direct image with compact support

> under construction

#Contents#
* table of contents
{:toc}

## Idea

Let $f:X\to Y$ be a morphism of [[locally compact topological spaces]]. Then there exist a unique [[subfunctor]] $f_!: Sh(X)\to Sh(Y)$ of the [[direct image]] functor $f_*$ such that for any [[abelian sheaf]] $F$ over $X$ the sections of $f_!(F)$ over $U^{open}\subset Y$ are those sections $s\in f_*(U)= \Gamma(f^{-1}(U),F)$ for which the restriction $f|_{supp(s)} : supp(s)\to U$ is a [[proper map]].

This is called the **direct image with compact support**.

It follows that $f_!$ is left exact.

Let $p:X\to {*}$ be the map into the one point space. Then for any $F\in Sh(X)$ the [[abelian sheaf]] $p_!F$ is the [[abelian group]] consisting of sections $s\in \Gamma(X,F)$ such that $supp(s)$ is compact. One writes $\Gamma_c(X,F):= p_! F$ and calls this group a group of sections of $F$ with compact support. If $y\in Y$, then the fiber $(f_! F)_y$ is isomorphic to $\Gamma_c(f^{-1}y,F|_{f^{-1}(y)})$.

## Related concepts

* [[exceptional inverse image]]

* [[Verdier duality]], [[six operations]]

* [[locally proper map]]

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[[!redirects direct images with compact support]]
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direct limit

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#### Category theory
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#### Limits and colimits
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=--

The term __direct limit__ is essentially another term for *[[colimit]]*, dating from a time before [[category theory|category theoretic]] terminology became established.

It is still used in various areas of mathematics.  ([[formal duality|Dually]], *[[inverse limit]]* is another term for *[[limit]]*.)  In this terminology, _direct system_ means *[[diagram]]*, and _direct cone_ means [[cocone]].

More specifically, the term *direct limit* is often understood by default to refer to [[colimits]] over [[directed sets]] (or [[filtered categories]]); see _[[directed colimit]]_  (or *[[filtered colimit]]*) for discussion of this case.

Especially common here is the directed set $(\mathbb{N},\leq)$ of [[natural numbers]], in which case the direct system may be called a _direct sequence_.

## Related concepts

* [[inductive limit]], another terminology for colimit

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direct product

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# Direct products
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{: toc}

## Definition

In [[universal algebra]], a __direct product__ is simply a [[product]] in a [[concrete category]] that is [[created limit|created]] by the [[forgetful functor]].

Compare the [[direct sum]], a more complicated concept.

## Examples

Trivially, a [[cartesian product]] of [[sets]] is a direct product in [[Set]].

One of the requirements of a [[topological category]] is that any family of objects must have a direct product, although the term 'direct product' is not used in topology.

Many [[algebraic categories]], such as [[Grp]], [[Ab]], [[Ring]], etc, also have all direct products; this is where the term 'direct product' originated.

The category of ($Set$-valued) models of any [[Lawvere theory]] has all direct products; this includes the examples from algebra above.

## Related concepts

* [[direct sum]]

[[!redirects direct products]]


direct product group

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#Contents#
* table of contents
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## Idea

The [[Cartesian product]] in the [[category]] of [[groups]] is often called the _direct product_ of groups. For [[abelian groups]] and a [[finite number]] of factors, this is also the [[direct sum]] of groups.

(Compare the [[formal duality|dual]] notion of "[[free products]]" of groups, which are really their [[category theory|category theoretic]] *[[coproducts]]*.)

## Properties

### Representations
 {#Representations}

+-- {: .num_prop #IrrepsOfDirectProductsAreExternalTensorProductsOfIrreps}
###### Proposition
**([[irreps]] of [[direct product groups]] are [[external tensor products]] of [[irreps]])**

Let $G_1, G_2$ be two [[groups]]. Then, over an [[algebraically closed field|algebraically closed]] [[ground field]], every [[irreducible representation]] $\rho \in (G_1 \times G_2) Rep_{irr}$ of their [[direct product group]] $G_1 \times G_2$ is the [[external tensor product]] of [[irreducible representations]] $\rho_i \in G_i Rep_{irr}$ of the two groups separately:

$$
  \rho \;=\; \rho_1 \boxtimes \rho_2
  \,.
$$

Here the [[external tensor product]] has as underlying [[vector space]] the corresponding [[tensor product of vector spaces]], equipped with the evident [[action]]

$$
  (g_1, g_2)( (v_1 \otimes v_2) ) \;=\; ( g_1(v_1) \otimes g_2(v_2) )
  \,.
$$

=--

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

By [[Schur's lemma]] see e.g. [here](https://math.stackexchange.com/a/2503141/58526).

=--

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

The statement of Prop. \ref{IrrepsOfDirectProductsAreExternalTensorProductsOfIrreps} is in general false if the ground field is not algebraically closed. A counterexample is given im [Kowalski 13, Example 2.7.31](#Kowalski13).

Also the converse to Prop. \ref{IrrepsOfDirectProductsAreExternalTensorProductsOfIrreps} is _false_ in general. The [[external tensor product]] of irreducible representations need not be irreducible itself. For more see [Fein 67](#Fein67).

=--

## Related concepts

* [[semidirect product group]]

* [[central product of groups]]

* [[free product of groups]]

## References

* {#Kowalski13} E. Kowalski, around p. 41 of _Representation theory_, 2013 ([[KowalskiReptheory2013.pdf:file]])

* {#Fein67} Burton Fein, _Representations of direct products of finite groups_, Pacific J. Math. Volume 20, Number 1 (1967), 45-58 ([Euclid:1102992967](https://projecteuclid.org/euclid.pjm/1102992967))
See also

* Wikipedia, _[Direct product of groups](https://en.wikipedia.org/wiki/Direct_product_of_groups)_

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direct sum

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# Direct sums and weak direct products
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## Idea

The notion of __direct sum__, or __weak direct product__, is a concept from [[algebra]] that actually makes sense in any [[category]] $C$ with [[zero morphisms]] (that is, any category [[enriched category|enriched]] over the [[closed monoidal category]] of [[pointed sets]]), as long as the needed ([[colimit|co]])[[limits]] exist.

A basic and familiar example is the direct sum $V_1 \oplus V_2$  of two [[vector spaces]] $V_1$ and $V_2$ over some [[field]], or, more generally of two [[modules]] over some [[ring]]. Generally, for $I$ a set and $\{V_i\}_{i \in I}$ an $I$-indexed family of vector space or modules, their direct sum $\bigoplus_{i \in I} V_i$ is the collection of [[formal linear combinations]] of elements in each of the $V_i$. This may in part motivate the terminology: _an element in a direct sum is a sum of elements_, at least in these cases.

This generalises in two distinct ways, which we call _direct sums_ and _weak direct products_.  In many cases (as in the example above), these coincide, but not always.  Also in many cases, direct sums will be the same as [[coproducts]].  In any case, finitary weak direct products are the same as [[products]] but the infinitary versions are (almost always) different.

## Terminology

The name 'weak direct product' comes from the concept of [[direct product]] in [[algebra]] for a [[product]] in a [[concrete category]] that is created by the [[forgetful functor]]; the weak direct product will be a [[subobject]] of the direct product (and the entire direct product in finitary cases).  But here we will not restrict ourselves to the context of such a concrete category.

The term 'direct sum' comes from the finitary [[biproduct]] (simultaneously product and coproduct) in [[additive category|additive categories]].  The additive character of these biproducts extends in the infinitary case (where biproducts generally no longer appear) to the coproduct rather than to the product.  Even when the direct sum is not the same as the coproduct, it still retains some of this flavour.

In the classical examples of $C$, the direct sum and weak direct product are the same.  However, the general definitions below distinguish them in some cases, and we use the terms 'direct sum' and 'weak direct product' to best evoke the 'like a coproduct' and 'part of a product' senses.

## Definitions

Let $\mathcal{C}$ be a [[category]] with [[products]] and [[coproducts]], as well as [[zero morphisms]].  Let $I$ be a [[set]], and let $(A_i)_{i \in I}$ be an $I$-indexed [[family]] of [[objects]] in $\mathcal{C}$, hence a [[function]] $A : I \to Obj(\mathcal{C})$.

We now define both the direct sum and weak direct product of this family.  The $A_i$ will be called the __direct summands__ or (weak) __direct factors__.

### Direct sum

Here we must assume moreover that $\mathcal{C}$ is a [[regular category]] (or otherwise has a good concept of [[image]]).

