distributive lattice

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#### $(0,1)$-Category theory
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#Contents#
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## Definition

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

A __distributive lattice__ is a [[lattice]] in which [[join]] $\vee$ and [[meet]] $\wedge$ _distribute_ over each other, in that for all $x,y,z$ in the lattice, the _distributivity laws_ are satisfied:

*  $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$,
*  $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$.

=--

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

The nullary forms of distributivity hold in any lattice:

*  $x \vee \top = \top$,
*  $x \wedge \bot = \bot$.

Distributive lattices and [[lattice]] [[homomorphisms]] form a [[concrete category]] [[DistLat]].

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###### Remark

Any lattice that satisfies one of the two binary distributivity laws must also satisfy the other; isn\'t that nice? (This may safely be left as an [[exercise]].) This also means that a distributive lattice is precisely a lattice (or indeed a [[poset]]) which is a [[distributive category]] (when viewed as a [[thin category]].)

This convenience does *not* extend to infinitary distributivity, however.

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###### Remark

Each distributivity law holds in one direction in any lattice (the proofs of which may be left as an exercise):

* $x \vee (y \wedge z) \leq (x \vee y) \wedge (x \vee z)$
* $x \wedge (y \vee z) \geq (x \wedge y) \vee (x \wedge z)$

As a result, the real content of the definition consists of the converse directions (and as remarked above, either one suffices):

* $x \vee (y \wedge z) \geq (x \vee y) \wedge (x \vee z)$
* $x \wedge (y \vee z) \leq (x \wedge y) \vee (x \wedge z)$

=--

## Alternative characterizations

As mentioned above, the theory of distributive lattices is self-dual, something that is proved in almost any account (or left as an exercise), but which is not manifestly obvious from the standard definition which chooses one of the two distributivity laws and goes from there. In this section we provide some other characterizations or axiomatizations which *are* manifestly self-dual.

Here is one such characterization:

+-- {: .num_prop #sd1}
###### Proposition
A lattice is distributive if and only if the identity

$$(a \wedge b) \vee (a \wedge c) \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) \wedge (b \vee c)$$

is satisfied.
=--

Again this may be left as a (somewhat mechanical) exercise.

Perhaps more useful in practice is the characterization in terms of "forbidden sublattices" due to Birkhoff. Namely, introduce the "pentagon" $N_5$ as the 5-element lattice $\{\bot \leq a, b, c \leq \top\}$ where $b \leq c$ and $a$ is incomparable to $b, c$, so that $\bot = a \wedge c$ and $a \vee b = \top$. Introduce the "thick diamond" $M_3$ as the 5-element lattice $\{\bot \leq a', b', c' \leq \top\}$ with $a', b', c'$ pairwise incomparable. Both $N_5$ and $M_3$ are self-dual. Birkhoff's characterization is the following (manifestly self-dual) criterion.

+-- {: .num_theorem}
###### Theorem
A lattice $L$ is distributive if and only if there is no [[injective function|embedding]] of $N_5$ or $M_3$ into $L$ that preserves binary meets and binary joins.
=--

This can be useful for determining distributivity or its failure, especially in cases where one can visualize a lattice via its [[Hasse diagram]].

The necessity of the forbidden sublattice condition is clear in view of the fact that the cancellation law stated in the next result fails in $N_5$ and $M_3$. This result gives another self-dual axiomatization of distributive lattices.

+-- {: .num_prop #sd2}
###### Proposition
A lattice $L$ is distributive if and only if the cancellation law holds: for all $y, z \in L$, we have $y = z$ whenever $x \wedge y = x \wedge z$ and $x \vee y = x \vee z$ (for some $x$).
=--

+-- {: .proof}
###### Proof
"Only if": if $x \wedge y = x \wedge z$ and $x \vee y = x \vee z$, then

$$y = y \vee (x \wedge y) = y \vee (x \wedge z) = (y \vee x) \wedge (y \vee z) = (x \vee z) \wedge (y \vee z) = (x \wedge y) \vee z$$

which implies $z \leq y$, and similarly we have $y \leq z$.

"If": this is harder. Assuming the cancellation law for $L$, we first show $L$ is modular. Recall from [[modular lattice]] that for any lattice $L$ and $a, b \in L$, there is a covariant [[Galois connection]]

$$([a \wedge b, b] \to [a, a \vee b]: x \mapsto a \vee x) \; \dashv \; ([a, a \vee b] \to [a \wedge b, b]: y \mapsto b \wedge y)$$

and that $L$ is *modular* if this Galois connection is a [[Galois correspondence]] (or [[adjoint equivalence]]) for all $a, b$. Now, if $x \in [a \wedge b, b]$, then $a \vee x = a \vee (b \wedge (a \vee x))$ because $f = f g f$ for any Galois connection $f \dashv g$. From $a \wedge b \leq x \leq b$ we also have

$$a \wedge x = a \wedge b \wedge x = a \wedge b = a \wedge b \wedge (a \vee x)$$

and so by cancellation of the $a$\'s, we conclude $x = b \wedge (a \vee x)$. Similarly (dually), for $y \in [a, a \vee b]$, we have $y = a \vee (b \wedge y)$. Hence $L$ is modular.

Now we show $L$ is distributive. Let $x, y, z \in L$ and consider the three elements

$$\array{
u & \coloneqq & [x \wedge (y \vee z)] \vee (y \wedge z) & = & [x \vee (y \wedge z)] \wedge (y \vee z) \\
v & \coloneqq & [y \wedge (x \vee z)] \vee (x \wedge z) & = & [y \vee (x \wedge z)] \wedge (x \vee z) \\
w & \coloneqq & [z \wedge (x \vee y)] \vee (x \wedge y) & = & [z \vee (x \wedge y)] \wedge (x \vee y)
}$$

where the non-definitional equalities follow from modularity. Using the first expressions, we compute

$$\array{
u \vee v & = & [x \wedge (y \vee z)] \vee (y \wedge z) \vee [y \wedge (x \vee z)] \vee (x \wedge z) \\
 & = & [x \wedge (y \vee z)] \vee [y \wedge (x \vee z)] \\
 & = & [(x \wedge (y \vee z)) \vee y] \wedge (x \vee z) \\
 & = & (x \vee y) \wedge (x \vee z) \wedge (y \vee z)
}$$

where the third and fourth lines use modularity. By symmetry in the letters $x, y, z$, we also have $u \vee w = v \vee w = (x \vee y) \wedge (x \vee z) \wedge (y \vee z)$. Now the second expressions are dual to the first, so by duality we compute

$$u \wedge v = u \wedge w = v \wedge w = (x \wedge y) \vee (x \wedge z) \vee (y \wedge z).$$

Now by cancellation of the $u$\'s, we may conclude $v = w$, but in that case we obtain

$$(x \vee y) \wedge (x \vee z) \wedge (y \vee z) = v \vee w = v \wedge w = (x \wedge y) \vee (x \wedge z) \vee (y \wedge z)$$

so that $L$ is distributive by Proposition \ref{sd1}.
=--

+-- {: .num_remark}
###### Remark
While the expressions for $u, v, w$ in the preceding proof may look as though they come out of thin air, the underlying idea is that the sublattice of $L$ generated by $x, y, z$ is the [[image]] of a lattice map $F(3) \to L$ out of the free modular lattice $F(3)$ on three elements. The only obstruction to distributivity in $F(3)$ is the presence of an $M_3$-sublattice appearing in the center of its [Hasse diagram](https://golem.ph.utexas.edu/category/2015/09/the_free_modular_lattice_on_3.html#c049685). The middle elements of that sublattice correspond to the formal expressions for $u, v, w$ given above, and the proof shows that under the cancellation law, we have $u = v = w$ in $L$, making the thick diamond collapse to a point in $L$ and removing the obstruction to distributivity.
=--

From Proposition \ref{sd2}, it is not very hard to deduce Birkhoff's theorem. The presence of a copy of $M_3$ or $N_5$ in a non-distributive lattice $L$ is deduced from a failure of the cancellation law where we have three elements $x, y, z$ with $x \wedge y = x \wedge z$, $x \vee y = x \vee z$, and $y \neq z$. If $y, z$ are comparable, say $y \leq z$, then the set $\{x \wedge y, x, y, z, x \vee y\}$ forms an $N_5$. If $y, z$ are incomparable, then we have either $y \vee z \lt x \vee y$, or $y \wedge z \gt x \wedge y$, or both $y \vee z = x \vee y$ and $y \wedge z = x \wedge y$; in the first two cases we get an $N_5$ (e.g., $\{x \wedge y, x, y, y \vee z, x \vee y\}$ for the first case), and in the third case the set $\{x \wedge y, x, y, z, x \vee y\}$ forms an $M_3$.

## Examples

Any [[Boolean algebra]], and even any [[Heyting algebra]], is a distributive lattice.

Every [[frame]] and every [[sigma-frame|$\sigma$-frame]] is a distributive lattice.

Any [[linear order]] is a distributive lattice.

An [[integral domain]] is a [[Prüfer domain]] iff its lattice of [[ideals]] is distributive. The classical example is $\mathbb{Z}$; equivalently, the (opposite of the) multiplicative monoid $\mathbb{N}$ ordered by divisibility, with $1$ at the bottom and $0$ at the top.

The lattice of [[Young diagrams]] ordered by inclusion is distributive.

## Infinitely distributive property

A distributive lattice that is complete (not necessarily [[completely distributive lattice |completely distributive]]) may be **infinitely distributive** or said to satisfiy the **infinite distributive law** :

  $$
    x \wedge (\bigvee_i y_i) = \bigvee_i (x\wedge y_i)
  $$

This property is sufficient to give the lattice [[Heyting algebra]] stucture where the implication $a\Rightarrow b$ (or  [[exponential object]] $b^a$) is:

$$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x$$

Note that this property does not imply the dual  **co-infinitely distributive** property:

  $$
    x \vee (\bigwedge_i y_i) = \bigwedge_i (x\vee y_i)
  $$

Instead this dual gives the lattice [[co-Heyting algebra|co-Heyting]] structure where the co-implication or "subtraction"  ($\backslash$) is

$$(u \backslash v) = \bigwedge_{u \leq v \vee x} x$$

If a lattice has both properties, as in a [[completely distributive lattice]], then it has [[bi-Heyting algebra|bi-Heyting]] structure (both  Heyting and co-Heyting) and the two exponentials are equal.

$$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x \qquad  = \qquad (u \backslash v) = \bigwedge_{u \leq v \vee x} x$$

> Does it make sense to define "infinitely distributive property" for non-complete lattices? (Something like: "Whenever the join exists, it satisfies the infinite distributive law.")

## Properties

### Finite distributive lattices
 {#OppositeCategory}

Since a finite distributive lattice is  [[completely distributive lattice |completely distributive]] it is a bi-Heyting lattice, as shown above.

Let $FinDistLat$ be the category of finite distributive lattices and lattice homomorphisms, and let $FinPoset$ be the category of finite [[posets]] and order-preserving functions.  These are contravariantly equivalent, thanks to the presence of an [[ambimorphic object]]:

**Proposition.** The [[opposite category]] of $FinDistLat$ is [[equivalence of categories|equivalent]] to $FinPoset$:

$$
  FinDistLat^{op} \simeq FinPoset.
$$

One direction of this equivalence is given by the hom-functor

$$
  [-,2] \;\colon\; FinDistLat^{op} \stackrel{\simeq}{\to} FinPoset
$$

where $2$ is the 2-element distributive lattice and for any $X \in  FinDistLat$, $[X,2]$ is the poset of distributive lattice morphisms from $X$ to $2$.  The other direction is given by

$$
  [-,2] \;\colon\; FinPoset^{op} \stackrel{\simeq}{\to} FinDistLat
$$

where $2 = \{0,1\}$ is the 2-element poset with $0 \lt 1$ and for any $Y \in \FinPoset$, $[Y,2]$ is the distributive lattice of poset morphisms from $Y$ to $2$.

This __Birkhoff duality__ is (in one form or another) mentioned in many places; the formulation in terms of hom-functors may be found in

* [[Gavin C. Wraith]], _Using the generic interval_, Cah. Top. Géom. Diff. Cat. **XXXIV** 4 (1993) pp.259-266. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1993__34_4/CTGDC_1993__34_4_259_0/CTGDC_1993__34_4_259_0.pdf))

The functorial nature of the correspondence means that morphisms of finite posets (i.e. order-preserving maps) naturally correspond to morphisms of finite distributive lattices (i.e. order-preserving maps that also respect meet and join).

