u$ the identity

$$\bar{u} u = (2\langle u, e \rangle e - u)u = N(u)e$$

so that every element $u$ of a composition algebra satisfies a quadratic equation

$$u^2 - 2\langle u, e \rangle u + N(u)e = 0.$$

This has as further consequence the fact that an algebra admits at most one norm making it a composition algebra (because the minimal monic polynomial of an element $u$ in a finite-dimensional algebra is uniquely determined; the norm of an element would the uniquely determined constant coefficient of its minimal polynomial).

A final corollary of **Reality** is

\begin{proposition} (Alternative law)
$u \cdot (u v) = u^2 \cdot v$ and $u \cdot v^2 = (u v) \cdot v$.
\end{proposition}

+-- {: .proof}
######Proof
We have $w(u v) = (w u)v$ if $w$ is either $e$ or $\bar{u}$, and $u$ is a linear combination of $e$ and $\bar{u}$. The other equation is proven similarly.
=--

These are the two axioms as given in [[alternative algebra]], but we remark that often a third alternative law is considered: $u (v u) = (u v) u$. For discussion of this in composition algebras, see the section on Moufang identities below.

### Cayley-Dickson doubling construction

This is essentially the same as the [[Cayley-Dickson construction]], but in this section it is applied specifically to composition algebras where we have to deal with a norm, whereas the general construction applies to general (nonassociative) algebras equipped with an anti-involution.

We begin with a simple observation:

+-- {: .num_prop}
######Proposition
Let $V$ be a finite-dimensional vector space with a nondegenerate bilinear form, and let $W$ be a subspace such that the form on $V$ restricts to a nondegenerate form on $W$. Then

$$V = W \oplus W^\perp$$

and the form on $V$ restricts to a nondegenerate form on $W^\perp$.
=--

+-- {: .proof}
######Proof
The fact that $W \cap W^\perp = \{0\}$ is immediate from nondegeneracy of the form on $W$, and that $W + W^\perp = V$ follows from this and the fact that $dim(W) + dim(W^\perp) = dim(V)$ (use $dim(W^\perp) = dim((V/W)^*) = dim(V/W)$ and $dim(V) = dim(W) + dim(V/W)$). For the second assertion, we know that for $v \in W^\perp$, the map $\langle v, - \rangle |_W: W \to k$ is zero; if also $\langle v, - \rangle |_{W^\perp}: W^\perp \to k$ is zero, then $\langle v, - \rangle: V \to k$ is zero because $V = W + W^\perp$, and $v = 0$ follows from nondegeneracy of the form on $V$.
=--

Thus, given a composition algebra $V$ and a composition subalgebra $W$ of $V$ (that is, a subspace closed under identity and multiplication, such that the norm on $V$ restricts to a nondegenerate form on $W$), the proposition shows there exists $\alpha \in W^\perp$ such that $N(\alpha) \neq 0$. This $\alpha$ is invertible, so $\alpha \cdot W$ has the same dimension as $W$. Moreover, for all $v, w \in W$ we have

$$\langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0$$

so that, by nondegeneracy of the form on $W$, $\alpha W \cap W = \{0\}$. Indeed, $\alpha W$ is orthogonal to $W$. It follows that $W + \alpha W$ has double the dimension of $W$.

Now let us fix such an $\alpha$, and put $\lambda = N(\alpha)$.

+-- {: .num_prop}
######Proposition
For elements $u, v, w, x \in W$,

$$\langle u + \alpha v, w + \alpha x \rangle = \langle u, w \rangle + \lambda \langle v, x \rangle.$$
=--

+-- {: .proof}
######Proof
This follows from the equations

$$\langle u, \alpha x \rangle = \langle u \bar{x}, \alpha \rangle = 0 \qquad \langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0 \qquad \langle \alpha v, \alpha x \rangle = N(\alpha)\langle v, x \rangle$$

plus bilinearity of the form.
=--

Consequently, if $\langle u + \alpha v, w \rangle = 0$ for all $w \in W$, we must have $u = 0$, and if $\langle u + \alpha v, \alpha x \rangle = 0$ for all $x \in W$, then $v = 0$. It follows that the form on $V$, when restricted to $W + \alpha W$, is nondegenerate.

Now we want to show that the double $W + \alpha W$ is closed under multiplication, hence forms a composition subalgebra. It follows immediately from all this that, starting from the trivial composition subalgebra $k \cdot e$ of dimension 1, $dim(V)$ must be a power of 2, and in fact we will see later that the only possible dimensions are 1, 2, 4, and 8. Indeed, the possible structures of composition algebras are very tightly constrained.

\begin{proposition} (Conjugation on the double)
We have $\widebar{u + \alpha v} = \bar{u} - \alpha v$. Consequently, $\alpha v = - \widebar{\alpha v} = - \bar{v} \bar{\alpha} = \bar{v} \alpha$, and $\widebar{\alpha} = -\alpha$.
\end{proposition}

+-- {: .proof}
######Proof
$\widebar{\alpha v} = 2\langle \alpha v, e \rangle e - \alpha v = -\alpha v$.
=--

\begin{thm} (Closure under multiplication)
For all $u, v, w, x \in W$, $(u + \alpha v)(w + \alpha x) = (u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x)$.
\end{thm}

+-- {: .proof}
######Proof
For all $y \in V$, we have the following sets of equations, using the previous proposition (Conj), the Exchange identity (Ex), and other identities frequently observed above (unlabeled as such).

$$\langle u (\alpha x), y \rangle = \langle \alpha x, \bar{u} y \rangle \stackrel{Ex}{=} 0 - \langle \alpha y, \bar{u} x \rangle = \langle y, \alpha (\bar{u} x) \rangle = \langle \alpha (\bar{u} x), y \rangle$$

$$\langle (\alpha v)w, y \rangle = \langle \alpha v, y \bar{w} \rangle \stackrel{Conj}{=} \langle \bar{v} \alpha, y \bar{w} \rangle \stackrel{Ex}{=} 0 - \langle \bar{v}\bar{w}, y \alpha \rangle = \langle (\bar{v}\bar{w})\alpha, y \rangle = \langle \widebar{w v} \alpha, y \rangle \stackrel{Conj}{=} \langle \alpha (w v), y \rangle$$

$$\langle (\alpha v)(\alpha x), y \rangle \stackrel{Conj}{=} -\langle \alpha v, y (\alpha x) \rangle \stackrel{Ex}{=} 0 + \langle \alpha (\alpha x), y v \rangle \stackrel{Conj}{=} -\langle \alpha x, \alpha (y v) \rangle = -\lambda \langle x, y v \rangle = \langle -\lambda x\bar{v}, y \rangle$$

These identities, combined with nondegeneracy of the form, give the result.
=--

#### Possible dimensions are 1, 2, 4, and 8.

The calculation expressed by the fundamental theorem just stated has some remarkable consequences:

* Suppose $V = W + \alpha W$. Then $W$ is an _associative_ composition algebra.

For, by starting from the identity

$$N(u + \alpha v)N(w + \alpha x) = N((u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x))$$

and expanding, one obtains

$$(N(u) + \lambda N(v))(N(w) + \lambda N(x)) = N(u w) - 2\lambda \langle u w, x\bar{v} \rangle + \lambda^2 N(x\bar{v}) + \lambda (N(w v) + 2\langle w v, \bar{u} x \rangle + N(\bar{u}x))$$

Using the fact that $N$ is a homomorphism, plus unitarity $N(u) = N(\bar{u})$, further expansions and cancellations yield

$$0 = -2\lambda \langle u w, x\bar{v} \rangle + 2\lambda \langle w v, \bar{u}x \rangle$$

which, by adjointness, yields

$$\langle (u w)v, x \rangle = \langle u(w v), x \rangle$$

which by nondegeneracy on $W$, yields associativity $(u w)v = u(w v)$.

* Suppose $V = W + \alpha W$ is an associative composition algebra. Then $W$ is a _commutative, associative_ composition algebra.

For clearly the subalgebra $W$ must be associative; it is also commutative via the following string of equations (using conjugation of the double):

$$\alpha(v w) = (\alpha v)w = (\bar{v}\alpha)w = \bar{v}(\alpha w) = \bar{v}(\bar{w} \alpha) = (\bar{v}\bar{w})\alpha = \widebar{w v}\alpha = \alpha (w v)$$

and cancelling out $\alpha$.

Conversely, a lengthy but straightforward calculation shows that if $W$ is commutative and associative, then $V$ is associative.

* Suppose $V = W + \alpha W$ is a commutative associative composition algebra. Then $W$ is purely real, i.e., is the trivial 1-dimensional associative commutative algebra $k \cdot e$.

This results from

$$\alpha w \stackrel{Conj}{=} \bar{w}\alpha \stackrel{comm}{=} \alpha \bar{w}$$

so that $w = \bar{w}$ for every $w \in W$, so that $w$ is real. Conversely, from

$$(u + \alpha v)(w + \alpha x) = (u w - \lambda x\bar{v}) + \alpha (w v + \bar{u} x)$$

$$(w + \alpha x)(u + \alpha v) = (w u - \lambda v\bar{x}) + \alpha (u x + \bar{w} v)$$

together with commutativity and trivial conjugation in $W$, we infer commutativity in $V$.

Hence the doubling process may be iterated three times at most.

This same result can also be proven using [[string diagram]] calculus.  See [this paper](http://math.ucr.edu/home/baez/Hurwitz.pdf) for a nice exposition of that route.

### Hurwitz's Theorem
 {#HurwitzTheoremSection}

The classification of composition algebras over specific fields (e.g., [[number field]]s, [[local fields]]) can be a bit intricate; in this section we concentrate solely on the classical case where $k = \mathbb{R}$ is the [[real numbers]], where the results have been known for a long time, known as the _[[Hurwitz theorem]]_.

A fundamental dichotomy is whether or not the composition algebra has zero divisors, i.e., elements $v$ such that $N(v) = 0$. If not, then the composition algebra is a division algebra (every nonzero element is invertible). If so, then the composition algebra is called a **split** composition algebra. We analyze each in turn.

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

In a division composition algebra, all nonzero elements have positive norm.