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

Let $r$ be the morphism from the [[coproduct]] $\coprod_i A_i$ to the [[product]] $\prod_i A_i$ characterized by having the following components

$$
  \left(
    A_i \to \coprod A \stackrel{r}{\to} \prod A \to A_j
  \right)
  =
  \left\{
    \array{
      Id_{A_i} & if\; i = j
      \\
      0_{ij} & if\; i \neq j
    ,}
  \right.
  \,
$$

where $0_{ij}$ is the [[zero morphism]] from $A_i$ to $A_j$.

The __direct sum__ over the family $\{A_i\}$ is the [[image]]

$$
  \coprod_i A_i \overset{\coim r}\to \bigoplus_i A_i \overset{\im r}\to \prod_i A_i
$$

of the morphism $r$.

=--

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

In [[constructive mathematics]], the definition of $r$ requires that the index set $I$ have [[decidable equality]], which is the case in most applications of interest.  An arbitrary index set will still work if $\mathcal{C}$ is [[enriched category|enriched]] over the category of sets and [[partial functions]]; this may be embedded as a [[full subcategory]] of the category of pointed sets, and the embedding is an [[equivalence of categories]] if and only if the law of [[excluded middle]] holds.  But the usual examples of $\mathcal{C}$ are not (constructively) so enriched.  Fortunately, the usual examples of $I$ have decidable equality.

=--

### Weak direct product

Here we consider the finitary [[products]]
$$ \prod_{i \in F} A_i $$
as $F$ varies over the [[finite set|finite subsets]] of the index set $I$.  (In [[constructive mathematics]], use 'finitely indexed' or '[[Kuratowski finite]]' here ... although if $I$ has [[decidable equality]], as is the case in the usual examples, then every finitely indexed subset of $I$ is actually finite in the strictest sense.)

These finite products form a [[directed limit|direct system]] indexed by the [[direction|directed set]] $\mathcal{P}_{fin}I$ of finite subsets of $I$ (ordered by inclusion) with the map
$$ \prod_{i \in F} A_i \to \prod_{i \in G} A_i ,$$
where $F \subseteq G$, given by
$$ \prod_{i \in F} A_i \cong \prod_{i \in F} A_i \times \prod_{i \in G \setminus F} 1 \stackrel{(id, 0)}{\to} \prod_{i \in F} A_i \times \prod_{i \in G \setminus F} A_i \cong \prod_{i \in G} A_i .$$

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

If it exists, the __weak direct product__ $\prod^wk_i A_i$ is defined to be the [[directed limit|directed colimit]] of this direct system.
=--

## Examples

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

In the categories [[Grp]] or [[Ab]] of ([[abelian group|abelian]]) [[groups]], the direct sum and weak direct product agree.  For finitely many objects, it is the same as the [[direct product]], which is the [[product]] in both categories.

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

In [[Ab]], where finite products are also finite [[coproducts]], the direct sum continues to be the coproduct, while in [[Grp]], it lies between the coproduct (the [[free product]]) and the product.

So in [[Ab]] the direct sum is the object equipped with a collection of morphisms

\begin{center}
\begin{tikzcd}
A_j \arrow[rd, "\iota_j"'] & \dots                   & A_k \arrow[ld, "\iota_k"] \\
                           & \bigoplus_{i \in I} A_i &
\end{tikzcd}
\end{center}

which is characterized up to unique [[isomorphism]] by the following [[universal property]]: for every other abelian group $K$ equipped with maps

\begin{center}
\begin{tikzcd}
A_j \arrow[rd, "f_j"'] & \dots & A_k \arrow[ld, "f_k"] \\
                       & K     &
\end{tikzcd}
\end{center}

there is a unique homomorphism $\phi : \bigoplus_{i \in I} A_i \to K$ such that
$ f_i = \phi \circ \iota_i$ for all $i \in I$.

=--

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

In these examples, the direct sum can also be described in more elementary terms as a [[subgroup]] of the direct product:

$$
  \bigoplus_{i: I} A_i =
  \left\{
    (a_i)_{i : I} \;|\; ess \forall (i: I),\; a_i = 0
   \right\}
  \,,
$$

where '$ess \forall$' means 'for all but finitely many'.  This makes it clear that the direct sum equals the direct product when there are only finitely many objects involved.

For $\mathcal{C} = $ [[Ab]], $R$[[Mod]] this is the group of [[formal linear combinations]] of elements in the summands.

=--

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

For $R$ a [[ring]], the direct sums in the category $R$[[Mod]] or [[modules]] over $R$ are given by those on the underlying abelian groups.

=--

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

In the category of [[pointed sets]], the direct sum and weak direct product are different.  The weak direct product is still given as a pointed subset of the direct product as above.  The direct sum, on the other hand, is the same as the [[wedge sum]], which is the same as the coproduct in this category.  Even for $2$ pointed sets, this is different from the weak direct product (which is, as always, the same as the product for finitely many objects).

=--

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

In the category of [[Banach spaces]] (with [[short linear maps]]), the direct sum is the $l^1$ direct sum, while the weak direct product is the $l^\infty$ direct sum.  (There is in fact a range of $l^p$ direct sums for $1 \leq p \leq \infty$, although I don\'t know what if any [[universal properties]] they all satisfy.)  In this case, the direct sum is the same as the coproduct, while the weak direct product is the same as the product even for infinitely many objects.  See [[direct sum of Banach spaces]].

=--

## Internal direct sums

Given an object $B$ and a [[family of subobjects]] $A_i$ of $B$ (or more generally a family of morphisms $A_i \to B$, or equivalently a map $\coprod_i A_i \to B$), suppose that the direct sum $\bigoplus_i A_i$ exists.  Suppose further that the map $\coprod_i A_i \to B$ factors through the map $\coprod_i A_i \to \bigoplus_i A_i$ (which means that it factors uniquely if $\coprod_i A_i \to \bigoplus_i A_i$ is [[epimorphism|epic]], as it must be in a regular category).  Finally, suppose that the (or a) quotient map $\bigoplus_i A_i \to B$ is [[isomorphism|isic]].  Then we say that $B$ is the __internal direct sum__ of the $A_i$.

In contrast, the abstractly defined direct sum $\bigoplus_i A_i$ may be called an __external direct sum__.  These terms are usually used with [[concrete categories]] where the $A_i$ may either be given independently (for an external direct sum) or as subsets of some ambient space (either $B$ or something of which $B$ is a subset) for an internal direct sum.  In too abstract a context, there is no difference: on the one hand, any internal direct sum is a fortiori isomorphic to any external direct sum; on the other hand, given an external direct sum, there is a natural map $\coprod_i A_i \to \bigoplus_i A_i$, relative to which the external direct sum is an internal direct sum.

## Related concepts

* [[direct sum of vector bundles]]/[[Whitney sum]]

* [[additive disjunction]], [[quantum parallelism]]

## References

* [[Masaki Kashiwara]], [[Pierre Schapira]]: Ntn. 8.2.5 in: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

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direct sum of Banach spaces

# Direct sums of Banach spaces
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## Idea

The concept of [[direct sum]] extends easily from [[vector spaces]] to [[topological vector spaces]]; we wish to explore a similar but more general notion in the case of [[Banach spaces]].

When taking the direct sum of two (or any [[finite set|finite number]]) of Banach spaces (i.e., in the category of Banach spaces and continuous linear maps), the only question is which [[norm]] to use; and we have a choice, entirely analogous to the choice of norms to put on the [[Cartesian space]] $\mathbb{R}^2$ (and its [[complexification|complexified]] variant $\mathbb{C}^2$): one for each [[extended real number]] $p \in [1, \infty]$ (and actually more choices than that).  In fact, this is a special case, the direct sum of two copies of the [[line]] $\mathbb{R}$ or $\mathbb{C}$.

For infinitely many summands, the naïve direct sum is not [[complete space|complete]] under any of these norms, so we must complete it, getting different results for each $p$; this is analogous to the different [[sequence space]]s $l^p$.  Again, this is a special case, a direct sum of infinitely many copies of the line.

In accordance with the last analogy, we speak of $l^p$-direct sums.  In fact, even more variety is possible, corresponding to other possible norms on standard Banach spaces.

## Definitions

Let $V$ be a [[Banach space]] equipped with a [[Schauder basis]] $B$, so that every element of $V$ may be written uniquely as an infinitary [[linear combination]] of elements of $B$.  Suppose also that that the basis is normal: the norm of any element of $B$ is $1$; and absolute:
$$ {\Big\| \sum_i a_i i \Big\|} = {\Big\| \sum_i {|a_i|} i \Big\|} $$
(where the $i$ are the elements of the basis and the $a_i$ are scalars).  The typical example is the [[sequence space]] $l^p$ (or a finitary or uncountablary version) with its usual basis.