It follows from Birkhoff's representation theorem that every finite distributive lattice can be seen as a lattice of sets (i.e. sets with join and meet given by union and intersection) -- in particular, sets whose elements are the join-irreducible elements of the lattice. Furthermore, a good intuition for why this duality holds is that either an element is generated as the join of existing elements, or it is join-irreducible. Hence given any existing poset, we can simply add all missing joins, and also a bottom (i.e. the nullary join). By general results (the adjoint functor theorem for posets) this suffices to ensure that all meets exist as well. This is analogous to the free colimit completion of a category, and indeed Birkhoff representation can be seen as a very special case of the Yoneda lemma as applied to (0,1)-category theory (i.e., order theory), since (0,1)-presheaves are functors into Bool rather than Set and hence correspond to [[lower set]]s.

Birkhoff duality does not hold for infinite distributive lattices. However, in the general case of not-necessarily-finite distributive lattices there is a correspondence not to posets, but instead to a class of spaces known as Priestley spaces. This is an instance of a general phenomena known as Stone-type duality.

### The free distributive lattice

Posets also give rise to a "free" distributive lattice, which is not the same as their Birkhoff dual. Instead, it is formed by the following procedure: First, take the poset of upsets with the reverse ordering (this is the free finite meet completion). Then form the distributive lattice of finitely generated downsets in that.

In the case that one begins with a discrete poset (i.e., a set) then the number of elements in the resultant free distributive lattice is known as a Dedekind number, which also counts the number of monotone Boolean functions in $n$ variables. Dedekind numbers increase extremely rapidly, and there is no good known closed-form summation to compute them. The first ten (and only known) Dedekind numbers are (starting at $n = 0$): 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, and 286386577668298411128469151667598498812366; see [OEIS A000372](#A000372).

### Categorification

Every distributive lattice, regarded as a [[category]] (a [[(0,1)-category]]), is a _[[coherent category]]_.  Conversely, the notion of coherent category may be understood as a [[categorification]] of the notion of distributive lattice.  A different categorification is the notion of [[distributive category]].

### Completion

The [[completely distributive lattice|completely distributive]] [[algebraic lattices]] (the [[frames of opens]] of [[Alexandroff locales]] ) form a [[reflective subcategory]] of that of all distributive lattices. The reflector is called _[[canonical extension]]_.

## References

* {#A000372} OEIS sequence [A000372](https://oeis.org/A000372)

[[!redirects distributive lattices]]


distributive law

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# Distributive laws

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## Idea

Sometimes in [[mathematics]] one considers objects equipped with two different types of [[extra structure]] which interact in a suitable way.  For instance, a [[ring]] is a [[set]] equipped with both (1) the structure of an (additive) [[abelian group]] and (2) the structure of a (multiplicative) [[monoid]], which satisfy the distributive laws $a\cdot (b+c) = a\cdot b + a\cdot c$ and $a\cdot 0 = 0$.

Abstractly, there are two [[monads]] on the [[category]] [[Set]], one (call it $\mathbf{T}$) whose [[algebra over a monad|algebras]] are abelian groups, and one (call it $\mathbf{S}$) whose algebras are monoids, and so we might ask "can we construct, from these two monads, a third monad whose algebras are rings?"  Such a monad would assign to each set $X$ the [[free object|free]] ring on that set, which consists of formal sums of formal products of elements of $X$---in other words, it can be identified with $T(S(X))$.  Thus the question becomes "given two monads $\mathbf{T}$ and $\mathbf{S}$, what further structure is required to make the composite $T S$ into a monad?"

It is easy to give $T S$ a unit, as the composite $Id \xrightarrow{\eta^S} S \xrightarrow{\eta^T S} T S$, but to give it a multiplication we need a transformation from $T S T S$ to $T S$.  We naturally want to use the multiplications $\mu^T\colon T T \to T$ and $\mu^S\colon S S \to S$, but in order to do this we first need to switch the order of $T$ and $S$.  However, if we have a transformation $\lambda\colon S T \to T S$, then we can define $\mu^{T S}$ to be the composite $T S T S \xrightarrow{\lambda} T T S S \xrightarrow{\mu^T\mu^S} T S$.

Such a transformation, satisfying suitable axioms to make $T S$ into a monad, is called a *distributive law*, because of the motivating example relating addition to multiplication in a ring.  In that case, $S T X$ is a formal product of formal sums such as $(x_1 + x_2 + x_3)\cdot (x_4 + x_5)$, and the distributive law $\lambda$ is given by multiplying out such an expression formally, resulting in a formal sum of formal products such as $x_1\cdot x_4 + x_1 \cdot x_5 + x_2 \cdot x_4 + x_2 \cdot x_5 + x_3\cdot x_4 + x_3 \cdot x_5$.

\begin{remark}
\label{TerminologyWhatDistributesOverWhat}
**(terminology -- what distributes over what)**
\linebreak
The eponymous example of distributivity in [[arithmetic]]
$$
  a \times \sum_i b_i \;=\; \sum_i a \times b_i
$$
hence
$$
  \big( a \times (-) \big)
  \circ
  \big(
    \sum (-)
  \big)
  \;\;
  =
  \;\;
  \big(
    \sum (-)
  \big)
  \circ
  \big( a \times (-) \big)
$$
(where, of course, the [[equality]] as such works in both directions, but the *distribution* of factors over summands is the step from left to right)
suggests that a suitable transformation of (co)monads of the form
$$
  T_1 \circ T_2 \longrightarrow T_2 \circ T_1
$$
should be referred to as *$T_1$ distributing over $T_2$* instead of the other way around.

However, already the original reference [Beck 1969 §1](#Beck69) uses the opposite terminology.

Authors sticking to this original but arguably reverse terminological convention include [Brookes & Van Stone 1993](#BrookesVanStone93), while other authors tacitly switch to the other terminological convention (eg. [Barr & Wells 1985 §9 2.1](#BarrWells85), [Power & Watanabe 2002 p. 138](#PowerWatanabe02)).
\end{remark}

## Big picture
 {#BigPicture}

[[monad|Monads]] in any [[2-category]] $C$ make themselves a 2-category $\mathrm{Mnd}$ in which [[1-morphisms]] are either lax or colax [[homomorphisms]] of monads (cf. *[[monad transformations]]*). By [[formal duality]] the analogue is true for [[comonads]].

Distributivity laws may be understood as
monads [[internalization|internal]] to this 2-category of monads.

In particular, distributive laws themselves make a 2-category.

There are other variants like distributive laws between a monad and an [[endofunctor]], "mixed" distributive laws between a monad and a comonad (the variants for algebras and coalgebras called [[entwining structures]]), distributive laws between actions of two different monoidal categories on the same category, for [[PROP]]s and so on.

Having a distributive law $l$ from one monad to another enables to define the composite monad $\mathbf T\circ_l\mathbf P$. This correspondence extends to a 2-functor $\mathrm{comp} \,\colon\, \mathrm{Mnd}(\mathrm{Mnd}(C))\to\mathrm{Mnd}(C)$. An analogue of this 2-functor in the mixed setup is a [[2-functor]] from the bicategory of entwinings to a bicategory of [[corings]].

## Explicit definition

### Monad distributing over monad

\begin{definition}
A __distributive law__ for a monad  $\mathbf{T} = (T, \mu^T, \eta^T)$ in $A$ over an endofunctor
$P$ is a [[2-morphism]] $l : T P \Rightarrow P T$ such that
$l \circ (\eta^T)_P = P(\eta^T)$ and
$l \circ (\mu^T)_P = P(\mu^T) \circ l_T \circ T(l)$. In diagrams:

\begin{tikzcd}
& P \ar{dl}[swap]{\eta^T P} \ar{dr}{P \eta^T} \\
T P \ar{rr}{l} && P T
\end{tikzcd}
\begin{tikzcd}
T T P \ar{d}{\mu^T P} \ar{r}{T l} & T P T \ar{r}{l T} & P T T \ar{d}{P \mu^T} \\
T P \ar{rr}{l} && P T
\end{tikzcd}

\end{definition}

Distributive laws for the monad $\mathbf{T}$ over the endofunctor $P$ are in a canonical bijection with lifts of $P$ to an [[endofunctor]] $P^{\mathbf T}$ in the [[Eilenberg-Moore category]] $A^{\mathbf T}$,
satisfying $U^{\mathbf T} P^{\mathbf T} = P U^{\mathbf T}$. Indeed, the endofunctor $P^{\mathbf T}$
is given by $(M,\nu) \mapsto (P M,P(\nu)\circ l_M)$.

\begin{definition}
A __distributive law__ for a monad  $\mathbf{T} = (T, \mu^T, \eta^T)$ over a monad
$\mathbf{P} = (P, \mu^P, \eta^P)$ in $A$
is a distributive law for $\mathbf T$ over the endofunctor $P$,
compatible with $\mu^P,\eta^P$ in the sense that
$l \circ T(\eta^P) = (\eta^P)_T$ and
$l \circ T(\mu^P) = (\mu^P)_T \circ P(l) \circ l_P$. In diagrams:

\begin{tikzcd}
& T \ar{dl}[swap]{T \eta^P} \ar{dr}{\eta^P T} \\
T P \ar{rr}{l} && P T
\end{tikzcd}

\begin{tikzcd}
T P P \ar{d}{T \mu^P} \ar{r}{l P} & P T P \ar{r}{P l} & P P T \ar{d}{\mu^P T} \\
T P \ar{rr}{l} && P T
\end{tikzcd}

\end{definition}
(due to [Beck 1969](#Beck69), review includes [Barr & Wells 1985 §9 2.1](#BarrWells85))

The correspondence between distributive laws and _endofunctor_ liftings extends to a correspondence between distributive laws and _monad_ liftings. That is, distributive laws $l \colon T P \Rightarrow P T$ from the monad $\mathbf{T}$ to the monad $\mathbf{P}$ are in a canonical bijection with lifts of the monad $\mathbf{P}$ to a monad $\mathbf{P}^{\mathbf T}$ in the [[Eilenberg-Moore category]] $A^{\mathbf T}$,
such that $U^{\mathbf T} \colon A^{\mathbf T}\to A$ preserves the monad structure.

Thus all together a distributive law for a monad over a monad is a 2-cell for which 2 triangles and 2 pentagons commute. In the entwining case,
Brzeziński and Majid combined the 4 diagrams into one picture which they call the _bow-tie diagram_.

\begin{remark}\label{DistributivityAsMonadsInMonads}
**(distributivity as monads in monads)**
\linebreak
As mentioned earlier, one can understand a distributivity law of a monad $(a,s)$ over another monad $(a,t)$ as displaying $s \colon a \to a$ as a monad on $(a,t)$ [[internalization|in]] the [[2-category]] $\mathbf{Mnd}(A)$ of monads in a 2-category $A$.

Specifically, a monad in $\mathbf{Mnd}(A)$ over $(a,t)$ (which is a monad in $A$!) is comprised of the following data:

1. A [[1-morphism]] $s \colon a \to a$ in $A$, together with an intertwiner $\lambda \colon t s \to s t$ satisfying the equations [[monad#BicategoryOfMonads|here]]

2. Two [[2-morphisms]] (in $\mathbf{Mnd}(A)$) $\sigma \colon 1 \Rightarrow (s,\lambda)$ and $\nu \colon (s,\lambda)(s,\lambda) \Rightarrow (s,\lambda)$ which correspond to two 2-morpisms in $A$ $\sigma \colon 1 \Rightarrow s$, $\nu \colon s s \Rightarrow s$ commuting with the intertwiners of $1$, $s$ and $ss$.

Then $\lambda$ is the distributive law sought, and the laws $\lambda$, $\sigma$ and $\nu$ satisfy correspond to those of a distributive law.
\end{remark}

### Monad distributing over a comonad

[van Osdol 1973 p. 456](#vanOsdol73)

(...)

### Comonad distributing over monad
 {#ComonadDistributingOverMonad}

The distributivity law of

* a comonad $\mathcal{C}$ over

* a monad $\mathcal{E}$

on the same [[category]] $\mathbf{C}$

is as follows ([Brookes & Van Stone 1993 Def. 3](#BrookesVanStone93), [Power & Watanabe 2002](#PowerWatanabe02)):

A [[natural transformation]]

$$
  distr^{\mathcal{C}, \mathcal{E}}_{(\text{-})}
  \;\;\colon\;\;
  \mathcal{C}
  \big(
    \mathcal{E}(-)
  \big)
  \longrightarrow
  \mathcal{E}
  \big(
    \mathcal{C}(-)
  \big)
$$

such that the following [[commuting diagram|diagrams commute]] for all objects $D$:

\begin{imagefromfile}
    "file_name": "MixedCoMonadDistributivity-230929.jpg",
    "width": 540,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

{#TwoSidedKleisliComposition} Given this distributivity structure, there is a two-sided ("double") [[Kleisli category]] ([Brookes & Van Stone 1993 Thm. 2](#BrookesVanStone93), [Power & Watanabe 2002, Prop. 7.4](#PowerWatanabe02)) whose [[objects]] are those of $\mathbf{C}$, and whose morphisms $D_1 \to D_2$ are morphisms in $\mathbf{C}$ of the form
$$
  prog_{12}
  \;\colon\;
  \mathcal{C}(D_1)
    \longrightarrow
  \mathcal{E}(D_2)
$$
with two-sided Kelisli composition

$$
  prog_{12} \text{>=>} prog_{23}
  \;\;
  \colon
  \;\;
  \mathcal{C}(D_1)
  \longrightarrow
  \mathcal{E}(D_3)
$$

given by the (co-)[[Kleisli extension|bind-operation]] on the factors connected by the distributivity transformation:

\begin{imagefromfile}
    "file_name": "CoMonKleisliComposition-230930b.pdf",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

##Examples##

### Products distributing over coproducts

In a _[[distributive category]]_ [[products]] distribute over [[coproducts]].