=--

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

If all elements $v$ orthogonal to the identity $e$ have positive norm, the result is immediate since

$$N(r v + s e) = r^2 N(v) + s^2 \geq 0$$

Otherwise, if some such element $v$ has $N(v) = \lambda \lt 0$, we may put $u = v/|\lambda|^{1/2}$ so that $N(u) = -1$. Then $u$ is orthogonal to $e$ and

$$N(u + e) = N(u) + N(e) = -1 + 1 = 0$$

which contradicts the assumption that all nonzero elements are invertible.

=--

In particular, any division composition algebra is a [[normed division algebra]].

Now let $V$ be a division composition algebra, with $V = W + \alpha W$, where $0 \neq \alpha \in W^\perp$. Put $j = \alpha/N(\alpha)^{1/2}$, so that $N(j) = 1$, $j \perp W$, and $V = W + j W$. We have the following possibilities.

* $dim(V) = 2$. In that case $W$ is purely real and $V$ is a commutative field over $\mathbb{R}$ with $-j^2 = j\bar{j} = N(j) = 1$. This is of course the complex numbers, with
$$N(s + j t) = s^2 + t^2$$
the usual norm. The conjugate of $s + j t$ is $s - j t$.

* $dim(V) = 4$. In that case $W$ is a 2-dimensional division composition algebra, hence isomorphic to $\mathbb{C}$, and $V$ is an associative division algebra over $\mathbb{R}$ given by $V = \mathbb{C} + j\mathbb{C}$, where again $j^2 = -1$. (Evidently $V$ is not commutative because $W$ is not purely real.) By conjugation of the double, we have
$$j i = -i j$$
where $i$ is an imaginary unit of $\mathbb{C}$, and we arrive at the algebra of quaternions $\mathbb{H}$ over $\mathbb{R}$, with orthonormal basis provided by $1, i, j, k = i j$. Conjugation is given by the usual operation
$$a + b i + c j + d k \mapsto a - b i - c j - d k$$

* $dim(V) = 8$. In that case $W$ is a 4-dimensional division composition algebra, hence isomorphic to $\mathbb{H}$, and $V$ is an alternative division algebra over $\mathbb{R}$ given by $V = \mathbb{H} + j\mathbb{H}$, with $j^2 = -1$. ($V$ is not associative because $W$ is not commutative.) The structure of multiplication is given by the theorem above and the resulting algebra is the algebra of octonions, with the standard norm and conjugation.

Thus, we have established the [[Hurwitz theorem]]

\begin{thm} (Hurwitz)
The only [[division algebra|division]] composition algebras over the [[real numbers]] $\mathbb{R}$ are the [[real numbers]], [[complex numbers]], [[quaternions]], and [[octonions]].

\end{thm}

#### Split composition algebras

Now we turn to split composition algebras $V$. It turns out that the structure of these is _not_ specific to the field $\mathbb{R}$: the classification of possible split composition algebras is the same over any field (see the text by Springer and Veldkamp), although we will continue to work over $\mathbb{R}$ as we describe them below.

Suppose $V = W + \alpha W$, where $\alpha \in W^\perp$, $N(\alpha) \neq 0$. Put $j = \alpha/|N(\alpha)|^{1/2}$, so $|N(j)| = 1$, $V = W + j W$. In addition to the trivial 1-dimensional case, we have the following possibilities.

* $dim(V) = 2$. In this case $N(j) = -1$ (else $V$ would be a division algebra, not a split composition algebra) and $j^2 = 1$ (we are now using $1$ to denote the identity). The elements
$$e_1 = \frac{1 + j}{2} \qquad e_2 = \frac{1-j}{2}$$
are primitive idempotents, conjugate to one another, and $V \cong \mathbb{R} e_1 \oplus \mathbb{R} e_2$ as a product ring. The norm of an element $x e_1 + y e_2$ is $N(x e_1 + y e_2) = x y$.

* $dim(V) = 4$. Let $i$ be an imaginary unit of $W$, so $\bar{i} = -i$ and $|N(i)| = 1$. Here either $N(i) = -1$ ($W$ is split), or $N(i) = 1$ ($W$ is isomorphic to $\mathbb{C}$). In the second instance, $N(j) = -1$, else $V$ would be a division algebra, and we may replace $W$ by the split algebra $W' = \mathbb{R} + \mathbb{R} i j$ and still have $V = W' + j W'$. So without loss of generality we may assume $W$ is split; therefore, there is up to isomorphism only one split composition algebra of dimension 4. This is the algebra of $2 \times 2$ matrices $A$, for which $N(A) = det(A)$ and $W$ is embedded as the subalgebra of diagonal matrices; the element $j$ may be taken to be the matrix $A$ with $a_{11} = a_{22} = 0$, $a_{12} = a_{21} = 1$. The conjugate of a matrix $A$ is $\bar{A} = Tr(A)I - A$, which leads to the familiar formula for $det(A) A^{-1}$ when $A$ is invertible.

* $dim(V) = 8$. Again, by an argument similar to the one used for the case of dimension 4, we may assume a maximal proper composition subalgebra $W$ is split, and up to isomorphism there is only one split composition algebra of dimension 8, aka the [[split octonions]]. The multiplication may be deduced from the fundamental result on doubling multiplication above, or may be expressed as follows. Denote scalars by letters like $r, s$ and 3-vectors by letters like $x, y$. Let $\langle x, y \rangle$ denote the standard inner product
$$x_1 y_1 + x_2 y_2 + x_3 y_3$$
and let $x \wedge y$ denote the standard cross-product, so that $\langle x \wedge y, z \rangle = det(x, y, z)$. Elements of $V$ are represented by $2 \times 2$ arrays
$$\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right)$$
and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:
$$\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right)
\cdot
\left(
\begin{aligned}
r' & x'\\
y' & s'
\end{aligned}
\right) =
\left(
\begin{aligned}
r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\
r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s'
\end{aligned}
\right)
$$
The norm is given by a kind of determinant formula
$$N\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right) = r s - \langle x, y \rangle
$$

### Moufang identities

Further consequences of the composition algebra axioms include the Moufang laws which are important in the study of octonions.

**Moufang identities**

* $(u v)(w u) = (u(v w))u) = u((v w)u)$

* $((u v)u)w = u(v(u w))$

* $((u v)w)v = u(v(w v))$

We will prove the first of these; the others are proven in similar style (see Springer-Veldkamp for details). (It may be tricky to remember how the bracketings go, but one thing to remember is that the bracketings shouldn't lead to proofs of general associativity when interpreted in a division algebra!)

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

We have

$$\array{
\langle (u v)(w u), x \rangle & = & \langle u v, x(\bar{u}\bar{w})\rangle\\ & \stackrel{Ex}{=} & 2\langle u, x\rangle\langle v, \bar{u}\bar{w} \rangle - \langle u(\bar{u}\bar{w}), x v \rangle \\
 & = & 2\langle u, x\rangle\langle v w, \bar{u} \rangle - \langle \bar{u}\bar{w}, \bar{u}(x v) \rangle \\
 & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \bar{w}\bar{v}, x \rangle \\
 & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \widebar{v w}, x \rangle
}$$

which makes it plain that $(u v)(w u)$ depends on $u$ and $v w$ only. Hence we get the same result if we replace $v$ and $w$ and any two elements whose product is $v w$, say $v w$ and $e$. In other words,

$$(u v)(w u) = (u(v w))(e u) = (u(v w))u, \qquad (u v)(w u) = (u e)((v w)u) = u((v w)u)$$

which completes the proof.

=--

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

For all $u$, $v$ in a composition algebra, the third alternative law holds: $u(v u) = (u v)u$.

=--

See also [[Moufang loop]].

## Generalizations

### From Ab to monoidal categories

This concept could be generalized from the [[category of abelian groups]] to any [[monoidal category]], since $k$-vector spaces are $k$-modules when $k$ is a field:

Let $(C, I, \otimes)$ be a [[monoidal category]], let $(k, 1, \pi_k)$ be a [[monoid object]] in $C$. $k$ itself is a $k$-module object with the [[action]] being represented by the [[monoid]] [[binary operation]] $\pi_k$. A **(nonunital) composition algebra object** [[internalization|in]] $C$ is a $k$-module object $(A, \rho)$ with

* an [[action]]-preserving [[morphism]] $f:V \otimes V \to k$ from the [[tensor product]] $V \otimes V$ to $k$: given morphisms $a:I \to k$, $v:I \to V$, and $w:I \to V$,

$$f \circ ((\rho \circ (a \otimes v)) \otimes w) = \pi \circ (a \otimes (f \circ (v \otimes w))$$

and

$$f \circ (v \otimes (\rho \circ (a \otimes w))) = \pi \circ (a \otimes (f \circ (v \otimes w))$$

* a morphism $\pi_A \colon A \otimes A \to A$

* such that for all morphisms $u \colon I \to A$ and $v \colon I \to A$,

\[
  \label{InternalCondition}
  f
    \circ
   \Big(
       \pi_A \circ (u \otimes u) \big)
      \otimes
     \big(
       \pi_A \circ (v \otimes v)
   \Big)
     \;\;=\;\;
   \pi_k
     \circ
   \Big(
     \big(
       f \circ (u \otimes v) \big)
      \otimes
     \big(
       f \circ (u \otimes v)
     \big)
  \Big)
  \,.
\]

A **paraunital composition algebra object** in $C$ is a composition algebra object $A$ which is also a [[paraunital algebra object]] with respect to the morphism $\pi_A:A \otimes A \to A$, and a **unital composition algebra object** in $C$ is a composition algebra object $A$ which is also a [[unital algebra object]] with respect to the morphism $\pi_A:A \otimes A \to A$.

### From Vect to monoidal categories

This concept may be generalized from the [[category of vector spaces]] to any [[monoidal category]], observing that the ground field $k$ is the [[tensor unit]] of the category of $k$-vector spaces:

Let $(C, I, \otimes)$ be a [[monoidal category]], where $I$ is a [[monoid object]] with unit $e:I \to I$ and binary operation $\pi_{I}:I \otimes I \to I$. Then a **(nonunital) composition algebra object** [[internalization|in]] $C$ is an [[object]] $A \in C$ with a morphism $\pi_A \colon A \otimes A \to A$ and $f \colon A \otimes A \to I$ such that for all morphisms $u:I \to A$ and $v \colon I \to A$,

\[
  \label{InternalCondition2}
  f
    \circ
   \Big(
     \big(
       \pi_A \circ (u \otimes u) \big)
      \otimes
     \big(
       \pi_A \circ (v \otimes v)
     \big)
   \Big)
     \;\;=\;\;
   \pi_{I}
     \circ
   \Big(
     \big(
       f \circ (u \otimes v) \big)
      \otimes
     \big(
       f \circ (u \otimes v)
     \big)
  \Big)
  \,.
\]

A **paraunital composition algebra object** in $C$ is a composition algebra object $A$ which is also a [[paraunital algebra object]] with respect to the morphism $\pi_A:A \otimes A \to A$, and a **unital composition algebra object** in $C$ is a composition algebra object $A$ which is also a [[unital algebra object]] with respect to the morphism $\pi_A:A \otimes A \to A$.