Then given a [[family]] $W$ of Banach spaces indexed by the [[set]] $B$, the __$V$-direct sum__ of this family is a [[subspace]] of the [[direct product]] of the family, consisting of those $w$ such that this sum converges:
$$ \sum_i {\|w_i\|} i $$
(where again the $i$ are the basis vectors and $w_i$ is in the space $W_i$, with the norm of $w_i$ taken in $W_i$ and the sum taken in $V$).  Then the norm of $w$ is the norm of this sum:
$$ {\|w\|} \coloneqq {\Big\| \sum_i {\|w_i\|} i \Big\|} .$$

We may succinctly write the $V$-direct sum as follows:
$$ \bigoplus^V_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; {\Big\| \sum_i {\|w_i\|} i \Big\|} \lt \infty \Big\} .$$
Strictly speaking, the only condition on the right-hand side is that the sum exists in $V$; then of course its norm will be finite.  However, often some sense can be established for the sum outside of $V$ but then it will have no (finite) norm.

In particular, if $V = l^p$ (or a finitary or uncountablary version of such) for $1 \leq p \lt \infty$, then
$$ \bigoplus^p_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sqrt[p] {\sum_i {\|w_i\|^p}} \lt \infty \Big\} ;$$
and if $V = l^\infty$, then
$$ \bigoplus^\infty_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sup_i {\|w_i\|} \lt \infty \Big\} .$$
These are the __$l^p$-direct sum__ and __$l^\infty$-direct sum__ (which is really a special case).  In particular, we have the __$l^1$-direct sum__:
$$ \bigoplus^1_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sum_i {\|w_i\|} \lt \infty \Big\} .$$

## Properties

If [[short linear maps]] are taken as the [[morphisms]] in the category of Banach spaces, then the $l^1$-direct sum is the [[coproduct]], and the $l^\infty$-direct sum is the [[product]].

We can also consider the abstract concepts of [[direct sum]] and [[weak direct product]]; here again the $l^1$-direct sum is the direct sum, and the $l^\infty$-direct sum is the weak direct product.  (It is quite common for coproduct and direct sum to be the same, but weak direct product usually diverges from the product for infinitely many objects.  That they match up here crucially depends on [[complete space|completeness]].)

If every Banach space in a direct sum is a [[Hilbert space]], then their $l^2$-direct sum is also a Hilbert space.  This is the standard notion of __direct sum of Hilbert spaces__.  In [[Hilb]], this the abstract [[direct sum]], the [[weak direct product]], and the [[coproduct]].  Thus for finitely many objects, it is a [[biproduct]] (so $Hilb$ behaves rather like [[Vect]]).

Any Banach space $V$ with basis $B$ is the $V$-direct sum of ${|B|}$ copies of the line ($\mathbb{R}$ or $\mathbb{C}$).

## Direct integrals

As $l^p$ is the [[Lebesgue space]] $L^p$ for a [[measure space]] with [[counting measure]], and infinitary sums are simply the [[integrals]] on such a measure space, we may generalise from direct sums of Banach spaces to their [[direct integral]]s.  This is particularly common (using $p = 2$) for [[Hilbert spaces]].

## References

Here\'s something about direct sums of finitely many Banach spaces using norms (on $\mathbb{C}^n$) *other* than the usual $l^p$-norms:

*  Kato, Saito, Timura (2003); On $\psi$-direct sums of Banach spaces and convexity; Journal of the Australian Mathematical Society 75, 413--422; [web](http://www.austms.org.au/Publ/Jamsa/V75P3/n57.html)

Here, $\psi$ is the norm, viewed as a [[convex function]] of multiple arguments.

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[[!redirects l_∞-direct sums]]
[[!redirects l? direct sum]]
[[!redirects l? direct sums]]
[[!redirects l-∞ direct sum]]
[[!redirects l-∞ direct sums]]
[[!redirects l^? direct sum]]
[[!redirects l^? direct sums]]
[[!redirects l_? direct sum]]
[[!redirects l_? direct sums]]

[[!redirects direct integral]]
[[!redirects direct integrals]]


direct sum of vector bundles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $E_1, E_2 \to X$ two [[vector bundles]], their [[direct sum]] over $X$, also called their **Whitney sum**, is the vector bundle $E_1 \oplus E_2 \to X$ whose [[fiber]] over any $x \in X$ is the [[direct sum]] of vector spaces of the fibers of $E_1$ and $E_2$ (the [fiber-wise](topological+vector+bundle#FiberwiseOperations) direct sum).

## Definition

+-- {: .num_defn #DirectSumOfTopologicalVectorBundlesViaTotalspaces}
###### Definition
**([[direct sum]] of [[topological vector bundles]] via total spaces)**

Let

1. $X$ be a [[topological space]],

1. $E_1 \overset{\pi_1}{\to} X$ and $E_2 \overset{\pi_2}{\to} X$ two [[topological vector bundles]] over $X$.

Then the _direct sum of vector bundles_ $E_1 \oplus_X E_2 \to E$ is the [[topological vector bundle]] whose total space is the [[topological subspace]]

$$
  E_1 \oplus_X E_2
  \;\coloneqq\;
  \left\{
    (v_1, v_2)
    \in
    E_1 \times E_2
    \,\vert\,
    \pi_1(v_1) = \pi_2(v_2)
  \right\}
  \;\subset\;
  E_1 \times E_2
$$

of the [[product topological space]] of the two total spaces, and whose projection map is

$$
  \array{
    E_1 \oplus_X E_2
      &\overset{\phantom{AA}\pi\phantom{AA}}{\longrightarrow}&
    X
    \\
    (v_1,v_2) &\overset{\phantom{AAA}}{\mapsto}& \pi_1(v_1) = \pi_2(v_2)
  }
  \,.
$$

For $x \in X$ the vector space structure on the [[fibers]]

$$
  (E_1 \oplus E_2)_x \simeq (E_1)_x \oplus (E_2)_x
$$

is the one on the [[direct sum]] of [[vector spaces]].

=--

+-- {: .num_defn #DirectSumOfTopologicalVectorBundlesViaTransitionFunctions}
###### Definition
**([[direct sum]] of [[topological vector bundles]] via [[transition functions]])**

Let $X$ be a [[topological space]], and let $E_1 \to X$ and $E_2 \to X$ be two [[topological vector bundles]] over $X$.

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] with respect to which both vector bundles locally trivialize (this always exists: pick a [[local trivialization]] of either bundle and form the joint [[refinement]] of the respective [[open covers]] by [[intersection]] of their patches). Let

$$
  \left\{
    (g_1)_{i j}
    \colon U_i \cap U_j \to GL(n_1)
  \right\}
  \phantom{AAA}
  \text{and}
  \phantom{AAA}
  \left\{
    (g_2)_{i j}
    \colon
     U_i \cap U_j \longrightarrow GL(n_2)
  \right\}
$$

be the [[transition functions]] of these two bundles with respect to this cover.

For $i, j \in I$ write

$$
  \array{
    (g_1)_{i j} \oplus (g_2)_{i j}
    &\colon&
    U_i \cap U_j
      &\longrightarrow&
    GL(n_1 + n_2)
    \\
    &&
    x
      &\overset{\phantom{AAA}}{\mapsto}&
    \left(
      \array{
         (g_1)_{i j}(x) & 0
         \\
         0 & (g_2)_{i j}(x)
      }
    \right)
  }
$$

be the pointwise [[direct sum]] of these transition functions

Then the _direct sum bundle_ $E_1 \oplus E_2$ is the one glued from this direct sum of the transition functions (by [this construction](topological+vector+bundle#TopologicalVectorBundleFromCechCocycle)):

$$
  E_1 \oplus E_2
  \;\coloneqq\;
  \left(
    \left( \underset{i}{\sqcup} U_i \right)
    \times
    \left(
      \mathbb{R}^{n_1 + n_2}
    \right)
  \right)/ \left( \left\{ (g_1)_{i j} \oplus (g_2)_{i j} \right\}_{i,j \in I} \right)
  \,.
$$

=--

## Examples

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

Let $X$ and $Y$ be [[topological spaces]], and write  $X \sqcup Y$
for their [[disjoint union space]].

Then every [[topological vector bundle]] on $X \sqcup Y$ is the direct sum of a vector bundle that has [[rank of a vector bundle|rank]] zero on $Y$ and one that has rank zero on $X$.

More explicitiy: let

$$
  i_X \colon Vect(X) \longrightarrow Vect(X \sqcup Y)
$$

and

$$
  i_Y \colon Vect(Y) \longrightarrow Vect(X \times Y)
$$

be the operations of extending a vector bundle on the other [[connected component]] by a rank-0 vector bundle, then

$$
  Vect(X) \times Vect(Y)
    \underoverset{\simeq}{ i_X \oplus_{(X \sqcup Y)} i_Y }{\longrightarrow}
  Vect(X \sqcup Y)
$$

is an [[isomorphism]] of [[isomorphism classes]] of vector bundles (and an [[equivalence of categories]] of [[Vect(X)|categories of vector bundles]] before passing to isomorphism classes).