### In Cat

* There is a distributive law of the monad (on [[Set]]) for [[monoids]] over the monad for [[abelian groups]], whose composite is the monad for [[rings]].  This is the canonical example which gives the name to the whole concept.

#### Tensor products distributing over direct sums

For many standard choices of [[tensor products]] in the presence of [[direct sums]] the former distribute over the latter. See at _[[tensor product of abelian groups]]_ and _[[tensor product of modules]]_.

### In other 2-categories

* [[strict factorization systems]] can be identified with distributive laws between categories regarded as monads in [[span|Span(Set)]].

* More generally, [[factorization systems over a subcategory]] can be identified with distributive laws in [[Prof]].  Ordinary [[orthogonal factorization systems]] are a special case.  The latter can also be obtained by other weakenings; see for instance [this discussion](http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html).

## Related concepts

* [[weak distributive law]]

* [[iterated distributive law]]

* [[wreath]]

* [[distributivity for monoidal structures]]

* [[monad transformer]]

* [[pseudo-distributive law]]

## Literature
 {#Literature}

* {#Beck69} [[Jon Beck]], *Distributive Laws*, in: *[[Seminar on Triples and Categorical Homology Theory]]*, ETH 1966/67, Lecture Notes in Mathemativs, Springer (1969), Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

* {#vanOsdol73} [[Donovan van Osdol]], *Bicohomology Theory*, Transactions of the American Mathematical Society **183** (1973) 449-476 [[jstor:1996479](https://www.jstor.org/stable/1996479)]

* {#Street72} [[Ross Street]], §6 of: *The formal theory of monads*, Journal of Pure and Applied Algebra **2** 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

* {#BarrWells85} [[Michael Barr]], [[Charles Wells]], *[[Toposes, Triples, and Theories]]*, Springer (1985) republished in: [[TAC reprints series|Reprints in Theory and Applications of Categories]], **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html), [tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

* {#BrookesVanStone93} [[Stephen Brookes]],  [[Kathryn Van Stone]], *Monads and Comonads in Intensional Semantics* (1993) [[dtic:ADA266522](https://apps.dtic.mil/sti/citations/ADA266522), [pdf](https://www.cs.cmu.edu/~brookes/papers/MonadsComonads.pdf), [[BrookesVanStone-CoMonads.pdf:file]]]

* [[Martin Markl]], _Distributive laws and Koszulness_,  Ann. Inst. Fourier (Grenoble)  46  (1996),  no. 2, 307--323 ([numdam](http://www.numdam.org/numdam-bin/fitem?id=AIF_1996__46_2_307_0))

* T. F. Fox, [[Martin Markl]], _Distributive laws, bialgebras, and cohomology_,  Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995),   Contemp. Math. **202** AMS (1997) 167-205

* [[T. Brzeziński]], [[S. Majid]], _Coalgebra bundles_, Comm. Math. Phys.  **191** 2  (1998) 467--492 [[arXiv:q-alg/9602022](http://arxiv.org/abs/q-alg/9602022)]

* {#PowerWatanabe99} [[John Power]], [[Hiroshi Watanabe]], *Distributivity for a monad and a comonad*, Electronic Notes in Theoretical Computer Science **19** (1999) 102 [doi:10.1016/S1571-0661(05)80271-3, [[PowerWatanabe-Distributivity.pdf:file]]]

* [[Steve Lack]], _Composing PROPS_, [Theory Appl. Categ.](http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html) 13 (2004), No. 9, 147--163.

* [[Steve Lack]], [[Ross Street]], _The formal theory of monads II_, Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra **175** 1-3 (2002) 243-265

* {#PowerWatanabe02} [[John Power]], [[Hiroshi Watanabe]], *Combining a monad and a comonad*, Theoretical Computer Science **280** 1–2 (2002) 137-162 [doi:10.1016/S0304-3975(01)00024-X]

* [[T. Brzeziński]], [[Robert Wisbauer]], _Corings and comodules_, London Math. Soc. Lec. Note Series 309, Cambridge (2003)

* [[Gabi Böhm]], _Internal bialgebroids, entwining structures and corings_, AMS Contemp. Math. **376** (2005) 207-226 [[arXiv:math.QA/0311244](http://front.math.ucdavis.edu/math.QA/0311244)]

* [[Zoran Škoda]], _Distributive laws for monoidal categories_ ([arXiv:0406310](http://front.math.ucdavis.edu/math.CT/0406310)); _Equivariant monads and equivariant lifts versus a 2-category of distributive laws_ ([arXiv:0707.1609](http://front.math.ucdavis.edu/0707.1609)); _Bicategory of entwinings_  ([arXiv:0805.4611](http://arxiv.org/abs/0805.4611))

* R. Wisbauer, _Algebras versus coalgebras_,  Appl. Categ. Structures __16__  1-2 (2008) 255--295 [[doi:10.1007/s10485-007-9076-5](https://doi.org/10.1007/s10485-007-9076-5)]

* [[Zoran Škoda]], _Some equivariant constructions in noncommutative geometry_, Georgian Math. J. 16 (2009) 1; 183--202 ([arXiv:0811.4770](http://front.math.ucdavis.edu/0811.4770))

* Bachuki Mesablishvili, [[Robert Wisbauer]], *Bimonads and Hopf monads on categories*, Journal of K-Theory **7** 2 (2011) 349-388  [[arXiv:0710.1163](https://arxiv.org/abs/0710.1163), [doi:10.1017/is010001014jkt105](https://doi.org/10.1017/is010001014jkt105)]

* [[Francisco Marmolejo]], Adrian Vazquez-Marquez, *No-iteration mixed distributive laws*, Mathematical Structures in Computer Science **27** 1 (2017) 1-16 [[doi:10.1017/S0960129514000656](https://doi.org/10.1017/S0960129514000656)]

* Liang Ze Wong, _Distributive laws_, post at $n$-[cafe](https://golem.ph.utexas.edu/category/2017/02/distributive_laws.html), Feb 2017

* Enrique Ruiz Hernández, *Another characterization of no-iteration distributive laws*, [arxiv](https://arxiv.org/abs/1910.06531)

* Werner Struckmann and Dietmar Wätjen, _A note on the number of distributive laws_, Algebra universalis 21 (1985): 305-306.

On distributive laws for [[relative monads]]:

* [[Gabriele Lobbia]], *Distributive Laws for Relative Monads*, Applied Categorical Structures **31** 19 (2023)  [[doi:10.1007/s10485-023-09716-1](https://doi.org/10.1007/s10485-023-09716-1), [arXiv:2007.12982](https://arxiv.org/abs/2007.12982)]

[[!redirects distributive laws]]
[[!redirects mixed distributive law]]
[[!redirects mixed distributive laws]]

[[!redirects distributivity law]]

[[!redirects distributivity]]


distributive monoidal category

> For the cartesian case see at *[[distributive category]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# Distributive monoidal categories

* table of contents
{: toc}

## Idea

A _distributive monoidal_ category is a [[monoidal category]] whose [[tensor product]] [[distributive law|distributes]] over [[coproducts]].

## Definition

A **distributive monoidal category** (this is not entirely standard terminology) is a [[monoidal category]] with [[coproducts]] whose [[tensor product]] [[preserved colimit|preserves]] [[coproducts]] in each variable: i.e. such that the canonical morphisms
$$\coprod_i (X\otimes Y_i)\to X \otimes \coprod_i Y_i$$
$$\coprod_i (X_i\otimes Y)\to \coprod_i X_i \otimes Y$$
are [[isomorphisms]].

Depending on the arity of the coproducts in question, we may speak of a **finitary** or **infinitary** distributive monoidal category.

The special case of distributive [[cartesian monoidal categories]] is known simply as  *[[distributive categories]]*. Conversely, distributive monoidal categories are a special case of [[rig categories]].

See also at *[[distributivity for monoidal structures]]*.

A more abstract way to say this, due to Weber and Batanin, is that if $M$ is the free monoidal category [[monad]] and $M V \xrightarrow{\otimes} V$ is the structure map of a monoidal category $V$, then $V$ is distributive if it admits [[left Kan extensions]] along functors $f:A\to B$ between [[discrete categories]] (of some size), and moreover if
$$\array{
  A && \xrightarrow{f} && B\\
  & \searrow & \xRightarrow{\phi} & \swarrow\\
  && V
}$$
is such a Kan extension, then so is
$$\array{
  M A && \xrightarrow{M f} && M B\\
  & \searrow & \xRightarrow{M \phi} & \swarrow\\
  && M V\\
  && \downarrow^\otimes\\
  && V.
}$$

## Examples
 {#Examples}

### Various

Beyond [[distributive categories]], examples of distributive monoidal categories include the following:

* [[Ab]], the category of [[abelian groups]] equipped with the [[tensor product of abelian groups]]

* $R$[[Mod]], the category of [[modules]] over a [[commutative ring]] $R$, equipped  with the [[tensor product of modules]]

* $k$[[Vect]] = $k$[[Mod]], the category of [[vector space]] over some [[field]] $k$, equipped with the [[tensor product of vector spaces]],

* $k$[[Vect(X)]], the category of ([[topological vector bundles|topological]]) [[vector bundles]] over some ([[topological space|topological]]) [[space]], equipped with the [[tensor product of vector bundles]],

In all these cases the coproduct is the respective [[direct sum]] (e.g. [[direct sum of vector bundles]] in the case of vector bundles).

Also:

* the category of [[pointed sets]] with respect to forming [[smash product]] and [[wedge sum]],

* more generally, the category of [[pointed topological spaces]] with respect to forming [[smash product]] and [[wedge sum]] (e.g. [Hatcher, Section 4.F](algebraic+topology#Hatcher)).

* $k$[[VectBund]], the category of ([[topological vector bundles|topological]]) [[vector bundles]] over any ([[convenient category of topological spaces|convenient]]) [[topological spaces]], equipped with the [[external tensor product of vector bundles]] (more on this [below](#ExamplesBundlesWithExternalTensorProduct)).