In [[cartesian monoidal categories]] $(C, 1, \times)$, since any morphism $V \times V \to 1$ exists and is unique by the universal property of the [[terminal object]], the only monoid structure on $1$ is the [[trivial monoid]] structure given by the identity function on $1$ and the (left or right) [[unitor]] on $1$ respectively, and composition algebra objects are the same as [[magma]] objects.

## Related concepts

* [[division algebra]]

* [[alternative algebra]]

* [[Cayley-Dickson construction]]

* *un-related*: [[composition ring]]

## References
 {#References}

The term "composition algebra" refers to "composition of sums of squares", as in

* {#Taussky70} Olga Taussky, Section 6 of: *Sums of squares*, The American Mathematical Monthly **77** 8  (1970) 805-830 $[$[doi:10.2307/2317016](https://doi.org/10.2307/2317016)$]$

Textbook account:

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 1 of: _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

Survey:

* {#Elduque21} [[Alberto Elduque]], *Composition algebras*, Chapter 2 in:  Abdenacer Makhlouf (ed.), *Algebra and Applications I: Non-associative Algebras and Categories*, Sciences-Mathematics, ISTEWiley (2021) 27-57 [[arXiv:1810.09979](https://arxiv.org/abs/1810.09979), [ISBN:978-1-789-45017-0](https://www.wiley.com/en-ae/Algebra+and+Applications+1:+Non+associative+Algebras+and+Categories-p-9781789450170)]

A general abstract formulation of [Rost 96](#Rost96)

* {#Rost96} Markus Rost, _On the dimension of a composition algebra_, Documenta  Mathematica __1__ (1996), 209-214, [files](http://www.math.uni-bielefeld.de/documenta/vol-01/10.html) Abstract: "The possible dimensions of a composition algebra are 1, 2, 4, or 8. We give a tensor categorical argument."

in terms of [[string diagrams]] in [[additive category|additive]] [[braided monoidal categories]] is in

* {#Street18} [[Ross Street]], _Vector product and composition algebras in braided monoidal additive categories_ ([arXiv:1812.04143](https://arxiv.org/abs/1812.04143))

An exposition of the [[string diagram]] proof of the [Hurwitz' theorem](HurwitzTheoremSection) on the classification of compositon algebras is given in

* [[Bruce Westbury]], _Hurwitz' theorem on composition algebras_ ([arXiv:1011.6197](http://arxiv.org/abs/1011.6197))

* [[John Baez]]'s [comment](http://golem.ph.utexas.edu/category/2010/03/division_algebras_and_supersym.html#c032790)

See also

* [[John Conway]], Derek A. Smith, _On Quaternions and Octonions_, A.K. Peters, 2003.

* T.A. Springer, F.D. Veldkamp, _Octonions, Jordan algebras, and exceptional groups_, Springer Monographs in Mathematics, Springer-Verlag 2000.

* [wikipedia](http://en.wikipedia.org/wiki/Composition_algebra)

* [[eom]]: [Lie-admissible algebra](http://eom.springer.de/l/l058360.htm)

A proposed application of (non-unital) composition algebras in [[QCD]] is described in

* [[Alessio Marrani]], Daniele Corradetti, Francesco Zucconi. *Physics with non-unital algebras? An invitation to the Okubo algebra*. (2023) ([arXiv:2309.17435](https://arxiv.org/abs/2309.17435)).

[[!redirects composition algebra]]
[[!redirects composition algebras]]

[[!redirects split composition algebra]]
[[!redirects split composition algebras]]

[[!redirects composition algebra object]]
[[!redirects composition algebra objects]]


composition law for factorizations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

# Composition laws for factorizations
* table of contents
{:toc}

## Idea

A *composition law* for a [[functorial factorization]] is a functorial way to compose lifting structures for its algebraic right maps.  A functorial factorization whose pointed endofunctor extends to a monad over $cod$ is an [[algebraic weak factorization system]] if and only if it has a composition law.  Moreover, subject to smallness conditions, any functorial factorization with a composition law freely generates an algebraic weak factorization system.

## Definition

Suppose $\mathcal{C}$ is a category equipped with a [[functorial factorization]], sending every arrow $f:A\to B$ to a factorization $A \xrightarrow{f_L} E f \xrightarrow{f_R} B$.  As noted at [[functorial factorization]], a functorial factorization is equivalent to a pointed endofunctor $R$ on $\mathcal{C}^{\mathbf{2}}$ over $cod$, which maps each morphism $f$ (regarded as an object of the [[arrow category]] $\mathcal{C}^{\mathbf{2}}$) to its right factor $f_R$, the point being given by the left factor $f_L$ and the identity:

$$\array{ & \xrightarrow{f_L} & \\ ^f \downarrow & & \downarrow^{f_R} \\ & \xrightarrow{id}&. }$$

As with any pointed endofunctor, we can consider the category of [[algebra over a pointed endofunctor|algebras]] for $R$.  Such an $R$-algebra is an arrow $f:A\to B$ equipped with a map $s : E f \to A$ such that $s \circ f_L = id_A$ and $f \circ s = f_R$.  Equivalently, it is a diagonal lifting in the square

$$ \array{ A & \xrightarrow{id} & A \\ ^{f_L}\downarrow & & \downarrow^{f} \\ E f & \xrightarrow{f_R} & B.}$$

In particular, this means that if $(L,R)$ is a factorization for a [[weak factorization system]], then the arrows of $\mathcal{C}$ that admit some structure of $R$-algebra are precisely those in the right class of the weak factorization system.

The morphisms of $R$-algebras are commuting squares $g \circ h = k \circ f$ that additionally commute with the actions, i.e. $h \circ s_f = s_g \circ E(h,k)$.  This defines a category $R Alg$ with a forgetful functor $U : R Alg \to \mathcal{C}^{\mathbf{2}}$.

If it should happen that the pointed endofunctor $R$ is actually a [[monad]] over $cod$, i.e. it also has a multiplication $R R \to R$ that is also the identity on codomains, then we can also consider the smaller category $\mathbb{R} Alg$ of [[algebra over a monad|monad algebras]], the $R$-algebras as above such that $s$ also satisfies an associativity condition.

+-- {: .un_defn}
###### Definition
A **right weak composition law** for a functorial factorization is a functor $R Alg \times_{\mathcal{C}} R Alg \to R Alg$, where the pullback is over $dom \circ U : R Alg \to \mathcal{C}$ and $cod \circ U : R Alg \to \mathcal{C}$, lying over the composition functor $\mathcal{C}^{\mathbf{2}} \times \mathcal{C}^{\mathbf{2}}\to \mathcal{C}$.  If $R$ is a monad over $cod$, then a **right strong composition law** is defined analogously using $\mathbb{R} Alg$ instead.
=--

More explicitly, this means that

1. whenever $(f, s)$ and $(g, t)$ are $R$-algebras (resp. $\mathbb{R}$-algebras) such that $cod(f) = dom(g)$, we have a specified $R$-algebra structure (resp. $\mathbb{R}$-algebra structure) $t \bullet s$ for $g f$, such that
2. for any morphisms of $R$-algebras (resp. $\mathbb{R}$-algebras) $(u, v) : (f, s) \to (f' , s')$ and $(v, w) : (g, t) \to (g' , t')$ between composable pairs $(f, s), (g, t)$ and $(f' , s' ),(g' , t' )$, the pasted square $(u, w) : (g f, t \bullet s) \to (g ' f' , t' \bullet s ' )$ is also a map of $R$-algebras (resp. $\mathbb{R}$-algebras).

In other words, a composition law is an operation with the requisite shape to be the vertical composition in a [[double category]] whose vertical arrows are algebras, whose horizontal arrows are arbitrary arrows, and whose 2-cells are commutative squares.  Associativity is not assumed, but as noted below it often comes for free.

## Relation to awfs

+-- {: .un_theorem}
###### Theorem
Suppose $(L,R)$ is a functorial factorization whose underlying pointed endofunctor $R$ over $cod$ has the structure of a [[monad]] on $\mathcal{C}^{\mathbf{2}}$ over $cod$.  Then $(L,R)$ is an [[algebraic weak factorization system]] if and only if it admits a right strong composition law.
=--
+-- {: .proof}
###### Proof
See [Garner 09](#Garner09), [Garner 10](#Garner10), [Riehl 11](#Riehl11), and [Barthel-Riehl 13](#BR13) for proofs in varying degrees of explicitness.
=--

+-- {: .un_theorem}
###### Theorem
Suppose $(L,R)$ is a functorial factorization with a right weak composition law, and that the [[algebraically free monad]] on the pointed endofunctor $R$ exists and is over $cod$ (for instance if it can be constructed by a [[transfinite construction of free algebras]]).  Then the latter monad has a right strong composition law, hence underlies an algebraic weak factorization system whose right-monad-algebras coincide with the $R$-pointed-endofunctor-algebras, including their natural composition law.
=--
+-- {: .proof}
###### Proof
By definition, the algebraically-free monad $\mathbb{F}(R)$ satisfies $\mathbb{F}(R) Alg = R Alg$.  Thus, the weak composition law for $R$ extends to a strong one for $\mathbb{F}(R)$; now we can apply the previous theorem.
=--

## References

* {#Garner09} [[Richard Garner]]. *Understanding the small object argument*. Appl. Categ. Structures. 17(3) (2009) 247--285.

* {#Garner10} [[Richard Garner]]. *Homomorphisms of higher categories*. Adv. Math. 224(6) (2010), 2269--2311.

* {#Riehl11} [[Emily Riehl]].  *Algebraic model structures*. New York J. Math. 17 (2011) 173-231.