=--

## Properties

### Whitney summands of trivial vector bundles

+-- {: .num_prop #TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}
###### Proposition
**(sub-bundles over [[paracompact spaces]] are [[direct sum of vector bundles|direct summands]])**

Let

1. $X$ be a [[paracompact Hausdorff space]],

1. $E \to X$ a [[topological vector bundle]].

Then every vector subbundle $E_1 \hookrightarrow E$ is a direct vector bundle summand, in that there exists another vector subbundle $E_2 \hookrightarrow E$ such that their direct sum of vector bundles (def. \ref{DirectSumOfTopologicalVectorBundlesViaTotalspaces}) is $E$

$$
  E_1 \oplus E_2 \simeq E
  \,.
$$

=--

([e.g. Hatcher, prop. 1.3](#Hatcher))

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

Since $X$ is assumed to be paracompact Hausdorff, there exists a [[inner product on vector bundles]]

$$
  \langle -,-\rangle
  \;\colon\;
  E \oplus_X E
    \longrightarrow
  X \times \mathbb{R}
$$

(by [this prop.](inner+product+of+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces)). This defines at each $x \in X$ the [[orthogonal complement]] $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The [[subspace]] of these orthogonal complements is readily checked to be a [[topological vector bundle]] $(E')^\perp \to X$. Hence by construction we have

$$
  E \;\simeq\; E' \oplus_X (E')^\perp
  \,.
$$

=--

+-- {: .num_prop #OverCompactHausdorffSpacesEveryTopologicalVectorBundleIsDirectSummandOfATrivialBundle}
###### Proposition
**(over [[compact Hausdorff spaces]] every vector bundle is [[direct sum of vector bundles|direct summand]] of a trivial bundle)**

Let

1. $X$ be a [[compact Hausdorff space]];

1. $E \to X$ a [[topological vector bundle]].

Then there exists another topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles (def. \ref{DirectSumOfTopologicalVectorBundlesViaTotalspaces}) of the two is a trivial vector $X \times \mathbb{R}^n$:

$$
  E \oplus \tilde E
   \;\simeq\;
  X \times \mathbb{R}^n
  \,.
$$

=--

(e.g. [Hatcher, prop. 1.4](#Hatcher), [Friedlander, ptop. 3.1](#Friedlander))

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

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which $E \to X$ has a [[local trivialization]]

$$
  \left\{
    \phi_i
      \;\colon\;
     U_i \times \mathbb{R}^n
       \overset{\simeq}{\longrightarrow}
     E\vert_{U_i}
  \right\}_{i \in I}
  \,.
$$

By compactness of $X$, there is a [[finite cover|finite sub-cover]], hence a [[finite set]] $J \subset I$ such tat

$$
  \{U_i \subset X\}_{i \in J \subset I}
$$

is still an open cover over which $E$ trivializes.

Since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]] there exists a [[partition of unity]]

$$
  \left\{
    f_i \;\colon\; X \to [0,1]
  \right\}_{i \in J}
$$

with [[support]] $supp(f_i) \subset U_i$. Hence the functions

$$
  \array{
    E\vert_{U_i}
     &\overset{\phantom{AAAA}}{\longrightarrow}&
    U_i \times \mathbb{R}^n
    \\
    v
       &\overset{\phantom{AAA}}{\mapsto}&
    f_i(x) \cdot \phi_i^{-1}(v)
  }
$$

extend by 0 to vector bundle homomorphism of the form

$$
  f_i \cdot \phi^{-1}_i
  \;\colon\;
  E
    \longrightarrow
  X \times \mathbb{R}^n
  \,.
$$

The finite pointwise [[direct sum]] of these yields a vector bundle homomorphism
of the form

$$
  \underset{i \in J}{\oplus} f_i \cdot \phi_i
  \;\colon\;
  E
    \longrightarrow
  X \times \left(
    \underset{i \in J}{\oplus}
       \mathbb{R}^n
  \right)
  \simeq
  X \times \mathbb{R}^{n \dot {\vert J\vert}}
  \,.
$$

Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle.

With this the statement follows by prop. \ref{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}.

=--

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

Prop. \ref{OverCompactHausdorffSpacesEveryTopologicalVectorBundleIsDirectSummandOfATrivialBundle} is key for the construction of [[topological K-theory]] groups on compact Hausdorff spaces.

=--

Remark : Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be vector bundles over $M$. This gives product map $E_1\times E_2\rightarrow M\times M$ which is still a vector bundle. Consider diagonal map $d:M\rightarrow M\times M$ given by $m\mapsto (m,m)$. The Whitney sum of $E_1\rightarrow M$ and $E_2\rightarrow M$ is the pull back of $E_1\times E_2\rightarrow M\times  M$ along the diagonal map $d:M\rightarrow M\times M$ which is denoted by $E_1\oplus E_2\rightarrow M$.

### Characteristic classes of Whitney sums

+-- {: .num_prop #EulerClassOfWhitneySumIsCupProductOfEulerClasses}
###### Proposition
**([[Euler class]] takes [[Whitney sum]] to [[cup product]])**

The Euler class of the [[Whitney sum]] of two [[orthogonal group|oriented]] [[real vector bundles]] to the [[cup product]] of the separate Euler classes:

$$
  \chi( E \oplus F )
  \;=\;
  \chi(E)  \smile \chi(F)
  \,.
$$

=--

For details see at [[Euler class]], [this Prop.](Euler+class#EulerClassOfWhitneySumIsCupProductOfEulerClasses).

## Related concepts

* [[external direct sum of vector bundles]]

* [[inner product on vector bundles]]

* [[tensor product of vector bundles]], [[external tensor product of vector bundles]], [[dual vector bundle]],

* [[tensor category]]

* [[Stiefel-Whitney class]]

* [[virtual vector bundle]], [[topological K-theory]]

* [[framed manifold]], [[Thom spectrum]]

## References

Discussion in a context of [[topological K-theory]]:

* [[Max Karoubi]], §4.8 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226** Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

* {#Hatcher} [[Allen Hatcher]], section 1.1 of _Vector bundles and K-Theory_, (partly finished book) [web](http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html)

and with an eye towards [[algebraic K-theory]];

* {#Friedlander} [[Eric Friedlander]], _An introduction to K-theory_ (emphasis on [[algebraic K-theory]]), 2007 ([pdf](http://users.ictp.it/~pub_off/lectures/lns023/Friedlander/Friedlander.pdf))

[[!redirects Whitney sum]]

[[!redirects direct sums of vector bundles]]


directed (n,r)-graph

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

# Contents
* table of contents
{: toc}

## Idea

The underlying higher [[directed graph]] of an [[(n,r)-category|$(n, r)$-category]].

## Definition

### Directed (n,r)-pseudographs

For finite $r$, directed $(n,r)$-pseudographs are defined inductively as follows:

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

For $-2 \leq n \leq \infty$, a **directed $(n,0)$-pseudograph** is an [[∞-groupoid]] that is [[n-truncated]]: an [[n-groupoid]].

For $0 \leq r \lt \infty$, a **directed $(n+1,r+1)$-pseudograph** is an (n+1)-groupoid $G$ such that for every object $A$ and $B$,  called a **vertex**, in $G$, there is a directed $(n,r)$-pseudograph of **edges** $Edge(A, B)$ between $A$ and $B$.

where $\infty + 1 = \infty$
=--

For the case $r = \infty$, directed $(n, \infty)$-pseudographs are [[coinductive definition|defined coinductively]] as follows:

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

For $-2 \leq n \leq \infty$, a **directed $(n,\infty)$-pseudograph** is an [[n-groupoid]] $G$ such that for every object $A$ and $B$, called a **vertex**, in $G$, there is a directed $(n,\infty)$-pseudograph of **edges** $Edge(A, B)$ between $A$ and $B$.
=--

### Directed (n,r)-multigraphs

For finite $r$, directed $(n,r)$-multigraphs are defined inductively as follows:

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

For $-2 \leq n \leq \infty$, a **directed $(n,0)$-multigraph** is an [[∞-groupoid]] that is [[n-truncated]]: an [[n-groupoid]].