### Vector bundles with external tensor product
 {#ExamplesBundlesWithExternalTensorProduct}

We spell out in much detail the example of the [[category]] [[VectBund|$Vect_{Set}$]] of [[vector bundles]] over arbitrary base spaces and equipped with the [[external tensor product]] [[external tensor product of vector bundles|of vector bundles]] --- for the simple special case that the base spaces are [[discrete topological spaces]], i.e. plain [[sets]]:

\begin{definition}
  For any [[ground field]], write [[VectBund|$Vect_{Set}$]] for the [[category]] of [[indexed sets]] of [[vector spaces]].
\end{definition}
\begin{remark}
We may and will present [[objects]] $\mathcal{V}$ of [[VectBund|$Vect_{Set}$]] as [[pairs]] consisting of a [[set]] $S$ and a [[function]] $\mathcal{V}_{(-)}$ (really a [[functor]] on the [[discrete category]] on $S$) to [[Vect]]:
$$
  \left(
    \array{
      S &\longrightarrow& Vect
      \\
      s &\mapsto& \mathcal{V}_s
    }
  \right)
  \;\;
  \in
  \;\;
  Vect_{Set}
  \mathrlap{\,.}
$$
\end{remark}

\begin{definition}
\label{ExternalTensorProductInVectSet}
The "[[external tensor product|external]]" [[tensor product]] on [[VectBund|$Vect_{Set}$]] is the [[functor]]
$$
  \boxtimes
  \,\colon\,
  Vect_{Set} \times VectSet
    \longrightarrow
  Vect_{Set}
$$
given by
$$
  \big(
    \mathcal{V}_{(-)} \,\colon\, S \to \Vect
  \big)
  \,\boxtimes\,
  \big(
    \mathcal{V}'_{(-)} \,\colon\, S' \to \Vect
  \big)
  \;\;
  \coloneqq
  \;\;
  \left(
    \array{
      S \times S' &\longrightarrow& \Vect
      \\
      (s, s')
        &\mapsto&
      \mathcal{V}_s \otimes \mathcal{V}'_{s'}
    }
  \right)
  \mathrlap{\,.}
$$
\end{definition}

\begin{proposition}
\label{CoproductInVectSet}
  The [[coproduct]] in [[VectBund|$Vect_{Set}$]] is given by [[disjoint union]] of [[bundles]]:
$$
  \big(
    \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to \Vect
  \big)
  \;
  \sqcup
  \;
  \big(
    \mathcal{V}^{(1)}_{(-)} \,\colon\, S_2 \to \Vect
  \big)
  \;\;
  \simeq
  \;\;
  \left(
    \array{
      S_1 \sqcup S_2 &\longrightarrow& \Vect
      \\
      s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
    }
  \right)
$$
\end{proposition}
\begin{proof}
  It is immediate to check the [[universal property]] characterizing the coproduct.
\end{proof}

\begin{proposition}
The [[external tensor product]] $\boxtimes$
(Def. \ref{ExternalTensorProductInVectSet})
[[distributive monoidal category|distributes]] over the coproduct (Prop. \ref{CoproductInVectSet}):
$$
  \mathcal{E} \boxtimes ( \mathcal{V}^{(1)} \sqcup  \mathcal{V}^{(2)})
  \;\simeq\;
  \big(
    \mathcal{E} \boxtimes \mathcal{V}^{(1)}
  \big)
  \,\sqcup\,
  \big(
    \mathcal{E} \boxtimes \mathcal{V}^{(2)}
  \big)
$$
and hence gives a [[distributive monoidal category]]:
$$
  \big(
    Vect_{Set}
    ,
    \sqcup
    ,
    \boxtimes
  \big)
  \;\in\;
  DistMonCat
  \mathrlap{\,.}
$$
\end{proposition}
\begin{proof}
Unwinding the above definitions and using that [[Set]] is a [[distributive category]], we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{l}
    \mathcal{E}
    \,\boxtimes\,
    \big(
      \mathcal{V}^{(1)}
      \,\sqcup\,
      \mathcal{V}^{(2)}
    \big)
    \\
    \;\equiv\;
    \big(
      \mathcal{E}_{(-)} \,\colon\, S_E \to Vect
    \big)
    \,\boxtimes\,
    \Big(
      \big(
        \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to Vect
      \big)
      \,\sqcup\,
      \big(
        \mathcal{V}^{(2)}_{(-)} \,\colon\, S_2 \to Vect
      \big)
    \Big)
    \\
    \;\simeq\;
    \big(
      \mathcal{E}_{(-)} \,\colon\, S_E \to Vect
    \big)
    \,\boxtimes\,
    \left(
      \array{
        S_1 \sqcup S_2 &\longrightarrow& Vect
        \\
        s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        S_E \times (S_1 \sqcup S_2)
        &\longrightarrow& Vect
        \\
        (s_E , s_i) &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        (S_E \times S_1) \,\sqcup\, (S_2 \times S_2)
        &\longrightarrow& Vect
        \\
        (s_E , s_i) &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        S_E \times S_1
        &\longrightarrow& Vect
        \\
        (s_E , s_1)
         &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(1)}_{s_1}
      }
    \right)
    \,\sqcup\,
    \left(
      \array{
        S_E \times S_2
        &\longrightarrow& Vect
        \\
        (s_E , s_2)
         &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(2)}_{s_2}
      }
    \right)
    \\
    \;\equiv\;
    \mathcal{E} \boxtimes \mathcal{V}^{(1)}
    \,\sqcup\,
    \mathcal{E} \boxtimes \mathcal{V}^{(2)}
  \end{array}
$$
\end{proof}

## Properties

### The weakly semicartesian case

+-- {: .num_remark}
###### Remark
A monoidal category is finitary distributive if its tensor product preserves binary coproducts in each variable and the monoidal unit $I$ is [[weak limit|weakly]] [[terminal object|terminal]] (e.g., if there is a morphism $1 \to I$ out of a terminal object).
=--

+-- {: .proof}
###### Proof
There is a canonical isomorphism
$$x \otimes 0 + x \otimes y \to x \otimes (0 + y)$$
and thus a canonical isomorphism
$$\phi: x \otimes 0 + x \otimes y \to x \otimes y$$
whose restriction along the coproduct inclusion $x \otimes y \to x \otimes 0 + x \otimes y$ is the identity $1_{x \otimes y}$. Let $k: x \otimes 0 \to x \otimes y$ be the restriction of $\phi$ along the other coproduct inclusion. Then $\phi$ induces an evident bijection
$$\hom(x \otimes y, y) \stackrel{\langle [k], id \rangle}{\to} \hom(x \otimes 0, y) \times \hom(x \otimes y, y).$$
Since $\hom(x \otimes y, y)$ is inhabited for all $x, y$ (with the help of some map $x \to I$, there is some map $x \otimes y \to I \otimes y \cong y$), this forces $\hom(x \otimes 0, y)$ to be a singleton for any $y$, so that $x \otimes 0$ is initial.
=--

### Free monoids

A distinguishing feature of (infinitary) distributive monoidal categories is that the [[monad]] $T$ for [[monoid objects]] in such a category has a particularly simple expression:
$$ T X = \coprod_n X^{\otimes n}.$$
The same is true for the monad on enriched graphs whose algebras are categories enriched over such a monoidal category.  This also generalizes to [[lax monoidal categories]], a.k.a. "multitensors"; see [Weber 13](#Weber13).

## Related concepts

* [[distributive law]]

* [[distributivity for monoidal structures]]

  * [[distributive monoidal category]]

  * [[bipermutative category]]

  * [[linearly distributive category]]

  * [[rig category]]

## References

* [[Hans-Joachim Baues]], [[Mamuka Jibladze]], [[Andy Tonks]], first page of: *Cohomology of monoids in monoidal categories*, in: *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202**, AMS (1997) 137-166 [[doi:10.1090/conm/202](https://doi.org/10.1090/conm/202), preprint:[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/1484/1/preprint_1995_121.pdf)]

* [[Anna Labella]], *Categories with sums and right distributive tensor product*, Journal of Pure and Applied Algebra **178** 3 (2003) 273-296 [doi:10.1016/S0022-4049(02)00169-X]

* {#Weber13} [[Mark Weber]], *Multitensors and monads on categories of enriched graphs*, [[TAC]] **28** 26 (2013) [[tac:28-26](http://www.tac.mta.ca/tac/volumes/28/26/28-26abs.html)]

[[!redirects distributive monoidal categories]]


distributivity for monoidal structures

+-- {: .rightHandSide}
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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Distributivity for monoidal structures

There are many variations on what it means for one [[monoidal category|monoidal structure]] on a [[category]] to *distribute* over another.  Here we collect a list of them and remark on their relationships.  Note that our terminology is by no means universal.

The following notions of distributivity exist in a linear hierarchy of less to more general.

* A [[distributive category]] has [[finite product]]s and [[coproducts]] (hence is both [[cartesian monoidal category|cartesian]] and [[cocartesian monoidal category|cocartesian]] monoidal), and the former [[distributive law|distribute]] over the latter, in that the canonical [[morphism]] $(X\times Y) + (X\times Z) \to X\times (Y+Z)$ is an [[isomorphism]].

* A [[distributive monoidal category]] is a [[monoidal category]] with coproducts whose tensor product preserves coproducts in each variable separately.  If it is [[cartesian monoidal category|cartesian monoidal]], then it is exactly a distributive category as above.

* A [[rig category]] (also called a [[bimonoidal category]]) is a category with two monoidal structures, say $\otimes$ and $\oplus$, together with coherent natural distributivity isomorphisms such as $X\otimes (Y\oplus Z) \cong (X\otimes Y) \oplus (X\otimes Z)$.  Generally one requires $\oplus$ to be [[symmetric monoidal category|symmetric]].  If $\oplus$ is a cocartesian structure, then it is exactly a distributive monoidal category as above.

  * A [[bipermutative category]] is a "semistrict" symmetric rig category: both $\otimes$ and $\oplus$ are [[permutative categories]] (symmetric strict monoidal categories) and one of the distributivity isomorphisms is an identity.

* A [[colax-distributive rig category]] is like a rig category, but the distributivity morphisms are not assumed to be invertible.

There are also the following related notions which are not comparable in generality.

* [[2-fold monoidal categories]] and [[duoidal categories]] have two monoidal structures, but rather than a "distributivity" morphism as above, they have transformations $(X\star Y)\odot (Z\star W) \to (X\odot Z) \star (Y\odot W)$.

* A [[linearly distributive category]] also has two monoidal structures, but its comparison morphisms have the form $X\otimes (Y\bullet Z) \to (X\otimes Y)\bullet Z$.

* A [[2-rig]] can mean a lot of different things, perhaps a [[distributive monoidal category]] but perhaps also including [[additive and abelian categories|additivity]] or [[cocomplete category|cocompleteness]].

## Related entries

* [[distributive law]]

* [[distributivity for monoidal structures]]

  * [[distributive monoidal category]]

  * [[bipermutative category]]

  * [[linearly distributive category]]

  * [[rig category]]


distributivity of limits over colimits

+-- {: .rightHandSide}
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###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In elementary algebra (or more generally in a [[ring]]), [[multiplication]] distributes over [[addition]], meaning that

$$
  a \cdot (b + c)
  \;=\;
  (a\cdot b) + (a\cdot c)
$$
and dually.  Since multiplication and addition are a [[decategorifications]] of [[products]] and [[coproducts]] in [[category theory]], it is suggestive to ask for products to distribute over coproducts in any category.

More generally, one may consider arbitrary [[limits]] distributing over [[colimits]].  Note that this is *not* the same notion as limits *[[commutativity of limits and colimits|commuting]]* with colimits, although in some cases they are related (see at *[[permuting limits and colimits]]* for more).

## Definitions

### Explicitly using limits and colimits

Given [[small categories]] $I$ and $K$, for any [[diagram]]
$$
  D \,\colon\, I \times K \longrightarrow C
$$
in a [[category]] $C$ that admits all necessary [[limits]] and [[colimits]] we have a [[morphism]]
$$
  f
    \,\colon\,
  colim_K lim_I D
   \longrightarrow
  lim_I colim_K D
$$
induced using the [[universal property]] of $colim_K$ and $lim_I$ by the family of composites
$$
  lim_I D(-,k)
   \longrightarrow
  D(i,k)
   \longrightarrow
  colim_K D(i,-)
$$
of the limit [[projection]] for $i\in I$ and the colimit [[coprojection|injection]] for $k\in K$.

Recall that $I$-[[limits]] are said to [[commutativity of limits and colimits|commute]] with $K$-[[colimits]] if and only if $f$ is an [[isomorphism]] for all $D$.

Now we note that $f$ factors as a [[composition]]
$$colim_K lim_I D \xrightarrow{g} colim_{K^I} lim_I D' \xrightarrow{h} lim_I colim_K D.$$
Here $K^I$ is the [[functor category]] and $D'\colon I\times K^I\to C$ is obtained from $D$ by [[precomposition]] with the functor $(\pi_1,ev) : I\times K^I \to I\times K$ that sends a pair $(i,s)$ to $(i,s(i))$.  The map $g$ is induced by the diagonal functor $K\to K^I$ sending each $k$ to the corresponding [[constant functor]], while $h$ is defined by the universal properties of $colim_{K^I}$ and $lim_I$ from the family of composites
$$ lim_I D'(-,s) \to D'(i,s) = D(i,s(i)) \to colim_K D(i,-) $$
of the limit projection for $i\in I$ and the colimit injection for $c(i)\in K$.  We say that **$I$-limits distribute over $K$-colimits** if $h$ is an isomorphism for all $D$.

More generally, we can allow $K$ to vary with $i$, becoming a functor $K\colon I\to Cat$.  Now instead of $K^I$ we use the limit category $lim_I K$, and the diagram $D$ is defined on the [[Grothendieck construction]] ([[lax colimit]]) $\int^I K$.  We recover the above definition when $K$ is a [[constant functor]].

\begin{example}
\label{DistributivityOfProductsOverSums}
In the simple example $A \times (B + C) \cong (A \times B) + (A \times C)$ alluded to in the introduction,

* $I$ is a two-element [[discrete category]],

* $K(0)$ is the [[terminal category]]

* $K(1)$ is again a two-element discrete category.