* {#BR13} [[Tobias Barthel]] and [[Emily Riehl]].  *On the construction of functorial factorizations for model categories*.  Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124. [projecteuclid](https://projecteuclid.org/euclid.agt/1513715550)

[[!redirects composition law for factorization]]
[[!redirects composition law for factorizations]]
[[!redirects composition laws for factorization]]
[[!redirects composition laws for factorizations]]

[[!redirects composition law for a functorial factorization]]
[[!redirects composition law for functorial factorizations]]
[[!redirects composition laws for a functorial factorization]]
[[!redirects composition laws for functorial factorizations]]

[[!redirects composition law for a factorization system]]
[[!redirects composition law for factorization systems]]
[[!redirects composition laws for a factorization system]]
[[!redirects composition laws for factorization systems]]

[[!redirects composition law for a weak factorization system]]
[[!redirects composition law for weak factorization systems]]
[[!redirects composition laws for a weak factorization system]]
[[!redirects composition laws for weak factorization systems]]

[[!redirects composition law for an algebraic weak factorization system]]
[[!redirects composition law for algebraic weak factorization systems]]
[[!redirects composition laws for an algebraic weak factorization system]]
[[!redirects composition laws for algebraic weak factorization systems]]

[[!redirects composition law for wfs]]
[[!redirects composition laws for wfs]]
[[!redirects composition law for awfs]]
[[!redirects composition laws for awfs]]

[[!redirects weak composition law for factorization]]
[[!redirects weak composition law for factorizations]]
[[!redirects weak composition laws for factorization]]
[[!redirects weak composition laws for factorizations]]

[[!redirects weak composition law for a functorial factorization]]
[[!redirects weak composition law for functorial factorizations]]
[[!redirects weak composition laws for a functorial factorization]]
[[!redirects weak composition laws for functorial factorizations]]

[[!redirects weak composition law for a factorization system]]
[[!redirects weak composition law for factorization systems]]
[[!redirects weak composition laws for a factorization system]]
[[!redirects weak composition laws for factorization systems]]

[[!redirects strong composition law for factorization]]
[[!redirects strong composition law for factorizations]]
[[!redirects strong composition laws for factorization]]
[[!redirects strong composition laws for factorizations]]

[[!redirects strong composition law for a functorial factorization]]
[[!redirects strong composition law for functorial factorizations]]
[[!redirects strong composition laws for a functorial factorization]]
[[!redirects strong composition laws for functorial factorizations]]

[[!redirects strong composition law for a factorization system]]
[[!redirects strong composition law for factorization systems]]
[[!redirects strong composition laws for a factorization system]]
[[!redirects strong composition laws for factorization systems]]

[[!redirects right weak composition law for factorization]]
[[!redirects right weak composition law for factorizations]]
[[!redirects right weak composition laws for factorization]]
[[!redirects right weak composition laws for factorizations]]

[[!redirects right weak composition law for a functorial factorization]]
[[!redirects right weak composition law for functorial factorizations]]
[[!redirects right weak composition laws for a functorial factorization]]
[[!redirects right weak composition laws for functorial factorizations]]

[[!redirects right weak composition law for a factorization system]]
[[!redirects right weak composition law for factorization systems]]
[[!redirects right weak composition laws for a factorization system]]
[[!redirects right weak composition laws for factorization systems]]

[[!redirects right strong composition law for factorization]]
[[!redirects right strong composition law for factorizations]]
[[!redirects right strong composition laws for factorization]]
[[!redirects right strong composition laws for factorizations]]

[[!redirects right strong composition law for a functorial factorization]]
[[!redirects right strong composition law for functorial factorizations]]
[[!redirects right strong composition laws for a functorial factorization]]
[[!redirects right strong composition laws for functorial factorizations]]

[[!redirects right strong composition law for a factorization system]]
[[!redirects right strong composition law for factorization systems]]
[[!redirects right strong composition laws for a factorization system]]
[[!redirects right strong composition laws for factorization systems]]

[[!redirects left weak composition law for factorization]]
[[!redirects left weak composition law for factorizations]]
[[!redirects left weak composition laws for factorization]]
[[!redirects left weak composition laws for factorizations]]

[[!redirects left weak composition law for a functorial factorization]]
[[!redirects left weak composition law for functorial factorizations]]
[[!redirects left weak composition laws for a functorial factorization]]
[[!redirects left weak composition laws for functorial factorizations]]

[[!redirects left weak composition law for a factorization system]]
[[!redirects left weak composition law for factorization systems]]
[[!redirects left weak composition laws for a factorization system]]
[[!redirects left weak composition laws for factorization systems]]

[[!redirects left strong composition law for factorization]]
[[!redirects left strong composition law for factorizations]]
[[!redirects left strong composition laws for factorization]]
[[!redirects left strong composition laws for factorizations]]

[[!redirects left strong composition law for a functorial factorization]]
[[!redirects left strong composition law for functorial factorizations]]
[[!redirects left strong composition laws for a functorial factorization]]
[[!redirects left strong composition laws for functorial factorizations]]

[[!redirects left strong composition law for a factorization system]]
[[!redirects left strong composition law for factorization systems]]
[[!redirects left strong composition laws for a factorization system]]
[[!redirects left strong composition laws for factorization systems]]


composition operation

The **composition operations** give the action of the homotopy groups of spheres on the homotopy groups of an arbitrary pointed space.

Let $X$ be a pointed space then, of course, its $r^{th}$-homotopy group $\pi_r(X)$ can be defined as the group of pointed homotopy classes of pointed maps from $S^r$ to $X$. In particular $\pi_k(S^r)$ consists of classes of maps from $S^k$ to $S^r$. If $\alpha\in \pi_k(S^r)$ and is represented by $a: S^k \to S^r$, and $\phi\in \pi_r(X)$, represented by $f: S^r\to X$ then the composite $f\circ a: S^k \to X$ so represents a class $\phi\cdot \alpha\in \pi_k(X)$.

This composition operation is well defined.  It forms one of the [[primary homotopy operations]].  An abstraction of this is a component part of the definitional structure  of a [[Pi-algebra]].

[[!redirects composition operations]]


composition ring

> This entry is about [[rings]] which mimic the [[composition]] operation on endo-functions on the [[ground ring]]. For the unrelated notion of [[algebras]] mimicking the *composition of sums of squares* see at *[[composition algebra]]*.

***

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of *composition rings* is an abstraction of the [[structure]] present in a [[ring of functions]] from the [[ground ring]] to itself: In addition to (pointwise) [[addition]] and [[multiplication]], there is a compatible operation of *[[composition]]*.

## Definition

A *composition ring* is a [[commutative ring]] $R$ equipped with an operation
$$(-)\circ(-) \colon R \times R \to R$$
such that for all [[elements]] $f \in R$, $g \in R$, and $h \in R$, we have:

1. $(f + g) \circ h = (f \circ h) + (g \circ h)$,

1. $(f \cdot g) \circ h = (f \circ h) \cdot (g \circ h)$,

1. $f \circ (g \circ h) = (f \circ g) \circ h$

   ([[associativity]] of composition).

## Examples

* The [[endofunction|endo]-[[function algebra]] on a [[commutative ring]] is a composition ring, with "$\circ$" being the [[composition]] of functions.

* In particular, every [[polynomial ring]] is a composition ring with "$\circ$" being the actual [[composition]] of polynomials regarded as [[functions]] from the [[ground ring]] to itself.

* Every commutative ring becomes a composition ring by setting $f \circ g \coloneqq f$.

## See also

* [[commutative ring]]

* [[polynomial ring]]

* [[function algebra]]

* *un-related*: [[composition algebra]]

## References

The concept is due to:

* Irving Adler, *Composition rings*, Duke Mathematical Journal, **29** 4 (1962) 607–623, $[$[doi:10.1215/S0012-7094-62-02961-7](https://projecteuclid.org/journals/duke-mathematical-journal/volume-29/issue-4/Composition-rings/10.1215/S0012-7094-62-02961-7.short), ISSN 0012-7094, MR 0142573$]$

See also:

* Wikipedia, *[Composition ring](https://en.m.wikipedia.org/wiki/Composition_ring)*

* Erhard Aichinger, *The Structure of Composition Algebras*(1998) $[$[pdf](http://www.algebra.uni-linz.ac.at/~erhard/Diss/diss.pdf)$]$

[[!redirects composition rings]]


composition series

#Contents#
* table of contents
{:toc}

##For groups

A _composition series_ for a [[group]], $G$,  is a subnormal series, (that is, a sequence of [[subgroups]], each a [[normal subgroup]] of the next one) such that each factor group $H_{i+1} / H_i$ is a [[simple group]].

##In an abelian category

An object $X$ of an [[abelian category]]  has a [[composition series]] if there is a chain  of [[subobjects]]

$$0= X_0 \lt X_1 \lt\ldots  \lt X_{n-1} \lt X_n = X$$

such that $X_i / X_{i-1}$ is [[simple object|simple]] for $1\leq i\leq n$.

See at _[[length of an object]]_ for more

##Related concept

* [[object of finite length]]


compositionality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Linguistics
+-- {: .hide}
[[!include linguistics - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

##Idea

\begin{center}_Ordo et connexio idearum idem est ac ordo et connexio rerum._[^ref]
\end{center}

[^ref]: [[Baruch Spinoza|Baruch de Spinoza]], Ethices Pars Secunda, propositio VII: "The order and connection of the thought is identical to with the order and connection of the things." ([buboquote](https://www.buboquote.com/en/quote/10756-spinoza-the-order-and-connection-of-the-thought-is-identical-to-with-the-order-and-connection-of))

The **principle of compositionality** asserts that the nature of complex structures is entirely determined by that of their simpler parts and the way these are composed.

In the context of the [[semantics]] of languages, this is sometimes called **Frege's principle** after [[Gottlob Frege]], although it was arguably assumed in [Boole 1854](#Boole1854), decades before Frege's work, and possibly not even embraced by Frege (see [Pelletier 01](#Pelletier01)).

Taking up [[Alfred Tarski|Alfred Tarski's]] work on the theory of  _[[truth]]_ in [[object languages]] with respect to [[metalanguages]] ([Tarski 35](#Tarski35)), the concept became popular in [[linguistics]] with the formalization of [[semantics]] of formal and informal languages via [[Montague semantics]]
([Montague 70a](#Montague70a),
[Montague 70b](#Montague70b),
[Montague 73](#Montague73)), where the principle asserts that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them.