For $0 \leq r \lt \infty$, a **directed $(n+1,r+1)$-multigraph** is an (n+1)-groupoid $G$ such that for every object $A$ and $B$,  called a **vertex**, in $G$, there is a directed $(n,r)$-multigraph of **edges** $Edge(A, B)$ between $A$ and $B$ such that given any vertex $A$, $Edge(A, A)$ is equivalent to the empty $\infty$-groupoid.

where $\infty + 1 = \infty$
=--

For the case $r = \infty$, directed $(n, \infty)$-multigraphs are [[coinductive definition|defined coinductively]] as follows:

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

For $-2 \leq n \leq \infty$, a **directed $(n,\infty)$-multigraph** is an [[n-groupoid]] $G$ such that for every object $A$ and $B$, called a **vertex**, in $G$, there is a directed $(n,\infty)$-multigraph of **edges** $Edge(A, B)$ between $A$ and $B$ such that given any vertex $A$, $Edge(A, A)$ is equivalent to the empty $\infty$-groupoid.
=--

### Note on terminology

Some authors use "directed (n,r)-graph" to mean what we refer here as a "directed (n,r)-pseudograph".

## The periodic table

### Directed $(n,r)$-pseudographs

There is a [[periodic table]] of directed $(n,r)$-pseudographs:

$r$↓\$n$→	$-2$	$-1$	$0$	$1$	$2$	...	$\infty$

$0$	trivial	[[truth value]]	[[set]]	[[groupoid]]	[[2-groupoid]]	...	[[infinity groupoid]]

$1$	"	"	[[directed loop graph]]	[[directed pseudograph]]	directed (2,1)-pseudograph	...	directed (infinity,1)-pseudograph

$2$	"	"	"	directed loop 2-graph	directed (2,2)-pseudograph	...	directed (infinity,2)-pseudograph

$3$	"	"	"	"	directed loop 3-graph	...	directed (infinity,3)-pseudograph

⋮	⋮	⋮	⋮	⋮	⋮	⋱	⋮

$\infty$	trivial	truth value	directed loop graph	directed loop 2-graph	directed loop 3-graph	...	directed loop infinity-graph

### Directed $(n,r)$-multigraphs

There is a [[periodic table]] of directed $(n,r)$-multigraphs:

$r$↓\$n$→	$-2$	$-1$	$0$	$1$	$2$	...	$\infty$

$0$	trivial	[[truth value]]	[[set]]	[[groupoid]]	[[2-groupoid]]	...	[[infinity groupoid]]

$1$	"	"	[[directed graph]]	[[directed multigraph]]	directed (2,1)-multigraph	...	directed (infinity,1)-multigraph

$2$	"	"	"	directed 2-graph	directed 2-multigraph	...	directed (infinity,2)-multigraph

$3$	"	"	"	"	directed 3-graph	...	directed (infinity,3)-multigraph

⋮	⋮	⋮	⋮	⋮	⋮	⋱	⋮

$\infty$	trivial	truth value	directed graph	directed 2-graph	directed 3-graph	...	directed infinity-graph

## See also

* [[directed pseudograph]]

* [[pseudograph]]

* [[n-graph]]

* [[(n,r)-category]]

* [[coinduction]]

[[!redirects directed (n,r)-graphs]]
[[!redirects directed (n,r)-multigraph]]
[[!redirects directed (n,r)-multigraphs]]
[[!redirects directed (n,r)-pseudograph]]
[[!redirects directed (n,r)-pseudographs]]


Directed Algebraic Topology

[[!redirects Directed Homotopy Theory]]
This entry is about the book

* [[Marco Grandis]], _Directed algebraic topology, Models of non-reversible worlds_, New Mathematical Monographs __13__, Cambridge Univ. Press 2009 (x+434 pages)

is the first monograph on the subject of [[directed space]]s and [[directed homotopy theory]].

See CUP [ad](http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=978052176036); [pdf excerpt](http://assets.cambridge.org/97805217/60362/excerpt/9780521760362_excerpt.pdf) of the first 9 (well, 10) pages of intro; and [pdf of the index](http://assets.cambridge.org/97805217/60362/index/9780521760362_index.pdf).

A [free online version](http://www.dima.unige.it/~grandis/BkDAT_page.html) of the book is now also available (as of November 2011) by agreement with CUP.

_Contents_

* Introduction;

* Part I. First Order Directed Homotopy and Homology: 1. Directed structures and first order homotopy properties; 2. Directed homology and noncommutative geometry; 3. Modelling the fundamental category;

* Part II. Higher Directed Homotopy Theory: 4. Settings for higher order homotopy; 5. Categories of functors and algebras, relative settings; 6. Elements of weighted algebraic topology;

* Appendix A. Some points of category theory;

* References; Index of symbols; General index.

Prof. Grandis has posted few lines from introduction to the book  __'Directed Algebraic Topology' Models of non-reversible worlds__ to the categories list.

>Directed Algebraic Topology is a recent subject which arose in the 1990's, on the one hand in abstract settings for homotopy theory, and on the other hand in investigations in the theory of concurrent processes.
>Its general aim should be stated as modelling non-reversible phenomena. The subject has a deep relationship with category theory.

>The domain of Directed Algebraic Topology should be distinguished from the domain of classical Algebraic Topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. While the classical domain of Topology and Algebraic Topology is a reversible world, where a path in a space can always be travelled backwards, the study of non-reversible phenomena requires broader worlds, where a directed space can have non-reversible paths.

>The homotopical tools of Directed Algebraic Topology, corresponding in the classical case to ordinary homotopies, the fundamental group and fundamental $n$-groupoids, should be similarly non-reversible: directed homotopies, the fundamental monoid and fundamental $n$-categories.
>Similarly, its homological theories will take values in directed algebraic structures, like preordered abelian groups or abelian monoids. Homotopy constructions like mapping cone, cone and suspension, occur here in a directed version; this gives rise to new shapes, like (lower and upper) directed cones and directed spheres, whose elegance is strengthened by the fact that such constructions are determined by universal properties.

>Applications will deal with domains where privileged directions appear, such as concurrent processes, rewrite systems, traffic networks, space-time models, biological systems, etc. At the time of writing, the most developed ones are concerned with concurrency.

**Review**

A review of this book can be found in the [CMS Notes](https://cms.math.ca/notes/v43/n2/Notesv43n2.pdf) (2011).

category: reference


This entry is about the book

* [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], [[Samuel Mimram]], [[Martin Raussen]], __Directed Algebraic Topology and Concurrency__, Springer International Publishing
 2016 (xi+167 pages)

which presents applications of [[directed space]]s and [[directed homotopy theory]] to concurrent programs in computer science.

It is [available from the editor](http://www.springer.com/book/9783319153971).

## Abstract

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis.

Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs.

The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than "just" the topological properties have to be considered: execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account.

The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.


directed colimit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Directed colimits
* table of contents
{:toc}

## Abstract definition

A __directed colimit__ is a [[colimit]] $\underset{\to}\lim F$ of a functor $F\colon J \to C$ whose [[source]] [[category]] $J$ is an (upward)-[[direction|directed set]].

More generally, for $\kappa$ a [[cardinal number|regular cardinal]] say that a **$\kappa$-directed set** $J$ is a [[poset]] in which every subset of cardinality $\lt \kappa$ has an upper bound. Then a colimit over a functor $J \to C$ is called **$\kappa$-directed colimit**.

If the directed set is an [[ordinal]], one speaks of a [[sequential colimit]].

The [[duality|dual]] notion is that of [[codirected limit]], a [[limit]] of a functor whose source is a downward-directed set.

## Terminology

Note that the terminology varies.  Especially in algebra, a directed colimit may be called an '[[inductive limit]]' or '[[direct limit]]'; it\'s also possible to distinguish these so that a direct limit may have an arbitrary (possibly undirected) [[partial order|poset]] as its source.  On the other hand, both terms are often used for arbitrary [[colimits]] as an alternative terminology.  (The corresponding dual terms are '[[projective limit]]' and '[[inverse limit]]' for [[limits]].)

Directed (co)limits were studied in algebra (as inductive and projective limits) before the general notion of limit in category theory.  The elementary definition still seen there follows.

## Concrete definition

Let $C$ be a [[category]].

An __inductive system__ in $C$ consists of a [[direction|directed set]] $I$, a family $(A_i)_{i: I}$ of objects of $C$, and a family $(f_{ij}: A_i \to A_j)_{i \leq j: I}$ of morphisms, such that:
* $f_{ii}: A_i \to A_i$ is the [[identity morphism]] on $A_i$;
* $f_{ik}: A_i \to A_k$ is the [[composition|composite]] $f_{ij} ; f_{jk}$.