The [[Grothendieck construction]] $\int^I K$ is a three-element discrete category, and the diagram $D \colon \int^I K \to \mathcal{C}$ picks the objects $A$, $B$ and $C$.
\end{example}

If $\mathcal{K}$ is a class of small categories rather than a single one, we say that **$I$-limits distribute over $\mathcal{K}$-colimits** in a category $C$ if the corresponding map is an isomorphism for any $K \,\colon\, I\to Cat$ taking values in $\mathcal{K}$.
For instance, if $\mathcal{K}$ is the class of all [[discrete categories]], we obtain the notion of $I$-limits distributing over coproducts, and so on.

If instead $I$-limits distribute over $K$-colimits for any single $K\in \mathcal{K}$, we may say instead that "$I$-limits distribute over uniform $K$-limits."

+--{: .standout}
The author of the previous paragraph doesn't know whether distributivity over uniform colimits is actually any weaker, but if it is, then the non-uniform version seems more correct.
=--

### For Kan extensions

Yet more generally, let $u : I\to J$ be any [[Grothendieck fibration]] and $v : K\to I$ a Grothendieck opfibration.  Since fibrations are [[exponentiable functors]], we can form the [[distributivity pullback]]
$$ \array{ X & \xrightarrow{p} & K & \xrightarrow{v} & I \\ ^q\downarrow &&&& \downarrow^u \\ Y && \xrightarrow{r} && J } $$
Thus $Y$ is the [[dependent product]] $\Pi_I K$, and $X = I\times_J \Pi_I K$, while the functor $p:X\to K$ is "evaluation".  Now in any sufficiently complete category, there is a [[Beck-Chevalley]] isomorphism
$$ r^* \circ Ran_u \cong Ran_q \circ p^* \circ v^*. $$
This is because any [[pullback]] of a fibration is an [[exact square]].  This isomorphism has a [[mate]]
$$ Lan_r \circ Ran_q \circ p^* \to Ran_u \circ Lan_v $$
and we say that **right Kan extensions along $u$ distribute over left Kan extensions along $v$** if this mate is an isomorphism.

If $J=1$ then this reduces to the above non-uniform notion, where $K\to I$ is the [[Grothendieck construction]] of a functor $I\to Cat$.  Moreover, since isomorphisms of $J$-diagrams are pointwise, right and left Kan extensions along fibrations and opfibrations respectively are given by fiberwise limits and colimits respectively, and dependent exponentials are preserved by pullback (the Beck-Chevalley condition), we can say that right Kan extensions along $u$ distribute over left Kan extensions along $v$ if and only if the above non-uniform distributivity condition holds for the fibers of $v$ over each object of $J$.

The formulation in terms of Kan extensions generalizes naturally to [[derivators]], and can also be [[internalization|internalized]] to [[internal categories]] and [[fibrations]] in any [[locally cartesian closed category]].

### For sound doctrines

(See Section 6 in [ABLR](#ABLR).)

Given a [[sound doctrine]] $D$ of limits,
we have the associated notions of $D$-limits,
$D$-filtered colimits,
and
$D$-filtered cocompletion $D-Ind(C)$ of a category $C$
together with the associated colimit functor $colim\colon D-Ind(C)\to C$.

We say that some class of limits _distributes over $D$-filtered colimits_ if the functor colim preserves these limits.

See Appendix A in [AR](#AR) for a comparison of this definition to the above explicit definition in the special case of distributivity of [[filtered colimits]]
over [[small limits]] in [[locally finitely presentable categories]]
and distributivity of [[sifted colimits]] over [[small limits]]
in [[varieties of algebras]].

## Relation to commutativity of limits and colimits

Observe that distributivity is asymmetric with respect to limits and colimits, whereas [[commutativity of limits and colimits]] is a symmetric notion.  In some cases, however, distributivity and commutativity are equivalent.  This happens exactly when the above functor $g$ is an isomorphism, which is to say that the [[diagonal]] map $K\to K^I$ is a [[final functor]].

Thus, for instance, finite limits distribute over (uniform) filtered colimits if and only if finite limits commute with filtered colimits.  The same is true for finite products and sifted colimits.

## Examples

### Finite products

The distributivity of [[finite products]] over arbitrary [[coproducts]] is the most classical version.  See [[distributivity for monoidal structures]] for various generalizations.

More generally, finite products distribute over arbitrary colimits in any [[cartesian closed category]], such as [[Set]].

### Filtered colimits

If $D$ is the doctrine of finite limits,
then $D$-filtered colimits are precisely filtered colimits
and the $D$-filtered cocompletion of $C$ is $Ind(C)$,
the category of [[Ind-objects]] in $C$.

According to Definition 5.11 in [ALR](#ALR),
a category is _precontinuous_
if it admits small limits and filtered colimits,
and the functor $colim: Ind(C)\to C$ is continuous.

According to Theorem 2.1 in [ARVloc](#ARVloc),
a category is precontinuous if and only if it has small [[limits]]
and [[filtered colimits]], [[filtered colimits]] commute with finite [[limits]],
and small [[products]] distribute over [[filtered colimits]].

In particular, any [[locally finitely presentable category]], equivalently the category of algebras
over some finitary [[essentially algebraic theory]],
is precontinuous.
See Theorem 5.13 and Lemma 5.14 in [ALR](#ALR).

In fact, as shown in [ALR](#ALR), precontinuous categories with continuous functors that preserve filtered colimits
form the accessible [[equational hull]] of the bicategory
of [[locally finitely presentable categories]] and continuous functors
that preserve filtered colimits.

### Sifted colimits

If $D$ is the doctrine of finite products,
then $D$-filtered colimits are precisely [[sifted colimits]]
and the $D$-filtered cocompletion of $C$ is $Sind(C)$,
the [[nonabelian derived category]] of $C$.

According to Definition 4.5 in [ALRalg](#ALRalg),
a category is _algebraically exact_
if it admits small limits and sifted colimits,
and the functor $colim: Sind(C)\to C$ is continuous.
In particular (Example 5.3 in [ALRalg](#ALRalg)),
any [[variety of algebras]]
is an algebraically exact category.
In particular, this includes [[Set]].

In fact, as shown in [ALRalg](#ALRalg), algebraically exact
categories with continuous functors that preserve sifted colimits
form the accessible [[equational hull]] of the bicategory
of [[varieties of algebras]] and continuous functors
that preserve sifted colimits.

### Indexed products and coproducts

When the definition for Kan extensions is internalized to internal categories and fibrations, and then specialized to the case of discrete categories, the notion of distributivity of dependent products over dependent sums encoded by a [[distributivity pullback]] becomes the statement that in the [[codomain fibration]], [[indexed products]] distribute over [[indexed coproducts]] in this sense.

### Pullback-stable colimits
 {#PullbackStableColimits}

As a nontrivial example of the $K\colon I\to Cat$ case, let $I$ be the [[walking structure|walking]] [[cospan]] $0 \to 1 \leftarrow 2$, and let $K(0) = J$ for some category $J$ while $K(1)=K(2)=\ast$, the [[terminal category]].  Then $\int^I K$ is $J$ together with a new terminal object $1$ and a new object $2$ having only a map to $1$, and so a diagram $D: \int^I K \to C$ consists of a $J$-diagram $D_0$ in a slice category $C/Z$ over some object $Z$ along with a morphism $f:Z\to Y$.  The limit category $\lim_I K$ is isomorphic to $J$, and the functor $D'\colon I\times \lim_I K\to C$ has $D'(0,j) = D_0(j)$, with $D'(1,j) =Z$ and $D'(2,j)=Y$.  Therefore, $\colim_{\lim_I K} \lim_I D'$ is the $J$-colimit of the pullback of $D_0$ along $f$, while $\lim_{i\in I} \colim_{K(i)} D$ is the pullback along $f$ of the colimit of $D_0$.  Thus, the distributivity condition for this $K$ holds if and only if $J$-colimits are [[universal colimit|universal]] (stable under pullback).

### For formal filtered limits and colimits

In the case of presentation of [[ind-object]]s and [[pro-object]]s as formal small filtered colimits and cofiltered limits of (co)representables, a corollary is that there is a functor
$$
Ind(Pro(C))\to Pro(Ind(C)).
$$
In a number of interesting cases this functor is inclusion.

## Related entries

* [[commutativity of limits and colimits]], [[permuting limits and colimits]]
* [[distributivity pullback]]
* [[distributive law]]
* [[distributivity for monoidal structures]]
* [[universal colimit]]

## References

The original definition is due to [[Jan-Erik Roos]], see Definition 1 in

* {#Roos64} [[Jan-Erik Roos]], [*Introduction à l'étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)*](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f1011.item), Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 259 (1964), 969–972 ([[roos-distributivity.pdf:file]]),

which is the first in a series of 3 papers, the others being

* [[Jan-Erik Roos]], _[Sur la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)](https://gallica.bnf.fr/ark:/12148/bpt6k40130/f1673.item)_, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 259 (1964), 1605–1608

* [[Jan-Erik Roos]], *[Complément à l'étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos)](https://gallica.bnf.fr/ark:/12148/bpt6k40149/f128.item)*, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 259 (1964), 1801–1804.

Additional references:

* {#ARV} [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico Vitale]], _Algebraic theories.  A categorical introduction to general algebra_ CUP 2011.

* {#ALR} [[Jiří Adámek]], [[William Lawvere]], [[Jiří Rosický]], [_Continuous Categories Revisited_](http://www.tac.mta.ca/tac/volumes/11/11/11-11abs.html) TAC 11, 2003.

* {#ALRalg} [[Jiří Adámek]], [[William Lawvere]], [[Jiří Rosický]], [_How algebraic is algebra_](http://www.tac.mta.ca/tac/volumes/8/n9/8-09abs.html) TAC 8, 2001.

* {#ABLR} [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiří Rosický]], [_A classification of accessible categories_](http://www.sciencedirect.com/science/article/pii/S0022404902001263) JPAA 175, 2002.

* {#AR} [[Jiří Adámek]], [[Jiří Rosický]], [_Algebra and local presentability: how algebraic are they? (A survey)_](https://doi.org/10.1515/tmj-2017-0113).

* {#ARVloc} [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico Vitale]],
[_On Algebraically Exact Categories and Essential Localizations of Varieties_](https://doi.org/10.1006/jabr.2000.8577)

* {#Garner.2011} [[Richard Garner]] and [[Stephen Lack]], [_Lex colimits_](http://dx.doi.org/10.1016/j.jpaa.2012.01.003), JPAA 216.6, 2012.

[[!redirects distributivity of limits and colimits]]
[[!redirects distributivity of colimits over limits]]
[[!redirects distributivity of colimits and limits]]
[[!redirects distributivity of products over colimits]]
[[!redirects distributivity of products and colimits]]
[[!redirects products distribute over colimits]]
[[!redirects limits distributive over colimits]]
[[!redirects precontinuous category]]
[[!redirects precontinuous categories]]
[[!redirects algebraically exact category]]
[[!redirects algebraically exact categories]]


distributivity pullback

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

# Distributivity pullback

* table of contents
{: toc}

## Idea

A **distributivity pullback** is the data which encodes a particular [[exponentiable morphism|exponentiation]] along a [[morphism]] in a [[category]].  In other words, it is a [[cofree object]] with respect to a [[pullback]] functor.

## Definition

+-- {: .un_defn}
###### Definition
For morphisms $g:Z\to A$ and $f:A\to B$ in a category, a **pullback around $(f,g)$** is a diagram
$$\array{ X & \xrightarrow{p} & Z & \xrightarrow{g} & A\\
  ^q\downarrow &&&& \downarrow^f\\
  Y && \xrightarrow{r} && B}
$$
in which the outer rectangle is a [[pullback]].
=--

A morphism of pullbacks around $(f,g)$ consists of $s:X\to X'$ and $t:Y\to Y'$ such that $p's=p$, $q s=t q'$, and $r = r's$.

+-- {: .un_defn}
###### Definition
For $g:Z\to A$ and $f:A\to B$ as above, a **distributivity pullback** around $(f,g)$ is a terminal object of the category of pullbacks around $(f,g)$.
=--

If the above diagram is a distributivity pullback, we say that it exhibits $r$ as a *distributivity pullback of $g$ along $f$*.