### In computation

From [Compositionality 16](#Comp16):

> The compositional description of complex objects is a fundamental feature of the logical structure of [[computation]]. The use of logical languages in database theory and in algorithmic and finite model theory provides a basic level of compositionality, but establishing systematic relationships between compositional descriptions and complexity remains elusive. Compositional models of probabilistic systems and languages have been developed, but inferring probabilistic properties of systems in a compositional fashion is an important challenge. In quantum computation, the phenomenon of entanglement poses a challenge at a fundamental level to the scope of compositional descriptions. At the same time, compositionally has been proposed as a fundamental principle for the development of physical theories.

## Related concepts

In [[linguistics]]:

* [[Montague semantics]]

* [[categorical compositional distributional semantics]]

## References

### General

The principle is regarded as originating, at least implicitly, in

* {#Boole1854} [[George Boole]], _An Investigation of the Laws of Thought On Which Are Founded the Mathematical Theories of Logic and Probabilities_, Cambridge University Press 1854 ([doi:10.1017/CBO9780511693090](https://doi.org/10.1017/CBO9780511693090), [Wikipedia](https://en.m.wikipedia.org/wiki/The_Laws_of_Thought))

The common attribution to [[Gottlob Frege]] has been called into question:

* {#Pelletier01} Francis Jeffry Pelletier, _Did Frege Believe Frege's Principle?_, Journal of Logic, Language and Information,  10, 87–114 (2001) ([doi:10.1023/A:1026594023292](https://doi.org/10.1023/A:1026594023292))

The principle was promoted in [[linguistics]] via [[Montague semantics]]:

* {#Montague70a} [[Richard Montague]], _English as a formal language_, Linguaggi nella societa e nella tecnica. Edizioni di Communita. pp. 188-221 (1970) ([[MontagueEnglishFormalLanguage.pdf:file]])

* {#Montague70b} [[Richard Montague]], _Universal Grammar_, Theoria 36 (3):373--398 (1970) ([doi:10.1111/j.1755-2567.1970.tb00434.x](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1755-2567.1970.tb00434.x))

* {#Montague73} [[Richard Montague]], _The Proper Treatment of Quantification in Ordinary English_, Approaches to Natural Language (1973) ([[MontagueQuantification.pdf:file]])

Montague relied on

* {#Tarski35} [[Alfred Tarski]], 1935, _The Concept of Truth in Formalized Languages_, in Logic, Semantics, Metamathematics, Indianapolis: Hackett 1983, 2nd edition, 152–278.

A concise and formal discussion of this tradition is

* Theo Janssen, _Compositionality_ , ch.7 of van Benthem/ter Meulen (eds.), _Handbook of Logic and Language_ , North-Holland Amsterdam 1997.

A somewhat critical reassessment of the concept from a mathematical POV is the monograph

* {#Kracht11}[[Marcus Kracht]], _Interpreted Languages and Compositionality_ , Springer Heidelberg 2011.

See also

* Wikipedia, _[Principle of Compositionality](https://en.m.wikipedia.org/wiki/Principle_of_compositionality)_

* SEP, _[Compositionality](https://plato.stanford.edu/entries/compositionality/)_

### Activities

* {#Comp16} _Compositionality_, [workshop](https://simons.berkeley.edu/workshops/logic2016-3) hosted at UC Berkeley in 2016.

* _Compositionality_  is also the name of a [journal](http://www.compositionality-journal.org/) whose aim is to publish papers about compositional phenomena, construed broadly.

[[!redirects Compositionality]]
[[!redirects Frege's principle]]


compositions in cubical sets

The [[cubical set|cubical singular complex]] $K X$, or $S^\square X$, of a topological space $X$ has an additional structure of compositions.

Let $(m) = (m_1, \ldots\, , m_n)$ be an $n$-tuple of positive integers
and

$$\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0,
m_n]$$

be the map $(x_1 , \ldots \, , x_n) \mapsto (m_1 x_1, \ldots  \, ,
m_n x_n).$ Then a _subdivision of type $(m)$_  of a map $\alpha : I^n
\rightarrow X$ is a factorisation $\alpha = \alpha' \circ
\phi_{(m)}$; its  _parts_ are the cubes $\alpha_{(r)}$ where
$(r) = (r_1, \ldots \,, r_n)$ is an $n$-tuple of integers with $1
\leq r_i \leq m_i$, $i = 1, \ldots\, , n,$ and where
$\alpha_{(r)} : I^n \rightarrow X$ is given by

$$(x_1, \ldots \, , x_n) \mapsto
\alpha'(x_1 + r_1 - 1, \ldots \, , x_n + r_n - 1).$$

We then say that $\alpha$ is the _composite_  of the cubes $\alpha_{(r)}$ and write
$\alpha = [\alpha_{(r)}]$. The _domain_ of $\alpha_{(r)}$ is
then the set $\{(x_1,\ldots,x_n) \in I^n : r_i-1 \leq x_i
\leq r_i, 1 \leq i \leq n\}$.

The composite is _in direction_ $j$  if $m_j$ is the only
$m_i \gt 1,$ and we then write $\alpha = [\alpha_1 , \ldots\, ,
\alpha_m{_j} ]_j;$ the composite is _in the directions_ $j$,
$k$ $(j \neq k)$  if $m_j$, $m_k$ are the only $m_i \gt 1,$ and we then write

$$\alpha = [ \alpha_{rs}]_{j,k} \quad  or \quad  [ \alpha_{rs}] \quad  _{j}\!\downarrow ^{\textstyle\to ^k} $$

for $r = 1 , \ldots\, , m_j$ and $s = 1, \ldots \,, m_k.$ The aim is to follow matrix conventions in writing double compositions.

These definitions and notations are useful for showing how the singular cubical complex allows expression for _algebraic inverses to subdivision_, something seemingly very difficult either simplicially or globularly. The implications of this advantage for weak category theory seem not to have been investigated.

A _cubical set with connections and compositions and inverses_ is a cubical
set $K$ with connections in which each $K_n$ has $n$ partial
compositions $+_i$ and $n$ unary operations $-_i$, $i =
1,2,\ldots \, ,n$ satisfying the following axioms.

 If $a,b \in K_n$, then $a+_i b$ is defined if and only if  $ \partial^+_i a=\partial^-_i  b
$, and then for $\alpha=\pm$:

$$ \begin{cases} \partial^-_i (a+_i b) = \partial^-_i a & \\
         \partial^+_i (a+_i b) = \partial^+_i b & \end{cases}$$

$$
 \partial^\alpha_i (a +_j b) = \begin{cases} \partial^\alpha_i a+_{j-1}\partial^\alpha_i b &(i \lt j) \\
         \partial^\alpha_i a+_j \partial^\alpha_i b& (i \gt j), \end{cases}
          $$

 If $a \in K_n$, then $-_i a$ is defined and

$$
\begin{cases}\partial^-_i (-_i a)=\partial^+_i a & \\ \partial^+_i(-_i a)=\partial^-_i a &
\end{cases} $$

$$
 \partial^\alpha_i(-_j a) = \begin{cases} -_{j-1}\partial^\alpha_i a & (i\lt j) \\
                     -_{j}\partial^\alpha_i a & (i \gt j)
\end{cases}
$$

$$      \varepsilon_i(a+_j b) = \begin{cases}
      \varepsilon_i a +_{j+1} \varepsilon_i b & (i \leq j) \\
      \varepsilon_i a +_j\varepsilon_i b & (i \gt j) \end{cases}$$

$$      \varepsilon_i (-_j b) =
                  \begin{cases}
     -_{j+1} \varepsilon_i a & (i \leq j) \\
     -_j \varepsilon_i a & (i \gt j) \end{cases}$$

We have for $i \ne j$ and whenever both sides are defined:

$$
   (a+_i b) +_j (c+_i d) = (a+_j c) +_i (b+_j d)
$$

These relations are called the _interchange laws_, and both sides of this equation may be written:

$$\begin{bmatrix} a& c\\ b & d  \end{bmatrix} \quad _{i}\!\downarrow ^{\textstyle\to ^j}$$

 Further:

$
-_i (a+_j b) = (-_i a) +_j (-_i b)$    and  $ -_i (-_j a) =
-_j (-_i a)$  if $i \ne j$

$-_j(a+_j b)  = (-_j  b)+_j(-_j a)$  and  $ -_j (-_j a) =a$.

If further $K$ is a cubical set with [[connection on a cubical set|connections]] then we also require

$$    \Gamma^\alpha _i (a+_j b) = \begin{cases}
      \Gamma_i^\alpha a +_{j+1} \Gamma_i^\alpha b & (i \lt j) \\
      \Gamma_i^\alpha  a +_j\Gamma_i\alpha  b & (i \lt j) \end{cases}$$

$$    \Gamma_j^+(a+_j b)= (\Gamma_j^+ a +_{j} \varepsilon_j a) +_{j+1}
    (\varepsilon_{j+1} a +_{j} \Gamma_j^+ b)$$

$$    \Gamma_j^-(a+_j b)= (\Gamma_j^- a +_{j} \varepsilon_{j+1}  b) +_{j+1}
    (\varepsilon_{j} b +_{j} \Gamma_j^- b)$$

These last two equations are  called the _transport laws_ and the right hand sides are also written respectively

$$ \begin{bmatrix} \Gamma^+_j a & \varepsilon_j a \\
\varepsilon_{j+1} a & \Gamma^+_j b  \end{bmatrix} \quad _{j+1}\!\downarrow ^{\textstyle\to ^j} $$

$$ \begin{bmatrix} \Gamma^-_j a & \varepsilon_{j+1} b \\
\varepsilon_{j} b & \Gamma^-_j b  \end{bmatrix} \quad _{j+1}\!\downarrow ^{\textstyle\to ^j} $$

They can be interpreted as saying that turning left, or right, with your arm outstretched, is the same as turning left, or right.

 It is easily verified that the singular cubical set $KX$ of a
space $X$  satisfies these axioms if $+_j , -_j$  are defined by

$$
   (a+_j b)(t_1 ,t_2 ,\ldots \, ,t_n ) = \begin{cases}
   a(t_1 ,\ldots \, , t_{j-1},2t_j,t_{j+1} ,\ldots, ,t_n) &(t_j \leq
   \frac{1}{2})\\
   b(t_1 ,\ldots \, , t_{j-1},2t_j-1,t_{j+1} ,\ldots \, ,t_n) &(t_j \geq
   \frac{1}{2})\\
   \end{cases}
 $$

whenever  $ \partial^+_j a=\partial^-_j b$; and

$$
(-_j a)(t_1 ,t_2 ,\ldots,t_n ) = a(t_1 ,\ldots,
t_{j-1},1-t_j,t_{j+1} ,\ldots,t_n).
$$

The above list of relations may seem formidable, but they all express simple geometric ideas most of which have been well used in some form or another  in algebraic topology.

