cofree coalgebra

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#Contents#
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## Definition

Given a [[monoidal category]] $M$, let $Coalg(M)$ denote the category of coalgebras (i.e., [[comonoid|comonoids]]) with respect to the [[tensor product]] of $M$. Let $U \colon Coalg(M) \to M$ be the [[forgetful functor]]. The **cofree coalgebra** refers to a functor _[[right adjoint|right]]_ [[adjoint functor|adjoint]] to $U$, if it exists.

## Examples

For example, a cofree coalgebra over a [[vector space]] $V$ consists of a coalgebra $C(V)$ and a [[linear map]] $\varepsilon \colon C(V) \to V$ which is [[universal construction|universal]] in the sense that given a coalgebra $C$ and linear map $f \colon C \to V$, there exists a unique coalgebra map $g \colon C \to C(V)$ such that $f = \varepsilon \circ g$.

## Constructions

The [[forgetful functor]] $U \colon Coalg(M) \to M$ [[preserved colimit|preserves]] and [[reflected colimit|reflects]] ([[created limit|creates]]) any [[colimits]] which happen to exist, analogous to the fact that the forgetful functor $Alg(M) \to M$ preserves and reflects [[limits]]. Therefore, under hypotheses of an [[adjoint functor theorem]] (which almost always obtain in cases of practical interest), $U$ will indeed have a [[right adjoint]].

### Modules over a commutative ring

The special [[adjoint functor theorem]] (SAFT) may be applied to show that for any commutative ring $K$, the forgetful functor

$$K\text{-Coalg} \to K\text{-Mod}$$

has a right adjoint. An argument is given by Michael Barr in

* [Coalgebras over a Commutative Ring](http://www.math.mcgill.ca/barr/ftp/pdffiles/coalgebra.pdf); see theorem 4.1.

(This is somewhat at odds with an assertion made by Michiel Hazewinkel in [Witt Vectors, Part I](http://arxiv.org/ftp/arxiv/papers/0804/0804.3888.pdf), section 12.11 (page 58): "Whether the cofree coalgebra over an Abelian group always exists is unknown." But it seems to me ([[Todd Trimble]]) that there is nothing wrong with Barr's argument.)

### Cofree coalgebra over a vector space

In what follows, we give an explicit construction of the cofree coalgebra cogenerated by a vector space.

As a first step, recall the following result.

+-- {: .num_theorem #FundamentalTheorem}
###### Theorem
**(Fundamental theorem of coalgebras)**

Every [[coalgebra]] is the [[filtered colimit]] over the [[diagram]] of its [[finite number|finite]]-[[dimension|dimensional]] subcoalgebras and inclusions between them.

=--

This seems to be due to ([Sweedler 69](#Sweedler69)). For a proof of this result, see for example ([Michaelis 03](#WM)). For more see at _[coalgebra -- As filtered colimits](coalgebra#AsFilteredColimits)_.

We repeat that colimits of coalgebras are created by taking colimits of their underlying vector spaces.

Let for a vector space $V$, let $T(V)$ be the [[tensor algebra]] (or free algebra) generated by $V$. We may consider $T(V^\ast)$ as the algebra of (non-commutative) polynomial functions on $V$. The dual of $T(V^\ast)$ is not a coalgebra, but in accordance with the fundamental theorem above, we may consider instead the colimit over a system of finite-dimensional coalgebras obtained by dualizing the system of finite-dimensional algebra quotients of $T(V^\ast)$ and surjections between them (compare [[profinite completion]]), using the fact that the dual of a finite-dimensional algebra is a finite-dimensional coalgebra. Let $T(V^\ast)^\circ$ be this colimit coalgebra, embedded in $T(V^\ast)^\ast$. The dual of the inclusion $V^\ast \hookrightarrow T(V^\ast)$ gives a map $T(V^\ast)^\ast \to V^{\ast \ast}$, and this restricts to a map $T(V^\ast)^\circ \to V^{\ast \ast}$.