## Properties

### Connection to exponentiation

+-- {: .un_theorem}
###### Theorem
A morphism $f:A\to B$ is [[exponentiable morphism|exponentiable]] if and only if all distributivity pullbacks along $f$ exist.
=--
+-- {: .proof}
###### Proof
The universal property of a distributivity pullback says exactly that $Y\xrightarrow{r} B$ is the exponential of $g$ along $f$.
=--

### Connection to distributivity

In a category with pullbacks, for any pullback around $(f,g)$ with $f$ and $q$ exponentiable, we have a canonical Beck-Chevalley isomorphism
$$ r^* f_* \xrightarrow{\cong} q_* p^* g^* . $$
The [[mate]] of the inverse of this is a transformation
$$ \delta_{p,q,r}:r_! q_* p^* \to f_* g_! $$

+-- {: .un_theorem}
###### Theorem
A pullback around $(f,g)$ with $f$ and $q$ exponentiable is a distributivity pullback if and only if $\delta_{p,q,r}$ is an isomorphism.
=--
+-- {: .proof}
###### Proof
See [(Weber, Proposition 2.2.3)](#Weber).
=--

Invertibility of $\delta$ expresses that [[dependent products]] (the functors $f_*$ and $q_*$) distribute over [[dependent sums]] (the functors $g_!$ and $r_!$).

For instance, in the category of sets, if $(C_z)_{z\in Z}$ is a $Z$-indexed family of sets, then
$$ (f_* g_! C)_b = \prod_{f(a)=b} \sum_{g(z)=a} C_z $$
while
$$ (r_! q_* p^* C)_b = \sum_{r(y)=b} \prod_{q(x)=y} C_{p(x)} $$
As a very simple example, if $B=1$, $A=\{0,1\}$, and $Z=\{00,01,02,10,11\}$ with $g(i j) = i$, then $Y$ is the set of sections of $g$, $X$ the set of pairs of a section and an element of $A$, and $p$ the evaluation.  Then if $C = (C_{00}, C_{01}, C_{02}, C_{10}, C_{11})$, we have
$$ f_* g_! C = (C_{00} + C_{01} + C_{02}) \times (C_{10} + C_{11}) $$
and
$$ r_! q_* p^* C = (C_{00}\times C_{10}) + (C_{00}\times C_{11}) +
(C_{01}\times C_{10}) + (C_{01}\times C_{11}) +
(C_{02}\times C_{10}) + (C_{02}\times C_{11}).
$$

In the [[internal logic|internal]] [[dependent type theory]] of the ambient category, if we express $f$ and $g$ as [[dependent types]]

$$ b:B \vdash A(b) : Type \qquad and \qquad b:B, a:A(b) \vdash Z(b,a) : Type $$

then the exponential $r$ of $g$ along $f$ in a distributivity pullback is the [[dependent product type]]

$$ b:B \vdash \prod_{a:A(b)} Z(b,a) : Type. $$

and the distributivity isomorphism $\delta$ says that for any further dependent type

$$ b:B, a:A(b), z:Z(b,a) \vdash W(b,a,z) : Type$$

in the context of $b:B$ we have an isomorphism

$$ \sum_{\phi:\prod_{a:A(b)} Z(b,a) } \prod_{a:A(b)} W(b,a,\phi(a)) \qquad \cong \qquad  \prod_{a:A(b)} \sum_{z:Z(b,a)} W(b,a,z). $$

This isomorphism (or more specifically the left-to-right map) has traditionally been called the "axiom of choice" in [[Martin-Lof type theory]], since if $\sum$ and $\prod$ are interpreted according to [[propositions as types]] then it looks like the set-theoretic axiom of choice.  However, this is not really appropriate, since this is a provable statement rather than an additional axiom, and does not have any of the usual strong consequences of the set-theoretic axiom of choice.  See [[axiom of choice]] for further discussion.

### Connection to pullback complements

If $p$ is an isomorphism, then a distributivity pullback is also a [[final pullback complement]].

## References

* [[Mark Weber]], "Polynomials in categories with pullbacks", [arXiv](http://arxiv.org/abs/1106.1983), [TAC](http://tac.mta.ca/tac/volumes/30/16/30-16abs.html)
 {#Weber}

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divergence

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### Context
#### Riemannian geometry
+-- {: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

In [[Riemannian geometry]], the **divergence** of a
[[vector field]] $X$ over a [[Riemannian manifold]] $(M,g)$ is the real valued [[smooth function]] $div(X)$ defined by

$$
  div(X) = \star_g^{-1} d_{dR} \star_g g(X)
,$$

where $\star_g$ is the [[Hodge star]] operator of $(M,g)$,

$$
\star_g\colon \Omega^i(M;\mathbb{R}) \to \Omega^{dim M-i}(M;\mathbb{R})
,$$

and $d_{dR}$ is the [[de Rham differential]].

Alternatively, the divergence of a vector field $\vec\mathcal{A}$ in some point $x\in M$ is calculated (or alternatively defined) by the integral formula

$$
div \vec\mathcal{A} = \lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n}\cdot \vec\mathcal{A} d S
$$

where $D$ runs over the [[open submanifold]]s containing point $x$ and with smooth boundary $\partial D$ and $\vec{n}$ is the unit vector of outer normal to the [[hypersurface]] $S$. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a [[coordinate chart]] and [[balls]] with decreasing radius in this particular coordinate chart.

Although an [[orientation]] is required for the usual notion of Hodge star as given above, we may take it as valued in [[pseudoforms]] to show that the orientation (or even orientability) of $M$ is irrelevant (since the Hodge star is applied twice, returning us to untwisted forms, and since a bounding hypersurface has a natural 'outwards' pseudoorientation). However, the metric, which is hidden in the volume form and in the "dot product", is relevant.

## Example

If $(M,g)$ is the [[Cartesian space]] $\mathbb{R}^n$ endowed with the canonical Euclidean metric, then the divergence of a vector field $X^i \partial_i$ is

$$
div(X) = \sum_{i=1}^n\frac{\partial X^i}{\partial x^i}
.$$

## Remarks

The divergence was first developed in [[quaternion]] analysis, where its opposite appeared most naturally, called the _convergence_ $con(X) = - div(X)$.  In many applications of the divergence to the successor field, classical [[vector analysis]], the metric is irrelevant and we may use [[differential forms]] instead: we translate a vector field $X$ into the $(n-1)$-form $\star_g g(X)$ and a scalar field $f$ into the $n$-form $\star_g f$, so that the divergence *is* simply the [[de Rham differential]], and simply use the differential forms from the start.

## Related concepts

* [[nabla]]

* [[gradient]]

  * [[gradient flow]]

  * [[symplectic gradient]]

* [[Hamiltonian flow]]

* [[curl]]

* **divergence**

[[!redirects divergence]]
[[!redirects divergences]]


divergence theorem

# The Divergence Theorem
* table of contents
{: toc}

## Idea

The _Divergence Theorem_ is a generalization of the classical _Ostrogradsky--Gauss Theorem_ to arbitrary dimensions.  As such, it is also a generalization of the (second) [[Fundamental Theorem of Calculus]] and a special case of the [[Stokes Theorem]].  The value of picking out this particular case is its formulation using [[vector fields]] in any number of dimensions.

## Statement

Let $n$ be a [[natural number]], and let $S$ be a [[continuously differentiable map|continuously differentiable]] simple [[closed manifold|closed]] [[hypersurface]] in $\mathbb{R}^n$, in other words the image of a continuously differentiable [[immersion]] of the $(n-1)$-[[sphere]].  By the [[Jordan–Brouwer separation theorem]], $S$ is the boundary of some [[bounded subspace|bounded]] [[open subset|open]] region $U$ in $\mathbb{R}^n$.

Let $F$ be a continuously differentiable [[vector field]] defined on a neighbourhood of $cl(U)$ (so defined on both $U$ and $S$).  We can integrate $F$ outwards across $S$ by taking the dot product $F \cdot \mathbf{n}$, where $\mathbf{n}$ is the unit normal vector field on $S$ (perpendicular to $S$ and pointing outwards, that is away from $U$) since $S$ is continuously differentiable, and integrating this with respect to hypersurface area on $S$.  Equivalently, form an exterior differential [[pseudoform]] of rank $n - 1$ by taking the dot product of $F$ with the [[Hodge dual]] of the identity vector-valued $1$-form $\mathrm{d}\mathbf{x}$ and integrate that outwards across $S$.  We can also form the [[divergence]] of $F$, a scalar field, by differentiating each component of $F$ with respect to the corresponding coordinate and adding these, and then integrate this with respect to volume on $U$; equivalently, form an exterior differential pseudoform of rank $n$ by multiplying the divergence of $F$ by the [[volume pseudoform]].

The Divergence Theorem states that these two integrals are equal:
$$ \int_S F \cdot \mathbf{n} = \int_U div F ,$$
or
$$ \int_S F \cdot *\mathrm{d}\mathbf{x} = \int_U div F \vol .$$
This is a special case of the generalized [[Stokes Theorem]], since $div F \vol$ is the [[exterior differential]] of $F \cdot *\mathrm{d}\mathbf{x}$.

[[!redirects divergence theorem]]
[[!redirects divergence theorems]]
[[!redirects Divergence Theorem]]

[[!redirects Ostrogradsky theorem]]
[[!redirects Ostrogradsky Theorem]]
[[!redirects Ostrogradsky's theorem]]
[[!redirects Ostrogradsky's Theorem]]
[[!redirects Ostrogradsky’s theorem]]
[[!redirects Ostrogradsky’s Theorem]]
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[[!redirects Ostrogradsky\'s Theorem]]
[[!redirects Остроградский theorem]]
[[!redirects Остроградский Theorem]]
[[!redirects Остроградский's theorem]]
[[!redirects Остроградский's Theorem]]
[[!redirects Остроградский’s theorem]]
[[!redirects Остроградский’s Theorem]]
[[!redirects Остроградский\'s theorem]]
[[!redirects Остроградский\'s Theorem]]

[[!redirects Gauss theorem]]
[[!redirects Gauss Theorem]]
[[!redirects Gauss's theorem]]
[[!redirects Gauss's Theorem]]
[[!redirects Gauss’s theorem]]
[[!redirects Gauss’s Theorem]]
[[!redirects Gauss\'s theorem]]
[[!redirects Gauss\'s Theorem]]
[[!redirects Gauß theorem]]
[[!redirects Gauß Theorem]]
[[!redirects Gauß's theorem]]
[[!redirects Gauß's Theorem]]
[[!redirects Gauß’s theorem]]
[[!redirects Gauß’s Theorem]]
[[!redirects Gauß\'s theorem]]
[[!redirects Gauß\'s Theorem]]

[[!redirects Ostrogradsky-Gauss theorem]]
[[!redirects Ostrogradsky-Gauss Theorem]]
[[!redirects Ostrogradsky–Gauss theorem]]
[[!redirects Ostrogradsky–Gauss Theorem]]
[[!redirects Ostrogradsky--Gauss theorem]]
[[!redirects Ostrogradsky--Gauss Theorem]]


divergent series

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[series]] is called __divergent__ if it is not [[convergence|convergent]] in the usual strong senses of pointwise or, sometimes, [[uniform convergence]].

In asymptotic analysis, and in physics, especially in [[perturbative QFT]] etc., often a divergent series is still a very useful concept if regarded as an [[asymptotic series]], as a sum of some such series can be defined in various weaker senses, like [[Borel summability]].

## Related $n$Lab entries

* [[resummation]], [[regular summation method]], [[regularization]], [[renormalization]], [[Borel summability]], [[asymptotic series]], [[Stokes phenomenon]]

## References

* [[G. H. Hardy]], _Divergent series_, Clarendon Press, Oxford, 1949.
* [[A. N. Kolmogorov]], _Sur la possibilité de la définition générale de la dérivée, de l'intégrale et de la sommation de séries divergentes_, C. R. Acad. Sci. Paris 180 (1925), 362–364.
* Yu. I. Lyubich, _Axiomatic theory of divergent series and cohomological equations_, Fundamenta Math. 198 (2008) [doi](https://dx.doi.org/10.4064/fm198-3-5)
[arxiv/0705.1578](https://arxiv.org/abs/0705.1578)
* В. В. Козлов, _Инвариантные меры гладких динамических систем, обобщенные функции и методы суммирования_, Изв. РАН. Сер. матем., 80:2 (2016) 63–80  mathnet  [MR3507379](http://www.ams.org/mathscinet-getitem?mr=3507379) eng. transl.: V. V. Kozlov, _Invariant measures of smooth dynamical systems, generalized functions and summation methods_, Izv. Math. 80:2 (2016) 342–358
* [[Leonhard Euler]], _De seriebus divergentibus_, Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 205-237. Collected in _Opera Omnia_: Series 1, Volume 14, pp. 585 - 617. German translation in arXiv:[1202.1506](https://arxiv.org/abs/1202.1506). See also the [Euler archive](http://eulerarchive.maa.org/pages/E247.html)
* Johann Boos, _Classical and modern methods in summability_, Oxford Mathematical Monographs, 2001

category: analysis

[[!redirects diverging series]]


divided power algebra

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *divided power algebra* is a [[commutative ring]] $A$ together with an [[ideal]] $I$ and a collection of operations $\{\gamma_{n}\colon I\to A\}_{n\in\mathbb{N}}$ which behave like operations of taking divided powers $x\mapsto x^{n}/n!$ in [[power series]].