Notice also that in the singular cubical complex of a space, the interchange and transport laws hold **exactly**.

We also get a (strict) notion of [[cubical omega-category]] with connections by assuming that  all compositions $+_i$ give a category structure with source and target maps $\partial^-_i, \partial^+_i:K_n \to K_{n-1}$ and identity maps $\varepsilon_i: K_{n-1} \to K_n$, and also for all $a$

$$\Gamma^+_i a \circ_i\Gamma^-_i a = \varepsilon _{i+1} a, \quad
\Gamma^+_i a \circ_{i+1}\Gamma^-_i a = \varepsilon_{i}a.  $$

These are important **cancellation** laws for the connections. They can be interpreted as saying that turning left and then right, or vice versa, leaves you facing the same way.  They were introduced by C.B. Spencer for double categories (see below).

In the [[omega-groupoid]] case, the $\Gamma^+_i$ can be recovered from the $\Gamma^-_i$, and vice versa,  by using the inverses, assumed to arise from the groupoid structure.

The main result of the second paper below is that (strict) cubical omega-groupoids with connections are equivalent to crossed complexes. It is easy to construct a functor from the former to the latter; the hard work is to show that such an omega-groupoid may be functorially reconstructed from the crossed complex it contains.

This work is used in the third paper to construct and apply a strict cubical homotopy groupoid of a filtered space.

The main result of the fourth   paper below is that (strict) cubical omega-categories  with connections are equivalent to strict globular omega-categories.

For more information, see [[Nonabelian Algebraic Topology]]..

##References##

* C.B. Spencer,  "An abstract setting for homotopy pushouts and pullbacks", Cahiers Topologie G\'eom. Diff\'erentielle,
18 (1977),  409-429.

* R. Brown and P. J. Higgins, The algebra of cubes, _J. Pure
Appl. Alg._, 21 (1981), 233--260.

* R. Brown and P. J. Higgins, Colimit theorems for relative homotopy groups.
_J. Pure Appl. Algebra_ 22 (1) (1981) 11--41.

* F. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence   between a globular and cubical approach, _Advances in Mathematics_, 170,   (2002), 71--118.

[[!redirects cubical omega-category]]
[[!redirects cubical ∞-category]]


comprehension > history

* see at _[[comprehension]]_


comprehensive factorization system

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

##Statement
 {#Statement}

+-- {: .num_theorem Comprehensive Factorization System }
###### Theorem

Let $E$ be the class of [[final functors]] and $M$ be the class of [[discrete fibrations]]. Then $(E,M)$ is an [[orthogonal factorization system]] of [[Cat]], called the **comprehensive factorization system**.

=--

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

Let $F:C\to D$ be a functor.  Define $K:D\to Set$ as the left Kan extension of the constant presheaf $C\to Set$ at the singleton along $F$.  Explicitly, $K(d)$ is the set of connected components of $F/d$. Let $E=\int K$, so an object of $E$ is an ordered pair $(d, [\alpha:Fc\to d])$ where $[\alpha]$ denotes the connected component of $(c,\alpha)$.  Then it is not hard to verify that $e:C\to E$ mapping $c\mapsto (Fc,[id_{fc}])$ is final, the canonical $m:E\to D$ is a discrete fibration, and $F=me$.

Now we show that $E$ and $M$ are replete subcategories of $Cat$.  Clearly they include all isomorphisms.

If functors $F:C\to D$ and $G: D\to E$ are final, then we show that $G\circ F$ is final.  For $e\in E$, there is an element $(d,\alpha:e\to Gd)$ of $e/G$, and thence an element $(c,\beta:d\to Fc)$ of $d/F$, so we obtain an element $(c, e \stackrel{\alpha}{\to} Gd \stackrel{G\beta}{\to} GFc)$ of $e/GF$.  Now we must show that any two elements $(c,\gamma:e\to GFc),(c',\gamma':e\to GFc')$ are connected.  Since $G$ is final, elements $(Fc,\gamma)$ and $(Fc',\gamma')$ of $e/G$ are connected.  It suffices to consider the case of a zig-zag of length one: a morphism $f:Fc\to Fc'$ such that

\begin{xymatrix}
  e
    \ar[r]^\gamma
    \ar[dr]_{\gamma'}
  &
  GFc
    \ar[d]^{Gf}
  \\
  &
  GFc'
\end{xymatrix}

By finality of $F$, the elements $(c,id:Fc\to Fc)$ and $(c', f:Fc\to Fc')$ of $Fc/F$ are connected.  A zig-zag path between them, by precomposition with $\gamma$, becomes a zig-zag path between $(c,\gamma)$ and $(c',\gamma')$.  So $G\circ F$ is final.

The proof that discrete fibrations form a subcategory is omitted.

Now we must show that the lifting problem

\begin{xymatrix}
A \ar[r]^f \ar[d]_e & X \ar[d]^m \\
B \ar[r]_g \ar@{-->}[ru]^h & Y \\
\end{xymatrix}

has a unique solution $h$ when $e\in E$ and $m\in M$.

We prove uniqueness first.  For $b\in B$, let $(a,\alpha:b\to e(a))\in b/e$.  Then $h(\alpha)$ must be the unique lifting of $g(\alpha)$, and $h(b)$ the domain of this lifting, proving uniqueness of $h$ on objects.  For $\beta:b\to b'$ in $B$, $h(\beta)$ must be the unique lifting of $g(\beta)$, so $h$ is unique (if it exists).

Now we must show that this $h$ is well-defined, functorial, and a solution to the lifting problem.  If $(a',\alpha':b\to e(a'))$ is another element of $b/e$, then WLOG let $u:a\to a'$ such that

\begin{xymatrix}
  b
    \ar[r]^\alpha
    \ar[dr]_{\alpha'}
  &
  e(a)
    \ar[d]^{e(u)}
  \\
  &
  e(a')
\end{xymatrix}

Lifting this diagram, we see that $g(\alpha)$ and $g(\alpha')$ must lift to morphisms with identical domain, so $h$ is well-defined on objects.

For $\beta:b\to b'$ in $B$, let $\alpha:b'\in e(a)$, and by the diagram

\begin{xymatrix}
  b
    \ar[r]^\beta
    \ar[dr]_{\alpha\circ\beta}
  &
  b'
    \ar[d]^\alpha
  \\
  &
  e(a)
\end{xymatrix}

we see that $g(\beta)$ and $g(\alpha\circ\beta)$ must lift to morphisms with identical domains, so $h(\beta)$ has domain $h(b)$.

Functoriality now follows easily from uniqueness of lifting for a discrete fibration, and it is not hard to show that $h$ is a solution to the lifting problem.

=--

Dually, there is an orthogonal factorisation system $(E,M)$ on $Cat$ for which $E$ is the class of [[initial functors]] and $M$ is the class of [[discrete opfibrations]].

## References

* [[Ross Street]], [[R. F. C. Walters]], *The Comprehensive Factorization of a Functor*, Bulletin of the American Mathematical Society 79(5): 936-941 ([bams:1183534973](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-79/issue-5/The-comprehensive-factorization-of-a-functor/bams/1183534973.full), [doi:10.1090/S0002-9904-1973-13268-9](https://doi.org/10.1090/S0002-9904-1973-13268-9))

Note there is a mistake in the proof of the main theorem of the paper above, as noted on page 74 of:

* {#Kelly82} [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* [[Fosco Loregian]], [[Emily Riehl]], *Categorical Notions of Fibration*, Expositiones Mathematicae **38**, 2020. ([arXiv:1806.06129](https://arxiv.org/abs/1806.06129), [doi:10.1016/j.exmath.2019.02.004](https://doi.org/10.1016/j.exmath.2019.02.004))

* [[Clemens Berger]], [[Ralph M. Kaufmann]], *Comprehensive Factorization Systems*, Tbilisi Math. J. 10 (2017), 255-277 ([doi:10.1515/tmj-2017-0112](https://projecteuclid.org/journals/tbilisi-mathematical-journal/volume-10/issue-3/Comprehensive-factorisation-systems/10.1515/tmj-2017-0112.short))

* [[Paolo Perrone]], [[Walter Tholen]], *Kan extensions are partial colimits*, Kan Extensions are Partial Colimits. Applied Categorical Structures **30**, 685–753 (2022). ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531). [doi:10.1007/s10485-021-09671-9](https://doi.org/10.1007/s10485-021-09671-9))

Internal comprehensive factorisations (and [[torsors]]) are considered in:

* [[Ross Street]], [[Dominic Verity]], *The comprehensive factorization and torsors*, [[TAC]] **23** (2010)  42-75 [[tac:23-03](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html)]

[[!redirects comprehensive factorization systems]]
[[!redirects comprehensive factorization]]
[[!redirects comprehensive factorisation]]
[[!redirects comprehensive factorisation system]]
[[!redirects comprehensive factorisation systems]]


*Comptes rendus hebdomadaires des séances de l'Académie des sciences* (CRAS) is a journal that published, among other topics, articles in [[category theory]] (usually under *Algèbre des catégories*, *Théorie des catégories*, or *Logique et catégories*) around 1960s and 1970s.

Currently, it is published under the name *[Comptes Rendus. Mathématique](https://comptes-rendus.academie-sciences.fr/mathematique/journals/CRMATH/)* by the French Academy of Sciences.

The articles are up to 4 pages in length, which means that proofs are usually omitted. While CRAS articles have been archived, their text has not been indexed by search engines, which means it can be difficult to find copies of papers published in the journal. Therefore, here we list various articles pertaining to category theory that were published by CRAS.