Then an __inductive cone__ of this inductive system is an object $X$ and a family of __inductions__ $\iota_i: A_i \to X$ such that
$$ \iota_i = f_{ij} ; \iota_j .$$

Finally, an __inductive limit__ of the inductive system is an inductive cone $\underset{\to}\lim_i A_i$ (where both $f$ and $\iota$ are suppressed in the notation, each in its own way) which is [[universal property|universal]] in that, given any inductive cone $X$, there exists a unique morphism $u\colon \underset{\to}\lim_i A_i \to X$ such that
$$ \iota_i = \iota_i ; u $$
(where the left-hand $\iota$ is from the cone $X$ and the right-hand $\iota$ is from the limit).

Notice that an inductive system in $C$ consists precisely of a directed set $I$ and a (covariant) [[functor]] from $I$ (thought of as a category) to $C$, while an inductive cone or limit of such an inductive system is precisely a cocone or colimit of the corresponding functor.  So this is a special case of [[colimit]].

As with other colimits, an inductive limit, if any exists at all, is unique up to a given isomorphism, so we speak of [[generalized the|the]] inductive limit of a given inductive system.

## Properties {#properties}

According to 1.5 and 1.21 in the [[Locally Presentable and Accessible Categories|book]] by [[Jiří Adámek]] & [[Jiří Rosický]], a category has $\kappa$-directed colimits iff it has $\kappa$-[[filtered colimit|filtered]] ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones.

The fact that directed colimits suffice to obtain all filtered ones may be regarded as a convenience similar to the fact that all [[colimits]] can be constructed from [[coproducts]] and [[coequalizers]].

## Applications

### In algebra

An inductive limit in algebra is usually defined as a [[quotient object|quotient]] of a [[disjoint union]].  To be precise, $\underset{\to}\lim_i A_i$ is the disjoint union $\biguplus_{i: I} A_i$ with $x: A_i$ identified with $y: A_j$ if
$$ f_{ik}(x) = f_{jk}(y) $$
for some $k$.  Here it is important that $C$ is a [[concrete category]] and that $I$ is a directed set (rather than merely a [[partial order|poset]]); this construction doesn\'t generalise very well.

### In accessible category theory

The objects of an [[accessible category]] and of a [[presentable category]] are $\kappa$-directed limits over a given set of generators.

## Examples

A [[Pruefer group]] $Z_{p^\infty}$ (for $p$ a [[prime number]]) is an inductive limit of the [[cyclic group]]s $Z_{p^n}$ (for $n$ a [[natural number]]).  Here, $C$ is the category of [[group]]s, $I$ is the directed set of natural numbers, $A_i = Z_{p^i}$, and $f_{ij}: A_i \to A_j$ is induced by multiplication by $p$ (which must be proved well defined on $Z_{p^i}$ for $i \leq j$).

A [[stalk]] $F_x$ (for $F$ a [[sheaf]] on a [[topological space]] $S$ and $x$ an element of $S$) is an inductive limit of $F(U)$ (for $U$ an open neighbourhood of $x$).

## Related concepts

* [[filtered colimit]]

[[!redirects directed colimit]]
[[!redirects directed colimits]]
[[!redirects Directed colimits]]


directed graph

> This entry is a *disambiguation page*, plus some commentary that would not fit well on either of the pages [[digraph]] and [[quiver]], particularly on
related technical terms such as *oriented graph*, *bidirected graph*, and *signed graph*, with their standard meanings in modern combinatorics.

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

# Contents
* table of contents
{:toc}

The term _directed graph_ is used in both [[graph theory]] and [[category theory]]. The definition varies -- even within *one* of the two theories.

In [[graph theory]], _directed graph_ (often abbreviated to the contraction _digraph_) nowadays usually means a [[digraph]], while in [[category theory]], directed graph generally means a [[quiver]]. The basic difference is: [[quivers]] may have multiple arrows in the *same* direction (often called "parallel"), and also *loops*, while [[digraphs]] may not have any of those.

## Overview of usual technical terms

> We see one family, in the same room but hardly together. The family of signed, gain, and biased graphs is a lot like that. The Bibliography and Glossary are supposed to lead the family members to talk to one another. May they do so!  ([Thomas Zaslavsky: front page](#HomePageOfSignedGraphs))

The following table may help distinguish the various terminological conventions in current use:

| technical term | distinctive feature |  short verbal definition  | [[signature]]; and axioms, of the [[theory]] (in a narrow sense of "theory")| comments |
|---------------------|-------------|---------------|---------------|---------------|
|  directed graph |   n/a (context-dependent umbrella-term)       |  n/a |  n/a | n/a   |
|  [[quiver]] |     loops allowed, parallel arcs allowed     | presheaf on the walking quiver | [[equality| = ]], two function symbols; more than one axioms (how many is up to interpretation)  |  |
|  [[digraph]] |    no loops, no parallel arcs, but 2-cycles allowed     | binary [[irreflexive relation]]  | [[equality| = ]], one relation symbol; one axiom expressing irreflexivity; if [[constructive logic]] is required, signature might be larger, requiring an [[apartness relation]] expressing irreflexivity    |  |
|  oriented [[graph]] |   looks essentially like oriented one-dimensional [[simplicial complex]]   | undirected simple graph whose edges were replaced by one arrow each | dependent on the chosen formalization of [[graph]]; if [[graph]] is formalized as a [[relation|relation]], and the orientedness by [[antisymmetry]], then the signature is: [[equality| = ]], one relation symbol; and the axioms then are: one axiom expressing irreflexivity, one axiom expressing antisymmetry; if [[constructive logic]] is required, the signature might be larger, requiring an [[apartness relation]] expressing irreflexivity |  |
|  signed [[graph]] | undirected [[graph]] with a sign on each edge    | undirected simple [[graph]] together with a function $\{$edges$\}$ $\rightarrow$ $\{-,+\}$,not subject to any axioms | dependent on the chosen formalization of [[graph|undirected simple graph]]; if [[graph|undirected simple graph]] is formalized in one of the usual ways as a [[relation|symmetric irreflexive relation]], and the signs in the usual way as two [[signature|constant symbols]], then the signature is:  [[equality| = ]], one relation symbol, two constant symbols; and the axioms then are: one axiom expressing irreflexivity, one axiom expressing symmetry; note that no axiom to express distinctness of the two constant symbols is necessary, as this, in the usual approach, is built into the signature; if [[constructive logic]] is required, the signature might be larger, requiring an [[apartness relation]] to express irreflexivity  | beware that this should *not* in a natural way be regarded a "directed graph", though they *are* often used in *contexts* together with directed graphs and thus belong on this page; in a sense, "signed graphs" are a non-example, an *odd-one-out*, in particular in that they seem tricky to categorically construct |
|  bidirected [[graph]] |  [[quiver]] with a sign on each vertex-arrow-*[[incidence|incidence relation]]*   | quiver together with function $\{$vertices$\}$ $\times$ $\{$arrows$\}$ $\rightarrow$ $\{-,+\}$, subject to axioms |   |  |

Note that there are two points of view in model theory: one regards [[equality| = ]] as a [[logic|logical symbol]] of a meta-language (e.g., first-order logic with equality), the other regards  [[equality| = ]] as a binary relation symbol defined within the language under investigation. In the table, as a reminder that equality plays an important role in these theories, the latter view was taken.

## Table of definitions of directed graphs in various references

This is to give a compressed look-up table, quickly usable, telling you what definition of directed graph is being used in the reference you happen to be working with.