## Definition

A __divided power algebra__ is a triple $(A,I,\gamma)$ with

- $A$ a [[commutative ring]] (with identity);

- $I$ an [[ideal]] of $A$;

- $\gamma=\{\gamma_{n}\colon I\to I\}_{n\geq 1}$ an [[indexed set]] of [[functions]] (of [[underlying]] [[sets]]);

where we additionally adopt the convention $\gamma_0(x) = 1$ (which is usually not in $I$), and this data is required to satisfy the following conditions:

1. For each $x\in I$, we have $\gamma_{1}(x)=x$.

1. For each $x,y\in I$ and $n\geq 0$, we have
$$
\gamma_{n}(x+y)=\sum_{k=0}^{n}\gamma_{n-k}(x)\gamma_{k}(y),
$$

1. For each $\lambda\in A$, each $x\in I$ and $n\geq 0$, we have
$$
\gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x).
$$

1. For each $x\in I$ and each $m,n\geq 0$, we have
$$
\gamma_{n}(x)\gamma_{m}(x)=\frac{(n+m)!}{n!m!}\gamma_{n+m}(x).
$$

1. For each $x\in I$ and each $m\geq 0$, $n\geq 1$, we have
$$
\gamma_{m}(\gamma_{n}(x))=\frac{(n m)!}{(n!)^{m}m!}\gamma_{n m}(x).
$$

For a given $(A,I)$, a __divided power structure__ on $(A,I)$ is a $\gamma$ making $(A, I, \gamma)$ a divided power algebra.

If $A$ is an $R$-algebra for a ring $R$, we call it a __divided power $R$-algebra__ or __PD-$R$-algebra__.

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

Some sources include $\gamma_0$ in the data rather than as convention. Some sources give the data as $\gamma_n : I \to A$ typing while including $\gamma_n(x) \in I$ for $n \geq 1$ as an axiom.

=--

## Properties

Genuine powers can be constructed in the expected way from the divided powers, and when $A$ is torsion free, the reverse is true:

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

If $(A,I,\gamma)$ is a divided power algebra, then $n! \gamma_n(x) = x^n$ for every $x \in I$ and $n \geq 0$ (taking $x^0:=1$).
=--

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

It is true for $n=0$ and $n=1$ by definition.
For $n \geq 2$, this follows by [[induction]], since $n! \gamma_n(x) = (n-1)! \gamma_{n-1}(x) \cdot 1! \gamma_1(x) = x^{n-1} \cdot x$.

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

If $A$ is a commutative, torsion free ring with an ideal $I$ such that $x^n$ is an $(n!)$-th multiple for every $x \in I$ and $n \geq 0$, then $(A,I)$ has a unique divided power structure, and it is given by $\gamma_n(x) = x^n / n!$.
=--

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

The hypotheses imply the quotients $x^n / n!$ are unique and well-defined, and any divided power structure on $(A,I)$ must be given by that formula. It's straightforward to check the definition does give a divided power algebra.

=--

So in the torsion free case, the divided power algebras are precisely of the motivating form. In positive characteristic, though, examples can be somewhat more exotic.

## Categorically

We can define the concept of divided power in any symmetric monoidal category.

Let $\mathcal{C}$ be a symmetric monoidal category.

The $n^{th}$ divided power of an object $A$ is then defined as the equalizer of the following diagram:

\begin{tikzcd}
A^{\otimes n} \arrow[rr, "\sigma", shift left=2] \arrow[rr, "...", shift right=2] &  & A^{\otimes n}
\end{tikzcd}

where there is one arrow for every $\sigma \in \mathfrak{S}_{n}$. We write $\sigma$ for the natural transformation associated to $\sigma$, which is defined in the entry [[symmetric monoidal category]].

The $n-th$ divided power $\Gamma_{n}(A)$ is thus described by the following equalizer diagram:

\begin{tikzcd}
\Gamma_{n}A \arrow[rr, "s_{n}"] &  & A^{\otimes n} \arrow[rr, "\sigma", shift left=2] \arrow[rr, "...", shift right=2] &  & A^{\otimes n}
\end{tikzcd}

In the category of modules over some commutative ring, the $n-th$ divided power of a module $A$ is equivalently descibed as the space $(A^{\otimes n})^{\mathfrak{S}_{n}}$ of [[invariants]] under the action of $\mathfrak{S}_{n}$ on $A^{\otimes n}$ by permutation of the factors. Note that the $n-th$ symmetric power $S_{n}(A)$ of an object $A$ in a symmetric monoidal category is described as the coequalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$. In a category of modules, it is equivalently described as the space $(A^{\otimes n})_{\mathfrak{S}_{n}}$ of [[coinvariants]] by the action of $\mathfrak{S}_{n}$. We then have the relation $(S_{n}(A^{*}))^{*} = \Gamma_{n}(A)$ which probably means that these powers can be interpreted as [[graded modalities|graded exponential modalities]] in a kind of graded [[differential linear logic]]. In [[characteristic 0]], we have $\Gamma_{n}(A) \cong S_{n}(A)$. Nondegenerated models of such a logic in a category of vector spaces would thus be only in the case where the field is of [[positive characteristic]].

## Related concepts

* [[crystalline cohomology]]

## References

Divided power algebras were originally introduced in

* [[Henri Cartan]], _Puissances divisées_, Séminaire Henri Cartan, Tome 7 (1954–1955) no. 1, Exposé no. 7, 11 p. [Numdam:SHC_1954–1955__7_1_A7_0](http://www.numdam.org/item/SHC_1954-1955__7_1_A7_0/).

Their theory was further developed in [[Pierre Berthelot]]'s PhD thesis (in the context of [[crystalline cohomology]]), which was later published as:

* [[Pierre Berthelot]], _Cohomologie cristalline des schémas de caractéristique $p \gt 0$_, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin, 1974. ([doi:10.1007/BFb0068636](https://doi.org/10.1007/BFb0068636),  MR 0384804)

Review:

* [[Pierre Berthelot]], [[Arthur Ogus]], _Notes on crystalline cohomology_, Princeton Univ. Press 1978. vi+243, ([ISBN:0-691-08218-9](https://press.princeton.edu/books/hardcover/9780691648323/notes-on-crystalline-cohomology-mn-21))

* [[Aise Johan de Jong]] et al., [[The Stacks Project]], [Chapter 09PD](https://stacks.math.columbia.edu/tag/09PD).

Recent works on divided power algebras include:

* [[Sacha Ikonicoff]], _Divided power algebras over an operad_, Glasgow Math. J. **62** (2020) 477-517, [doi:10.1017/S0017089519000223](https://doi.org/10.1017/S0017089519000223), [pdf](https://arxiv.org/pdf/1712.03694.pdf)

* [[Sacha Ikonicoff]], _Divided power algebras and distributive laws_, 2021, [doi:10.48550/arXiv.2104.11736]( https://doi.org/10.48550/arXiv.2104.11736), [pdf](https://arxiv.org/pdf/2104.11736.pdf)

It is related to [[differential categories]] in:

* [[Sacha Ikonicoff]], [[Jean-Simon Pacaud Lemay]], _Cartesian Differential Comonads and New Models of Cartesian Differential Categories_, 2021, [doi:10.48550/arXiv.2108.04304](https://doi.org/10.48550/arXiv.2108.04304), [pdf](https://arxiv.org/pdf/2108.04304.pdf)

On divided powers:

* [[Luis Narváez Macarro]], _Hasse-Schmidt derivations, divided powers and differential smoothness_, 2009, [doi:10.5802/aif.2513](https://doi.org/10.5802/aif.2513), [pdf](https://aif.centre-mersenne.org/item/10.5802/aif.2513.pdf)

See also:

* Wikipedia, *[Divided power structure](https://en.wikipedia.org/wiki/Divided_power_structure)*

In relation to the [[sphere spectrum]]

* [[Sanath Devalapurkar]], _Divided powers and the sphere spectrum_, [blog post](https://sanathdevalapurkar.github.io/blog/algebraic-topology/2019/02/20/divided-powers.html)

[[!redirects PD-algebra]]
[[!redirects PD-algebras]]

[[!redirects divided power structure]]
[[!redirects divided power structures]]
[[!redirects divided power algebras]]

[[!redirects divided power]]
[[!redirects divided powers]]


divisible group

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###Context###
#### Group Theory
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=--
=--
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# Contents
* table of contents
{: toc}

## Definition

Let $G$ be a [[group]]. For the following this is often assumed to be (though is not necessarily) an [[abelian group]]. Hence we write here the group operation with a plus-sign

$$
  + : G \times G \to G
  \,.
$$

For $\mathbb{N}$ the [[natural numbers]], there is a [[function]]

$$
  (-)\cdot (-) : \mathbb{N} \times G \to G
$$

which takes a group element $g$ to

$$
  n \cdot g \coloneqq \underbrace{g + g + \cdots + g}_{n \; summands}
  \,.
$$

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

A [[group]] $G$ is called **divisible** if for every [[natural number]] $n$ (hence for every [[integer]]) we have that for every element $g \in G$ there is an element $h \in G$ such that

$$
  g = n \cdot h
  \,.
$$

=--

In other words, if for every $n$ the 'multiply by $n$' map $G \stackrel{n}{\to} G$ is a [[surjection]].

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

For $p$ a [[prime number]] a group is **$p$-divisible** if the above formula holds for all $n$ of the form $p^k$ for $k \in \mathbb{N}$.

=--

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

There is also an abstract notion of $p$-[[p-divisible group|divisible group]] in terms of [[group schemes]].

=--

## Properties

### Equivalent characterization

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

Let $A$ be an [[abelian group]].

Assuming the [[axiom of choice]], the following are equivalent:

1.  $A$ is divisible

1.  $A$ is [[injective object]] in the the [[category]] [[Ab]] of [[abelian groups]]

1. the [[hom functor]] $Hom_{Ab}(-,A) : Ab^{op} \to Ab$ is [[exact functor|exact]].

=--

This is for instance in ([Tsit-YuenMoRi,Proposition 3.19] (#Tsit-YuenMoRi)). It follows for instance from using [[Baer's criterion]].

### Vector space structure

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

The [[torsion-free]] and divisible abelian groups are precisely the [[rational vector spaces]], i.e. if $A$ is torsion-free and divisible, then it carries a unique vector space structure.

=--

### Stability under various operations

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

The [[direct sum]] of divisible groups is itself divisible.

=--

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

Every [[quotient group]] of a divisible group is itself divisible.

=--

## Examples

+-- {: .num_example }
###### Example

The additive group of [[rational number]] $\mathbb{Q}$ is divisible.
Hence also that underlying the [[real numbers]] $\mathbb{R}$ and the [[complex numbers]].

=--

Hence:

+-- {: .num_example }
###### Example

The underlying abelian group of any $\mathbb{Q}$-[[vector space]] is divisible.

=--

Also, by prop. \ref{QuotientOfDivisibleIsDivisible},

+-- {: .num_example #CircleGroupExample}
###### Example

The [[quotient groups]] $\mathbb{Q}/\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ are divisible (the latter is also written $U(1)$ (for _[[unitary group]]_) or $S^1$ (for _[[circle group]]_)).

=--

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

What is additionally interesting about example \ref{CircleGroupExample} is that it provides an [[injective object|injective]] [[cogenerator]] for the [[category]] [[Ab]] of abelian groups. Similarly, $\mathbb{R}/\mathbb{Z}$  is an injective cogenerator.

=--

+-- {: .num_example }
###### Counter-Example

The following groups are **not** divisible:

* the additive group of [[integers]] $\mathbb{Z}$.

* the [[cyclic group]] $\mathbb{Z}_n$ for $n \geq 1 \in \mathbb{N}$.

=--

## References

* {#Tsit-YuenMoRi}  [[Tsit-Yuen Lam]], _Lectures on modules and rings_, Graduate Texts in Mathematics No. 189, Berlin, New York (1999) ([doi:10.1007/978-1-4612-0525-8](https://link.springer.com/book/10.1007/978-1-4612-0525-8))

* Phillip A. Griffith (1970), Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7


division

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#Contents#
* table of contents
{:toc}

## Idea

The [[inverse function|inverse]] operation to [[multiplication]].

## Definition

### In a field

Given a [[Heyting field]] $F$, let us define the type of all terms in $F$ apart from 0:

$$F_{#0} \coloneqq \{a \in F \vert a # 0\}$$

The **division function** is a [[binary function]] $\frac{(-)}{(-)}:F \times F_{#0} \to F$ defined as

$$\frac{x}{y} \coloneqq x \cdot \frac{1}{y}$$

where

$$\frac{1}{y}$$

is the [[reciprocal function]].