## List of category theory articles published in CRAS

- *L'obstruction à la construction d'une section d'une fibré au sens de Kan*, Michel Zisman (1960), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3202p/f819))
- *Homologie à coefficients dans un antifaisceau*, Réne Deheuvels (1960), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k732p/f756))
- *Les précomplexes semi-simpliciaux*, Michel André (1962), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4003q/f534))
- *Catégories doubles et catégories structurées*, [[Charles Ehresmann]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246))
- *Catégories avec multiplication*, [[Jean Bénabou]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1965))
- *Catégorie double des quintettes; applications covariantes*, [[Charles Ehresmann]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1969))
- *Catégories structurées d'opérateurs*, [[Charles Ehresmann]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4005b/f16))
- *Sous-structures et applications $\mathscr{K}$-covariantes*, [[Charles Ehresmann]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4005b/f232))
- *Catégories localement de type fini et catégories localement nœthériennes*, Yvon Nouazé (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4007z/f863))
- *Complétion des catégories ordonnées*, [[Charles Ehresmann]] (1963), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4009k/f1388))
- *Un théorème de plongement des catégories*, André Roux (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4012p/f261))
- *Sur une équivalence de catégories abéliennes*, André Roux (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4012p/f1235))
- *Sur les décompositions de morphismes dans les catégories à sommes our à produits fibrés*, (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4012p/f2032))
- *Complétion des catégories sous-prélocales*, (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f723))
- _*Introduction à l'étude de la distributivé des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)*_, Roos (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f1011))
- _*Sur la distributivé des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)*_, Roos (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f1673))
- _*Complément à l'étude de la distributivité des foncteurs $\varprojlim$ par rapport aux $\varinjlim$ dans les catégories des faisceaux (topos)*_, Roos (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40149/f128))
- _*Expansion d'homomorphismes en foncteurs*_, (1964), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f1438))
- _*Expansion générale des foncteurs*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4016x/f30))
- _Homologie dans les catégories non abéliennes; Exactitude dans C*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4016x/f779))
- _*Critères de représentabilité des foncteurs*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4016x/f782))
- _*Homologie dans les catégories non abéliennes; définition de l'homologie*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4016x/f1101))
- _*Catégorie quotient d'une catégorie par une sous-catégorie*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40177/f833))
- _*Sous-jacence, union et intersection en catégories*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4018j/f16))
- _*Cohomologie non abélienne; préliminaires*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4018j/f19))
- _*Construction de catégories ordonnées, projections de graphes multiplicatifs ordonnés*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4018j/f920))
- _*Systèmes formels et catégories*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4018j/f1202))
- _*Catégories relatives*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37))
- _*Graphes et cographes sur une catégorie abstraite*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236863n/f391))
- _*Catégorie double projection d'un graphe multiplicatif double*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236863n/f395))
- _*Objets libres dans une catégorie primitive et comultiplications*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236863n/f435))
- _*Foncteur de séperation*_, (1965), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6238594s/f66))
- _*Sur la structure des objets de certaines catégories abéliennes*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6238594s/f327))
- _*Sur les notions d'image, noyau, coimage, conoyau*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6238594s/f413))
- _*Foncteur de séperation dans des catégories de type topologique*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6474412x/f21))
- _*Cohomologie des catégories de Banach*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6474412x/f289))
- _*Cohomologie des catégories de Banach : applications*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6426017v/f19))
- _*Les théorèmes d'isomorphismes dans les catégories à supports*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6248450j/f17))
- _*Adjonction de limites aux catégories structurées*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6248450j/f77))
- _*Quasi-élargissement d'un système de structures structuré*_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6248450j/f184))
- _Premier théorème d'expansion structurée_, (1966), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6248451z/f5))
- _Deuxième théorème d'expansion structurée_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f19))
- _Sur la notion de point de partage dans les ensembles ordonnés et son application aux limites dans les catégories_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f23))
- _Théorème de quasi-expansion régulière_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f70))
- _Problèmes universels relatifs aux catégories n-aires_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f287))
- _Sur la dimension des catégories de foncteurs d'un ensemble ordonné vers une catégorie_, ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6435215p/f291))
- _Produit tensoriel généralisé_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6431380c/f248))
- _Construction d'une catégorie n-uple projection d'un graphe multiplicatif n-uple_, ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6431380c/f252))
- _Expansions $p_{\mathscr{F}}$-structurées_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6431380c/f313))
- _Sur les structures algébriques_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62390736/f74))
- _Existence et construction de sommes finies dans un catégorie d'applications inductives entre classes locales complètes et dans la catégorie des applications «continues» entre paratopologies_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62390736/f158))
- _Catégorie double de fractions, perfectionnement $p_{\mathscr{F}}$-structuré d'une catégorie double_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236329w/f239))
- _Adjonctions de limites à un foncteur fidèle ou à une catégorie_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62358514/f22))
- _Sur le centre d'une catégorie de Grothendieck_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62358514/f99))
- _Construction explicite d'une complétion inductive libre pour un catégorie quelconque_, (1967), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6234140m/f278))
- _Sur les catégories fibrées_, (1968), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480294z/f412))
- _Catégories fibrées et cohomologie non abélienne de dimension deux_, (1968), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480294z/f456))
- _Sections initiales d'un ensemble ordonné_, (1969), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4802973/f116))
- _Structures uniformes et préuniformes dans une catégorie à involution_, (1969), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4802973/f270))
- _La duale de la catégorie $\mathscr{L}d(0, 1)$_, (1969), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4802973/f497))
- _Sur les catégories des $(t, T)$-faisceaux_, (1969), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4802973/f411))
- _Les faisceaux d'une théorie_, (1969), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4802973/f378))
- _Problèmes universels associés à quelques catégories d'applications_, ([[René Guitart]], 1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480298g/f1408.item))
- _Transformations H-naturelles et H-quintettes_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f9))
- _Structures discrètes et triples_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f13))
- _Réalisations dominées_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f17))
- _Complétion de groupoïdes uniformes_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f21))
- _Esquisse de topologie_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f24))

- _Morphismes et comorphismes de cotriples_, Laurent Coppey (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f27))

  > (on [morphisms of (co)monads](monad#BicategoryOfMonads))