| reference | kind of directed graph emphasized in the reference | formalization used; and exact reference to place in reference  | *word* used in reference | miscellaneous further comments |
|---------------------|---|-----|---------------|----------------------|
|  [BangJensenGutin2nd](#DG2nd) |    [[digraph]]               | the one in [[digraph]], except for a needless assumption of a vertex-set being non-empty: an empty vertex-set implies that the binary relation is the empty relation, but this *is* a [[digraph]] |  *digraph*  | beware that that book sometimes treats [[quivers]], but calls them *directed pseudographs* and formalizes those using [[multisets]] of ordered pairs |
| [BondyMurtyt1st](#BondyMurtyFirstEdition) | [[quiver]] | unusually hybrid functional-relational formalization via *families of ordered pairs* | *digraph*; beware that Bondy-Murty-digraphs are *not* [[digraphs]] | aversion to the void: needless assumption of vertex-sets being non-empty; essentially, Bondy-Murty-digraphs$\simeq$Schrijver-digraphs|
| [ChartrandLesniakZhang6th](#CLZ2016) | [[digraph]] | the one in [[digraph]], except for a needless assumption of a digraph being non-empty set: the empty set *is* a [[digraph]] | *digraph* | beware that what the authors call "The First Theorem of Digraph Theory" is a *not* a theorem of "Digraph Theory" in the strictest technical sense of "Digraph Theory"($=$the first-order theory of irreflexive binary relations) since it *uses the language of arithmetic*|
|  [CsabaEtAl2016](#DecompositionProof) |    [[digraph]]               | the one in [[digraph]] |  *digraph*  | |
|  [DiestelGraphTheory4th](#Diestel2010) |    [[quiver]]               | the usual one: cf. [Section 1.10](#Diestel2010), which is the "nuts-and-bolts-definition" in [[quiver]] | *directed graph*; wisely, the contraction *digraph* is used but once in main text, avoiding a clash with [[digraph]] | beware that *directed path* in the book means something different, type-theoretically at least, from *path in a [[digraph]]*: in the book, a *directed path* is itself a [[digraph]], not a sequence of vertices |
|  [HararyGraphTheory1969](#HararyGraphTheory1969) |   [[digraph]]               | the one in [[digraph]]; cf. [Chapter 16](#HararyGraphTheory1969), provided the word "collection" is read "set", which, as is abundantly clear from the context, is what Harary means; Harary is unambiguous that he does not allow digraphs to have loops; so Harary-digraphs are [[digraphs]] |  *digraph* | Harary uses [p. 199, second paragraph](#HararyGraphTheory1969) the unusual term "semiwalk" what is called "wealk walk" in [[digraph]]; note [p. 198, last paragraph](#HararyGraphTheory1969) that Harary's "path" is (modulo "paths" in [[digraph]] not being digraphs but only vertex-sequences) precisely the usual meaning of "path", although he only mentions that there be no point-repetitions: this evidently implies that there are no arc-repetitions; in summary, Harary-paths are [[digraph|paths]] |
|  [FlajoletSedgewick1st](#AnalyticCombinatorics2009) |   [[binary relation]]               | [[binary relations]], formalized in [[material set theory]]-style, cf. [V.5.1](#AnalyticCombinatorics2009) | both  *digraph* and *directed graph* used interchangeably | beware that Flajolet-Sedgwewick-digraphs are just [[binary relations]], *not* [[digraphs]] |
|  [SchrijverComOpt1st](#Schriver2003) |    [[quiver]]               | unusually hybrid formalization via *families of ordered pairs*; Bondy-Murty formalise Schrijver's *family* using a function called $\psi_D$ | *digraph*; beware that Schrijver-digraphs are *not* [[digraphs]] | no aversion to the void here; moreover, note that SchrijverDigraphs$\simeq$Bondy-Murty-digraphs  |
|  [West2002](#West2002) |    [[quiver]]           | unusually hybrid functional-relational formalization via *families of ordered pairs*: cf. [Definition 1.4.2](#West2002); West is very explicit about allowing both loops and multiple directed edges | *digraph* | beware that, as usual in the combinatorial literature, a "path" in [Definition 1.4.6](#West2002) is *itself* a quiver, i.e. does not carry any parametrization-information, i.e. is not a map like [[paths]] are  |

Note that "kind of directed graph" is a used in a loose, undefined sense, sufficiently clear from the context.

## Bidirected graphs

Roughly speaking, bidirected graphs are quivers in which *each non-loop arrow gets decorated with _two_ elements drawn from a two-element set fixed in advanced, one thought to be attached to the arrow's tail, the other to its head*.
Like for signed graphs, the elements of this two-element set are *sometimes* interpreted to be *signs* and equipped with arithmetical structure (essentially, the two-element set is taken to be a two-element [[group]]).

(The treatment of *loops* is slightly controversial in the combinatorical literature, prompting some ad-hoc-ery, and this is explicitly commented on at least once in the literature.)
The category-theoretic literature seems not to have picked up on this so far, and could clarify the issue.

Bidirected graphs are important in the theory of [[flows]] on [[graphs]].

## Connections to category theory

Herein some evident aspects are pointed out explicitly.

### Underlying quivers

A basic connection is that for any category, its [[forgetful functor|underlying]] [[quiver]] is a directed graph.

Another aspect is that a *nonempty* [[digraph]] never is isomorphic to the [[quiver]] [[forgetful functor|underlying]] any category: such a quiver *always* has a loop at each vertex, which is already enough to rule out any isomorphism.

### An application of directed graphs to category theory

Briefly, and from a slightly [[model theory|model theoretical]] point of view, the charm of this application is that [[pasting scheme|A. J. Power's pasting theorem]] is an example of a *transfer* of knowledge from a *theory without [[signature (in logic)|function symbols]]* to a *theory with [[signature (in logic)|function symbols]]*.
(The theory of [[digraphs]], strictly construed, is *purely relational*, while [[higher category theory]] makes essential use of several [[signature (in logic)|function symbols]]).

An example of a use of digraph theory in category theory is giving a rigorous justification of the  notational practice of [[pasting diagrams]].
This was achieved in the late 1980s (cf. e.g. [Johnson 1987](#Johnson1987), [Johnson 1989](#Johnson1989), [Power 1990](#Power1990)).
In this approach, parallel arcs or loops are of secondary importance, and sometimes expressly forbidden, so the work of Johnson and Power can be seen as a genuine application of graph theory to category theory.

### *All* quivers deemed irreflexive by Lawvere

There is an article of [[William Lawvere]] which is relevant in one precise respect to a disambiguation page such as the present one: *an unusual use of the adjective "irreflexive"*.
Therein, one reads (cf. ([Lawvere 1989](#GeneralizedGraphs), page 272))

> [..] the elementary "parallel process" $E= \bullet\overset{\rightarrow}{\rightarrow}\bullet$ is a reflexive graph which happens to admit only one definition of composition making it into a category $\mathbf{P}$. [..] Its [[actions]] $S^{\mathbf{P}^{\mathrm{op}}}$ are the **irreflexive** graphs (the negative is in a way appropriate even for those objects which happen to have loops at some point $p$, for morphisms are allowed to interchange *any* two such loops).

With "the negative is in a way appropriate" Lawvere explains the use of the strong negation *irr-*, despite such an "irreflexive graph" being permitted to contain loops at *some* of its vertices, which is obviously different from what one would expect from the usual sense of [[irreflexive relation]].
This is a usage in need of explanation, all the more against the backdrop of contemporary teaching where the term "simple undirected graph" is often literally defined as symmetric irreflexive relation (on the vertex set).

Some authors have adopted this interesting variant sense of "irreflexive", prompting an additional use of the a modifier "strict" to form a composite "strict irreflexive graph". (cf. ([Brown et al 2008](#GraphsOfMorphismsOfGraphs)), which uses "digraph" as a synonym for "strict irreflexive graph", and this sense of "digraph" is precisely the one in [[digraph]]).
In a sense, a [[digraph]] is a "strictly-Lawvere-irreflexive graph".

From a graph-theoretical point of view, every digraph is a quiver, but not every quiver is a digraph (again, in the sense of [Gutin and Bang-Jensen 2009](#DG2nd)).

## Related concepts

* [[digraph]]
* [[graph]]
* [[quiver]]
* [[n-quiver]]
* [[ribbon graph]]
* [[oriented graph]]
* [[signed graph]]
* [[loop digraph object]]

## References

* A. Bondy, U.S.R. Murty: Graph Theory with Applications. Fifth Printing. North Holland 1982.
{#BondyMurtyFirstEdition}

* André Bouchet: _Nowhere-Zero Integral Flows on a Bidirected Graph_. Journal of Combinatorial Theory, Series B 34, 279--292(1983)
{#Bouchet1983}

* Gary Chartrand, Linda Lesniak, Ping Zhang: _Graphs & Digraphs_. Sixth edition. CRC Press. 2016
{#CLZ2016}

* B. Csaba, D. Kühn, A. Lo, D. Osthus, A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures. Memoirs of the American Mathematical Society, 244 (2016), monograph 1154, 170 pages
{#DecompositionProof}

* [[Ronald Brown]], I. Morris, J. Shrimpton, C.D. Wensley: _Graphs of morphisms of graphs_, The Electronic Journal of Combinatorics 15 (2008), A1
{#GraphsOfMorphismsOfGraphs}

* {#BumbyLatch1986} Richard Bumby and Dana Latch, *Categorical constructions in graph theory*, lnternat. J. Math. and Math. Sci., Vol. 9 No. l (1986), 1-16. ([pdf](http://www.emis.de/journals/HOA/IJMMS/Volume9_1/791947.pdf))

* R. Diestel: _Graph Theory_. Fourth Edition. Springer (2010)
{#Diestel2010}

* [[Philippe Flajolet]], Robert Sedgewick, _Analytic Combinatorics_, 2009 ([pdf](http://algo.inria.fr/flajolet/Publications/book.pdf))
{#AnalyticCombinatorics2009}

* [[Gregory Gutin]], [[Jørgen Bang-Jensen]]: _Digraphs: Theory, Algorithms and Applications_. Springer Monographs in Mathematics. Second Edition (2009)
{#DG2nd}

* [[Frank Harary]]: _Graph Theory_. Addison Wesley. First Edition. (1969)
{#HararyGraphTheory1969}

* [[Frank Harary]]: _The notion of balance of a signed graph_.