## Related concepts

* [[arithmetic]]

* [[division ring]]

* [[division algebra]]

* [[localization of a ring]], [[field of fractions]]

* [[fraction]],  [[rational numbers]]

* [[reciprocal function]], [[rational function]]


division algebra

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#### Algebra
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#Contents#
* table of contents
{:toc}

## Definition

A **division algebra** is a possibly [[non-associative algebra]] $A$, typically over a [[field]] $k$, which has the property that for any nonzero element $a$ and any element $b$, the equations $a x=b$ and $x a=b$ each have a unique solution for $x$ in the algebra.

Usually the algebra is assumed to have a [[unital magma|multiplicative identity]], in which case this condition implies that each element has a left inverse and a right inverse. These inverses necessarily coincide if the algebra is [[associative algebra|associative]], but this may fail in the absence of associativity.

A sub-topic of interest is the existence of a multiplicative norm for the algebra, see at _[[normed division algebra]]_.

### Other definitions

An alternative definition is used in [Baez 02](#Baez02): The axioms given there require that the algebra is finite-dimensional over $k$ and that for any $a$ and $b$ in the algebra, $a b = 0$ implies at least one of $a$ or $b$ is already $0$.  This coincides with the definition given above for [[finite-dimensional vector space|finite dimensional]] algebras.

## Examples

* Perhaps the most famous division algebras are the [[real numbers]] $\mathbb{R}$, the [[complex numbers]] $\mathbb{C}$, the [[quaternions]] $\mathbb{H}$, and the [[octonions]] $\mathbb{O}$. According to the [[Hurwitz theorem]], these are the only normed, finite dimensional, real division algebras.

* Any [[division ring]] is an associative division algebra over its [[center]] and has identity, but it may not be finite dimensional over its center.

* The ring of rational [[polynomials]] $\mathbb{R}(X)$ is an infinite dimensional real associative division algebra.

* There exists a division algebra with identity that does not have two-sided inverses for every nonzero element.

## Counter-examples

* There exists a [[reciprocal algebra]] with nonzero zero divisors.

* The [[sedenions]] $\mathbb{S}$ are a finite-dimensional normed real algebra that is not a division algebra (it has zero divisors.)

* The ideal generated by $Y$ in the [[polynomial ring]] $\mathbb{R}[Y]$ is an associative, commutative, infinite dimensional real algebra without identity and without zero divisors. The equation $Yx=Y$ does not have a solution for $Y$ in the algebra.

## Related concepts

For many applications (also to [[physics]]) the most interesting division algebras are the [[normed division algebras]] over the [[real numbers]]: By the [[Hurwitz theorem]] these are the [[real numbers]], [[complex numbers]], [[quaternion|quaternions]] and [[octonion|octonions]].  These have important relations to [[division algebras and supersymmetry|supersymmetry]].

[[!include exceptional spinors and division algebras -- table]]

See at _[[division algebra and supersymmetry]]_.

* [[Hopf invariant one]]

* [[division ring]]

## References

See also

* [[eom]], _[Division algebra](https://encyclopediaofmath.org/wiki/Division_algebra)_

* Wikipedia, _[Division algebra](https://en.m.wikipedia.org/wiki/Division_algebra)_

* {#Baez02} [[John Baez]], Section 1.1 _Preliminaries_ In: _The octonions_, _Bull. Amer. Math. Soc._ **39** (2002), 145-205.   ([doi:10.1090/S0273-0979-01-00934-X](https://doi.org/10.1090/S0273-0979-01-00934-X), [website](http://math.ucr.edu/home/baez/octonions/node2.html))

[[!redirects division algebra]]
[[!redirects division algebras]]


division algebra and supersymmetry

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#### Exceptional structures
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[[!include exceptional structures -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There is a close relationship between

* the four real [[normed division algebras]]

* [[spin geometry]] and real (Majorana) [[spin representations]];

* the [[Lie algebra cohomology]] of the [[super Poincare Lie algebra]],

* and [[supersymmetry]] in [[quantum field theory]] and [[string theory]].

This is based on the fact that in certain dimensions, [[spin group]] [[representation]]s are naturally identified with a $\mathbb{K}^n$, for $\mathbb{K}$ one of the [[normed division algebras]], see at [[spin group]] _[The exceptional isomorphisms](spin%20group#ExceptionalIsomorphisms)_.

For detailed exposition see

* _[[geometry of physics -- supersymmetry]]_ the section _[Real spin representations via Real alternative division algebra](geometry+of+physics+--+supersymmetry##RealSpinRepresentationViaNormedDivisionAlgebra)_

* _[[geometry of physics -- fundamental super p-branes]]_

[[!include exceptional spinors and division algebras -- table]]

The structure of the [[normed division algebras]] also governs the existence of the _[[brane scan]]_ and the super-[[∞-Lie algebras]] such as the [[supergravity Lie 3-algebra]]. By the [[D'Auria-Fre formulation of supergravity]] the [[∞-Lie algebra valued forms]] with values in these constitute the field content of (11-dimensional) [[supergravity]].

Combining this, one finds that [[supergravity]] coupled to [[super Yang-Mills theory]] (super [[Einstein-Yang-Mills theories]]) are parameterized by triples of real normed division algebras, forming a "[[magic pyramid]]".

## Related concepts

* [[SL(2,H)]]

* [[Hopf invariant one]]

## References
 {#References}

The relation between [[supersymmetry and division algebras]] was gradually established by a variety of authors, including

* {#KugoTownsend82} [[Taichiro Kugo]], [[Paul Townsend]], _Supersymmetry and the division algebras_, Nuclear Physics B, Volume 221, Issue 2 (1982) p. 357-380. ([spire:181889](https://inspirehep.net/literature/181889), doi:10.1016/0550-3213(83)90584-9, [pdf](http://cds.cern.ch/record/140183/files/198301032.pdf))

* {#Sudbery84} [[Anthony Sudbery]], _Division algebras, (pseudo)orthogonal groups and spinors_, Jour. Phys. A17 (1984),
939–955 ([doi:10.1088/0305-4470/17/5/018](
https://iopscience.iop.org/article/10.1088/0305-4470/17/5/018))

* {#Evans88} [[Jonathan Evans]], _Supersymmetric Yang–Mills theories and division algebras_, Nucl. Phys. B298 (1988), 92–108 (https://doi.org/10.1016/0550-3213(88)90305-7)

* {#ChungSudbery87} K.-W. Chung, [[Anthony Sudbery]], _Octonions and the Lorentz and conformal groups of ten-dimensional space-time_, Phys. Lett. B 198 (1987), 161–164 (doi:10.1016/0370-2693(87)91489-4)

* {#ManogueSudbery89} [[Corinne Manogue]], [[Anthony Sudbery]], _General solutions of covariant superstring equations of motion_, Phys. Rev. D 12 (1989), 4073–4077 ([doi:10.1103/PhysRevD.40.4073](https://doi.org/10.1103/PhysRevD.40.4073))

* {#Schray96} Jörg Schray, _The general classical solution of the superparticle_, Class. Quant. Grav. 13 (1996), 27–38. ([arXiv:hep-th/9407045](https://arxiv.org/abs/hep-th/9407045))

* [[Tevian Dray]], J. Janesky, [[Corinne Manogue]], _Octonionic hermitian matrices with non-real eigenvalues_, Adv. Appl. Clifford Algebras 10 (2000), 193–216 ([arXiv:math/0006069](https://arxiv.org/abs/math/0006069))

Corresponding discussion of [twistor space](twistor+space#TwistorSpace):

* {#BengtssonCederwall88} [[Ingemar Bengtsson]], [[Martin Cederwall]], _Particles, Twistors and the Division Algebras_, Nucl.Phys. B302 (1988) 81-103 ([spire](http://inspirehep.net/record/247269))

* {#Cederwall93} [[Martin Cederwall]], _Introduction to Division Algebras, Sphere Algebras and Twistors_ ([arXiv:hep-th/9310115](https://arxiv.org/abs/hep-th/9310115))

and its [[anti de Sitter spacetime|AdS]] version

* [[Alex Arvanitakis]], Alec E. Barns-Graham, [[Paul Townsend]], _Twistor description of spinning particles in AdS_, JHEP 01 (2018) 059 ([arXiv:1710.09557](https://arxiv.org/abs/1710.09557))

Streamlined proof and exposition is in

* {#BaezHuerta09} [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry I_, in R. Doran, G. Friedman and [[Jonathan Rosenberg]] (eds.), _Superstrings, Geometry, Topology, and $C^*\ast$-algebras_, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 ([arXiv:0909.0551](http://arxiv.org/abs/0909.0551))

* {#BaezHuerta10} [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_, Adv. Math. Theor. Phys. 15 (2011), 1373-1410  ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))

The case of 2-component quaternionic spinors in 6d (see at [[SL(2,H)]]) is discussed in more detail in:

* [[Joás Venâncio]], [[Carlos Batista]], _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_ ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296))

Discussion with an emphasis on [[super Yang-Mills theory]] and [[U-duality]] in [[supergravity]] and the [[Freudenthal magic square]] is in

* [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], _A magic square from Yang-Mills squared_ ([arXiv:1301.4176](http://arxiv.org/abs/1301.4176))

* [[Alexandros Anastasiou]], [[Leron Borsten]], [[Mike Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], _Super Yang-Mills, division algebras and triality_ [[arXiv:1309.0546](http://arxiv.org/abs/1309.0546)]

* {#ABDHN} A. Anastasiou, [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, _A magic pyramid of supergravities_, [arXiv:1312.6523](http://arxiv.org/abs/1312.6523)

The relationship in [[string theory]] via [[octonion]] algebra between the NRS [[spinning string]] and the [[Green-Schwarz superstring]] [[sigma-models]] is discussed in

* Rafael I. Nepomechie, _Nonabelian bosonization, triality, and superstring theory_ Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210

* [[Itzhak Bars]], D. Nemschansky and S. Yankielowicz, SLACPub-3758.

* H. Tachibana, K. Imeda, _Octonions, superstrings and ten-dimensional spinors_ , Il nuovo cimento, Vol 104 B N.1

The relation of the division algebras to ordinary (Lie algebraic) extensions of the [[super Poincare Lie algebra]] is discussed in

* Jerzy Lukierski, Francesco Toppan, _Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory_ ([arXiv:hep-th/0203149](https://arxiv.org/abs/hep-th/0203149), [pdf](http://cbpfindex.cbpf.br/publication_pdfs/NF00102.2010_08_03_10_47_48.pdf))

* A. Anastasiou, L. Borsten, [[Mike Duff]], L. J. Hughes, S. Nagy, _An octonionic formulation of the M-theory algebra_ ([arXiv:1402.4649](http://arxiv.org/abs/1402.4649))

Normed division algebras are used to describe the construction of [[Lie 2-algebra]] extensions of the [[super Poincare Lie algebra]] in

* {#BH10} [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))

* [[John Huerta]], _Division Algebras, Supersymmetry and Higher Gauge Theory_, ([arXiv:1106.3385](http://arxiv.org/abs/1106.3385))

* [[John Huerta]], _Division Algebras and supersymmetry III_, ([arXiv:1109.3574](http://arxiv.org/abs/1109.3574))

This is made use of in the [[Hypothesis H|homotopy theoretic description of M-theory]] in:

* [[John Huerta]], [[Urs Schreiber]], _[[schreiber:M-Theory from the Superpoint]]_, Letters of Mathematical Physics, 2018 ([arXiv:1702.01774](https://arxiv.org/abs/1702.01774))

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

* [[Domenico Fiorenza]],  [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Super-exceptional M5-brane model -- Emergence of SU(2)-flavor sector]]_ ([arXiv:2006.00012](https://arxiv.org/abs/2006.00012))

[[!redirects division algebras and supersymmetry]]

[[!redirects supersymmetry and division algebra]]
[[!redirects supersymmetry and division algebras]]

[[!redirects real spin representations and division algebras]]
[[!redirects division algebras and real spin representations]]


division dagger 2-poset

[[!redirects division dagger 2-posets]]

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## Contents ##
* table of contents
{:toc}

## Definition ##

A division dagger 2-poset is a [[dagger 2-poset]] $C$ such that for every object $A \in Ob(C)$, $B \in Ob(C)$, and $C \in Ob(C)$ and morphisms $f \in Hom(A, B)$ and $g \in Hom(A, C)$ there is a morphism $g/f:Hom(B, C)$ such that for every morphism $h \in Hom(B, C)$, $(h \leq g/f) \iff (h \circ g = f)$.

## Examples ##

* The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a division dagger 2-poset.

## See also ##

* [[dagger 2-poset]]