- _Ensemble ordonné des atlas d'un groupoïde_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f31))
- _Foncteurs structurés compatiblement engendrants et à adjoints compatibles_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5728916p/f16))
- _Catégories discrèment structurées_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5728916p/f20))
- _Groupoïde uniforme_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f79))
- _Foncteurs structurés compatiblement engendrants et à adjoints compatibles_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f125))
- _Catégories discrèment structurées_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f129))
- _Factorisations d'un cotriple_, Laurent Coppey (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f133))
- _Transformations H-naturelles et H-quintettes_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f216))
- _Structures discrètes et triples_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f220))
- _Réalisations dominées_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f224))
- _Complétion de groupoïdes uniformes_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f228))
- _Esquisse de topologie_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f231))
- _Morphismes et comorphismes de cotriples_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f234))
- _Ensemble ordonné des atlas d'un groupoïde_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f238))
- _Commutations des limites généralisées_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5618529d/f5))
- _Equisses de types : catégories munies de limites_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56212371/f11))
- _Définitions de l'homotopie entre relations_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5620643t/f13))
- _Cohomologie non abélienne et 2-catégorie d'opérateurs sur une catégorie_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5620643t/f10))
- _Plongements, objets induits, co-induits et quotients dans une catégorie_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f1234))
- _Idées et maquettes de structures algébriques_, (1970), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480299v/f1238))
- _Pureté et platitude dans les catégories de modules généralisés_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f16))
- _Les faisceaux d'une classe des morphismes_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f104))
- _Cofaisceaux d'une catégorie_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f302))
- _Problème universel résolu par un foncteur d'interpolation_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f532))
- _La catégorie des espaces cohérents_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f859))
- _Sur les catégories doubles associatives et unitaires à des isomorphismes près_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f863))
- _Catégories exactes_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k480300n/f1510))
- _Algèbres de décompositions dans les catégories_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619155t/f24))
- _Sur les fonctions associées à la notion de sous-structure_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619155t/f28))
- _Foncteurs additifs : Algèbres et multi-algèbras_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191923/f15))
- _Foncteurs d'omission de structures algébriques_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191923/f19))
- _Structures n-uples de Hurewicz_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191671/f14))
- _Moyennes invariantes dans une catégorie_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191671/f18))
- _Conditions nécessaires et suffisantes pour qu'une catégorie soit fortement ou très fortement spécifiable_, (1971), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5620638h/f22))
- _Objets séparés ou compacts dans une catégorie_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5743515w/f26))
- _Construction de catégories auto-dominées_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5743515w/f30))
- _Triples p-structurés_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5794353k/f25))
- _Sur la notion de complétion d'un foncteur p-structuré_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5794353k/f28))
- _Complétion des catégories monoïdales fermées_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5794353k/f32))
- _La réflexivité des espaces vectoriels à génération compacte_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k57351665/f22))
- _Catégorie des foncteurs types et structures_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56207042/f26))
- _Sur les catégories produit-croisé_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56207042/f30))
- _Catégories localement cohérentes et objets α-injectifs_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619143m/f14))
- _Catégories hilbertiennes et modèles caractéristiques_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619100v/f7))
- _Sur les triples structurés_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191426/f23))
- _Sur les idempotents dans les triples et la description des structures_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191426/f27))
- _Complétion universelle et régularisation d'une petite catégorie préadditive_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5731002n/f7))
- _Groupe dans une catégorie_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7))
- _Sur le type simple d'homotopie des groupoïdes_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56189647/f9))
- _Catégories à sommes commutables_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k57310492/f19))
- _Sur les objets de Kleisli et d'Eilenberg-Moore dans une 2-catégorie_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k57310492/f23))
- _Au sujet de l'existence d'adjoints à droite aux foncteurs «image réciproque» dans la catégorie des catégories_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k56191352/f19))
- _Constructions relatives aux objets compacts d'une catégorie_, (1972), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k5619091b/f33))
- _Quasi-topos_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f251))
- _Algèbres relatives à une loi distributive_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217214v/f21))
- _Fermeture standard des catégories de structures algébriques_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217214v/f95))
- _Caractérisation de Dist_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217214v/f99))
- _Catégories relatives à une loi distributive_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217214v/f249))
- _2-catégories représentables associées à une catégorie cartésienne fermée_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6217214v/f253))
- _Monomorphismes et objets algébiquement et existentiellement clos dans les catégories localement présentables. Applications aux monoïdes, catégories et groupoïdes_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236938x/f19))
- _Lois distributives mixtes_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236938x/f193))
- _Complétion absolue universelle d'une petite catégorie_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236938x/f197))
- _Coalgèbre et extension de Kan_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236938x/f387))
- _Dominions de foncteurs relatifs_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236938x/f391))
- _Complétion monadique_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6424261b/f243))
- _Anadèses et catadèses naturelles_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6424261b/f247))
- _Structures algébriques duales, involutives et commutatives_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6424261b/f493))
- _Structures algébriques et extensions de Kan d'applications covariantes_, [[René Guitart]] (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62362677/f97))
- _Dominos_, ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62362677/f101))
- _Ensemble sous-jacent dans un topos_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62362677/f167))
- _Topos infinitésimal_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6312609m/f155))
- _Catégories algébriques comme catégories relatives_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k63126108/f165))
- _Sur les structures définies par des graphes multiplicatifs avec des cônes_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236335m/f35))
- _Les monades involutives en théorie élémentaire des ensembles_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236335m/f85))
- _Anadèses et catadèses naturelles entre 3-foncteurs_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236335m/f175))
- _Dominions de foncteurs relatifs_, (1973), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236335m/f307))
- _Analgèbres et catalgèbres_, [[Dominique Bourn]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f331))
- _Algorithme de $n$-arisation pour les structures algébriques esquissées, Christian Lair (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f233))
- _Syntaxes et catégories monadiques_, Jean-Pierre Barthelemy (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f213))
- _Systèmes $(K', p, K)$-injectifs et systèmes $(K', p, K)$-surjectifs_, Robert Durand (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236817d/f37))
- _Sous-catégorie classifiée d'une 2-catégorie_, [[Jacques Penon]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f489))
- _$J$-adjonction, limite $J$-absolue et $J$-théorie algébrique_, [[Yves Diers]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236817d/f375))
- _Construction d'un modèle booléen de la théorie des ensembles à partir d'un topos booléen_, Marie-Françoise Coste (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236817d/f441))
- _Catégories de fractions quasi-topologiques_, Françoise Berquier (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f57))
- _Un adjoint au foncteur Span_, Élisabeth Vaugelade (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f61))
- _Distributeurs (co) représentables_, Roland Grünig (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f117))
- _Constructions de monades involutives_, [[René Guitart]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k64180124/f155))
- _Traduction équationnelle de notions ensemblistes_, [[René Guitart]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k64180124/f205))
- _Sesquifibrations_, Dominique Prochasson (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f485))
- _Sur l'algébricité de la complétion à gauche des catégories_, Dominique Prochasson (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f327))
- _Catégories fibrées et homotopie_, Georges Hoff (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62369363/f237))
- _Catégories cofibrées, fibrations faibles_, (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236817d/f445))
- _Une notion de limite dans les 2-catégories_, (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f19))
- _Catégories localement internes_, (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f355))
- _Ditopos_, [[Dominique Bourn]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6226936t/f15))
- _Sur les ditopos_, [[Dominique Bourn]] (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6226936t/f195))
- _Une formalisation simple de la théorie des catégories_, Giorgio Germano (1974), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k64180124/f293))
- _Structures et opérations initiales et finales_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62266732/f255))
- _Préfiltres initiaux et finaux. Topologies initiales et finales_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6226674g/f17))
- _J-monades_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6216795q/f279))
- _Arithmétique de premier ordre dans les topos_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62167964/f163))
- _Théorème de Yoneda dans les bicatégories_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62167964/f245))
- _Caractérisation de Dist(X) pour une bicatégorie X à sommes_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6234154n/f145))
- _Les fins cartésiennes_, Syméon Bozapalidès (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62154338/f5))
- _Bon ordre dans l'objet des nombres naturels d'un topos booléen_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62154338/f9))
- _Théories relatives à un corpus_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f105))
- _Fibrations petites et localement petites_, (1975), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f171))
- _Les fins cartésiennes_, Jean-Marie Sirot (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6426770n/f42))
- _Théorèmes, exemples et contre-exemples pour certaines propriétés des entiers naturels dans les topos_, Louis Mahé (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62351984/f353))
- _Objets à limites dans les bicatégories_, Syméon Bozapalides (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6426770n/f372))
- _Monoïdes et catégories avec zéro algébriquement ou existentiellement clos_, Bernard Calvo (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6235850q/f247))
- _Bicatégories relatives à une 2-catégorie multiplicative_, Syméon Bozapalides (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62351984/f215))
- _Les quasi-quotients dans le transfert d'adjonctions aux catégories commas_, Luc Van den Bril (1976), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k62351984/f219))
- _Spectres et localisations relatifs à un foncteur_, [[Yves Diers]] (1978), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6238948x/f513))
- _Théories algébriques internes dans les topos_, Brigitte Lesaffre (1978), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f395))
- _Les théories convexes de Horn_, Jorge Herrera Miranda (1978), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k9813799h/f127))
- _La Théorie de Galois dans les Catégories monoïdales_, Thomas S. Ligon (1979), ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k98138170/f17))

## Related pages

* [[Esquisses mathématiques]]

[[!redirects CRAS]]

category: reference


Compton wavelength

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _Compton wavelength_ is a [[physical unit]] that governs the [[dispersion relation]]/[[energy-momentum relation]] of [[mass|massive]] [[field (physics)|fields]].

For a [[particle]]/[[field (physics)|field]] of [[mass]] $m$, its _Compton wavelength_ is the [[length]]

$$
  \ell_m \coloneqq \frac{2\pi \hbar}{m c}
$$

where $c$ denotes the [[speed of light]] and $2\pi\hbar$ denotes [[Planck's constant]]. Correspondingly $\frac{\hbar}{m c}$ is also called the "reduced Compton wavelength".

The inverse of the Compton wavelength appears as the [[mass]] term notably in the [[Klein-Gordon equation]] of the [[scalar field]] or the [[Dirac equation]] of the [[Dirac field]].

## Examples

* The Compton wavelength corresponding to the [[mass]] of the [[electron]] is about $\ell_{m_e} ~ 386$ [[fm]].

* Another length scale parameterized by a mass $m$ is the _[[Schwarzschild radius]]_ $r_m \coloneqq 2 m G/c^2$, where $G$ is the [[gravitational constant]]. Solving the [[equation]]

  $$
    \array{
      & \ell_m &=& r_m
      \\
      \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2
    }
  $$

  for $m$ yields the _[[Planck mass]]_

  $$
     m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}}
    \,.
  $$

  The corresponding Compton wavelength $\ell_{m_{P}}$ is given by the _[[Planck length]]_ $\ell_P$

  $$
    \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} }
    \,
  $$

## Related concepts

* [[wavelength]]

[[!include fundamental scales -- contents]]

## References

* Wikipedia, _[Compton wavelength](https://en.wikipedia.org/wiki/Compton_wavelength)_

[[!redirects Compton wavelength]]


computability

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Computability theory_ studies [[mathematics|mathematical]] entities that may be obtained by actual [[computation]] (instead of by less concrete proofs of existence). As such computability theory is similar to _[[constructive mathematics]]_ and to _[[realizability]]_; indeed (emphasized e.g. in ([Bauer 05](#Bauer05))):

Computable mathematics is the [[realizability]] [[interpretation]] of [[constructive mathematics]].

Computability theory deals only with computability in principle and disregards the complexity of computation, that is instead the topic of _[[complexity theory]]_.

The key concept in computability theory is that of a _[[computable function]]_, hence of a [[function]] whose output may be determined from its input by an actual [[computation]]. There are two main types of computability, depending on whether one takes the domain and codomain of computable functions to be _finite string_ from a finite alphabet, hence equivalently [[natural numbers]], or _infinite strings_ from a finite interval, hence sequence of natural numbers.

In the first case -- "type I computability" -- computable functions are [[partial recursive functions]]. In the second -- "type II computability" -- they are [[continuous functions]] on (quotients of) [[Baire space (computability)|Baire space]] (see at _[[computable function (analysis)]]_).

[[!include computable mathematics -- table]]

## Properties

### Relation to intuitionistic mathematics

Computable mathematics is an instance of [[intuitionistic mathematics]] (see e.g. ([Bauer 05, section 4.3.1](#Bauer05))).

## Related concepts

* [[synthetic computability theory]]

* [[computation]], [[hypercomputation]]

* [[computable real number]]

* [[computable set]]

* [[equilogical space]]

* [[halting theorem]]

* [[persistent homology]]

* [[computable physics]]

* [[descriptive set theory]]

## References

Textbook containing the basic notions:

* [[Michael Sipser]], *Introduction to the Theory of Computation*, Third edition (2012), Course Technology Inc.

The relation to [[constructive mathematics]] and [[realizability]] is discussed in

* {#Bauer05} [[Andrej Bauer]], _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

based on

* {#Bauer00} [[Andrej Bauer]], _The Realizability Approach to
Computable Analysis and Topology_, PhD thesis CMU (2000) ([pdf](http://math.andrej.com/wp-content/uploads/2006/04/thesis.pdf))

* Peter Lietz, _From Constructive Mathematics to Computable Analysis via the Realizability Interpretation_ ([pdf.gz](http://www.mathematik.tu-darmstadt.de/~streicher/THESES/lietz.pdf.gz))

For [[computable analysis]] see

* {#Weihrauch00} [[Klaus Weihrauch]], _Computable Analysis_ Berlin: Springer, 2000

[[!redirects computable mathematics]]

[[!redirects computability theory]]
[[!redirects computability theories]]

[[!redirects recursion theory]]
[[!redirects recursion theories]]


computable analysis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[computability]] in [[analysis]], [[constructive analysis]]

## Related concepts

[[!include computable mathematics -- table]]

* [[computable real number]]

* [[exact real computer arithmetic]]

* [[computable physics]]

## References

A standard textbook is

* {#Weihrauch00} [[Klaus Weihrauch]], _Computable Analysis_,  Springer Berlin (2000)

A survey is in

* Herman Geuvers, Milad Niqui, [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Science
Mathematical Structures in Computer Science / Volume 17 / Issue 01 / February 2007, pp 3-36 ([publisher](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=851596))

The Relation to [[realizability topos]] theory is discussed in

* P. Lietz, _From constructive mathematics to computable analysis via the
realizability interpretation_ , Ph.D. dissertation, Fachbereich Mathematik,
TU Darmstadt, Darmstadt, 2004.

* [[Andrej Bauer]], _The realizability approach to computable analysis and topology_, Ph.D. dissertation, School of Computer Science, Carnegie Mellon University, Pittsburgh, 2000.

* {#Bauer05} [[Andrej Bauer]], _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))