* [[Michael Johnson]]: _Pasting Diagrams in $n$-Categories with Applications to Coherence Theorems and Categories of Paths_, Doctoral Thesis, University of Sydney, 1987
{#Johnson1987}

* [[Michael Johnson]]: _The Combinatorics of $n$-Categorical Pasting_, Journal of Pure and Applied Algebra 62 (1989)
{#Johnson1989}

* [[William Lawvere]]: _Qualitative Distinctions Between Some Toposes of Generalized Graphs_, Contemporary Mathematics 92 (1989)
{#GeneralizedGraphs}

* [[Eugene L. Lawler]]: _Combinatorial Optimization: Networks and Matroids_, Holt, Rinehart and Winston. (1976)
{#Lawler1976}

* [[John Power]]: _A 2-Categorical Pasting Theorem_, Journal of Algebra 129 (1990)
{#Power1990}

* [[Alexander Schrijver]]: _Combinatorial Optimization_. Volume A. Springer. First  Edition (2003)
{#Schrijver2003}

* Douglas B. West: _Introduction to Graph Theory_. Second Edition. Pearson Education. (2002)
{#West2002}

* Thomas Zaslavsky: _A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas_, Electronic Journal of Combinatorics, Dynamic Surveys in Combinatorics, #DS8.
{#ZaslavskyBibliography}

* Thomas Zaslavsky: [The Home Page of Signed, Gain, and Biased Graphs](http://people.math.binghamton.edu/zaslav/Bsg/index.html)
{#HomePageOfSignedGraphs}

[[!redirects bidirected graph]]
[[!redirects directed graphs]]
[[!redirects Directed graph]]
[[!redirects Directed graphs]]


directed homotopy theory

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

# Contents
* table of contents
{:toc}

##Idea##

Directed Homotopy Theory is a variant of [[homotopy theory]] which aims to study the properties of [[directed spaces]].

Much of the impetus for the theory  comes from work on modelling concurrent process. It can also be seen as a way of studying an 'evolving' space.

The following examples illustrate the sort of problems that arise:

##Example##

The example uses two directed spaces that are slightly different and use a [[pospace]], i.e. a space with a closed partial order.
(Both these use a rectangle with order $(x,y)\leq (x\prime, y\prime)$ if and only if $|y\prime - y| \leq x\prime - x$, so the [[future cone]] of any point is a cone symmetric about a horizontal line through the point and with edges at $\pm 45$ degrees to that line.)

The two spaces are

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In both the space is the rectangle with two smaller rectangles removed.  The position of the upper left small rectangle is the same in both, but that of the righthand lower rectangle is shifted slightly to the right in the second picture.  The directed homotopy classes of d-paths from $p$ to $p^\prime$ in the two cases are different. The crucial point is that in the second there is such a class that was impossible in the first example, yet the spaces are homeomorphic, so classically would be 'the same'. The subtlety is in the order.

### Problem

The first problem is to find a small model of such structures.  The [[fundamental category]] would be a model, but unlike with the [[fundamental groupoid]] in the non-directed case, it is not sufficient to take a 'base point' in each connected component.  That would ignore the order.

## Related concepts
 {#RelatedConcepts}

* [[directed homotopy type theory]]

* [[models for concurrency]];

* [[topological data analysis]] and [[persistent homology]];

* Models for mixed temporal-spatial [[modal logics]];

* [[causal sets]], discrete models for space time and [[causal models]] in Computer Science, Physics and Systems Biology;

* models for multi-agent systems, [[multimodal logics]]

## References

(See also under [[directed space]].)

Foundational work was done by [[Eric Goubault]] and his collaborators.

* [[Eric Goubault]], [[Some geometric perspectives in concurrency theory]].

Categorical aspects are looked at in

* [[Lisbeth Fajstrup]] and [[Jiří Rosický]], *A convenient category for directed homotopy*, TAC 21 no. 1, [online](http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html)

For more on this see also at _[[Delta-generated space]]_.

Applications to computer science are presented in

* [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], [[Samuel Mimram]], [[Martin Raussen]], [[Directed Algebraic Topology and Concurrency]]

The fundamental category of a pospace is discussed in

* [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], and  [[Martin Raussen]], Components of the fundamental category. Appl.Cat. Struct. Vol. 12, pp.81-108, 2004

and the possibility of an analogue of covering spaces in

* [[Lisbeth Fajstrup]],  Dicovering spaces. Algebraic topological methods in computer science (Stanford, CA, 2001). Homology Homotopy Appl. 5 (2003), no. 2, 1-17
([HHA](http://www.intlpress.com/HHA/v5/n2/))

[[Philippe Gaucher]] ([[PPS]], Paris) has introduced an interesting related model, namely that of 'flows'.  These are, approximately, topological categories without identity arrows.  They are intended as another model of processes. One of his papers on this idea is at [Arxiv](http://arxiv.org/abs/math/0308054v1), published as

*  [[Philippe Gaucher]], A model category for the homotopy theory of concurrency, Homology Homotopy and Applications, vol. 5 (1):p.549-599, 2003.

Marco Grandis' work on the area is listed amongst his publications at his ([homepage](http://www.dima.unige.it/~grandis/rec.public_grandis.html)). Such as

* [[Marco Grandis]], Directed homotopy theory. I, Cah. Topol. G eom. Di  er. Cat eg. 44 (4) (2003) 281–316.

* [[Marco Grandis]], Directed homotopy theory. II. Homotopy constructs, Theory Appl. Categ. 10 (2002) No. 14, 369–391 (electronic).

* [[Marco Grandis]], The shape of a category up to directed homotopy, Theory Appl. Categ. 15 (2005/06) No. 4, 95–146 (electronic).

* [[Marco Grandis]], Modelling fundamental 2-categories for directed homotopy, Homology, Homotopy Appl. 8 (1) (2006) 31–70 (electronic)

* [[Marco Grandis]], _[[Directed Algebraic Topology]], Models of non-reversible worlds_ , Cambridge University Press, 2009.

 A websearch will find others.

Another approach to model category structures in this area is by Kahl, who uses a Baues type fibration category approach.

*  Thomas Kahl, Relative directed homotopy theory of partially ordered spaces,
Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 79-100,([JHRS](http://jhrs.rmi.acnet.ge/volumes/2006/n1a4/ )).

[[Krzysztof Worytkiewicz]] and [[Peter Bubenik]] have given a model category structure for local pospaces:

*  P. Bubenik and [[K. Worytkiewicz]], A model category structure for local po-spaces, Homology, Homotopy and Applications, Vol. 8 (2006), No. 1, pp.263-292. ([HHA](http://intlpress.com/HHA/v8/n1/a10/)).

Further related references are

* [[L. Fajstrup]], Loops, ditopology and deadlocks, Math. Structures Comput. Sci. 10 (4) (2000) 459–480, geometry and concurrency.

* [[L. Fajstrup]], [[M. Raussen]],  [[E. Goubault]], [[E. Haucourt]], Components of the fundamental category, Appl. Categ. Structures 12 (1) (2004) 81–108, homotopy
theory.

* [[L. Fajstrup]], Dihomotopy classes of dipaths in the geometric realization of a cubical set: from discrete to continuous and back again, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), _Spatial Representation: Discrete vs. Continuous Computational Models_, no. 04351 in Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, 2005.

* [[L. Fajstrup]], Dipaths and dihomotopies in a cubical complex, Adv. in Appl.
Math. 35 (2) (2005) 188–206.

* [[M. Raussen]], Deadlocks and dihomotopy in mutual exclusion models, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), _Spatial
Representation: Discrete vs. Continuous Computational Models, no. 04351 in
Dagstuhl Seminar Proceedings_, IBFI, Schloss Dagstuhl, Germany, 2005.

* U. Fahrenberg, [[M. Raussen]], Reparametrizations of continuous paths, available
as [preprint R-2006-22](http://www.math.aau.dk/research/reports/2006.htm)  (2006).

* [[E. Goubault]], [[E. Haucourt]], Directed algebraic topology and concurrency, ([web](http://iml.univ-mrs.fr/lafont/Geocal/goubault2.pdf))
(2006).23

* [[E. Goubault]], [[Emmanuel Haucourt|E. Haucourt]], Components of the fundamental category, II,
technical reports, CEA, Saclay (2006).

* [[M. Raussen]], Invariants of directed spaces, available as preprint R-2006-28 ([web](http://bib.mathematics.dk/preprint.php?id=DMF-2006-08-002 ))(2006).

[[!redirects directed algebraic topology]]