For finite-dimensional $V$, there is an isomorphism $V^{\ast \ast} \cong V$, and we have the following result.

+-- {: .num_lemma}
######Lemma
If $V$ is finite-dimensional, then the composite $T(V^\ast)^\circ \to V^{\ast \ast} \cong V$ exhibits $T(V^\ast)^\circ$ as the cofree coalgebra over $V$.
=--

+-- {: .proof}
######Proof
Suppose given a coalgebra $C$ and a linear map $f \colon C \to V$. We want to show this has a unique lift to a coalgebra map $\widehat{f} \colon C \to T(V^\ast)^\circ$. For $x \in C$, there is a minimal finite-dimensional subcoalgebra $C_x$ containing $x$, and if $f_x \colon C_x \to T(V^\ast)^\circ$ is the restriction of $f$, then the value $\widehat{f}(x)$ must match the value $\widehat{f_x}(x)$. In other words, in view of the fundamental theorem of coalgebras, there is no loss of generality in taking $C$ to be finite-dimensional.

In this case, the map $f^\ast \colon V^\ast \to C^\ast$ extends uniquely to an algebra map $T(V^\ast) \to C^\ast$, since the tensor algebra is the free algebra on $V$. This algebra map factors as

$$T(V^\ast) \stackrel{onto}{\to} Q \stackrel{into}{\to} C^\ast$$

where $Q$ is a finite-dimensional algebra. The dual $Q^\ast$ carries a coalgebra structure that embeds in $T(V^\ast)^\circ$, and by dualizing we obtain coalgebra maps

$$C \to Q^\ast \to T(V^\ast)^\circ$$

that provide the desired lift. The uniqueness of the lift is evident from the uniqueness of the extension $T(V^\ast) \to C$ (or equivalently, of $T(V^\ast)_{prof} \to C$, passing to the profinite completion) on the dual algebra side.
=--

Now let $V$ be an infinite-dimensional vector space. We may view $V$ as the union or filtered colimit over the system $\{V_\alpha\}$ of its finite-dimensional subspaces. Applying the cofree coalgebra construction to these finite-dimensional subspaces, we get a system of coalgebras.

+-- {: .num_theorem}
###### Theorem
The coalgebra colimit

$$K = colim_{V_\alpha \subset V} C(V_\alpha)$$

is the cofree coalgebra over $V$.
=--

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

Happily, this colimit is computed as in $Vect$, and more happily still -- since the underlying functor $Vect \to Set$ also preserves and creates filtered colimits -- this colimit is just a set-theoretic union of the $C(V_\alpha)$ along the canonical inclusions between them.

=--

Indeed, suppose given a coalgebra $C$ and a linear function $f \colon C \to V$. To construct a coalgebra map $C \to K$ which lifts $f$, we may argue just as we did in the proof of the lemma, and suppose without loss of generality that $C$ is finite-dimensional. Then of course the map $f \colon C \to V$ factors through a map $f_\alpha \colon C \to V_\alpha$ where $V_\alpha$ is a finite-dimensional subspace of $V$. This lifts uniquely to a coalgebra map $\widehat{f_\alpha} \colon C(V_\alpha)$, and the desired lift $\widehat{f}$ is the composite

$$C \stackrel{\widehat{f_\alpha}}{\to} C(V_\alpha) \hookrightarrow K$$

Any such lift is obtained in just this way.

It is easy to see that this description is equivalent to that described in a MathOverflow discussion [here](http://mathoverflow.net/questions/31109/is-there-an-explicit-construction-of-a-free-coalgebra/31126#31126). See also the nForum discussion [here](https://nforum.ncatlab.org/discussion/3467).

#### The cofree coalgebra over a 1-dimensional space

Specializing still further, we investigate the detailed structure of the cofree coalgebra over a 1-dimensional vector space (over a field) $k$.

Here, the cofree coalgebra $T(k)^\circ$ is the filtered colimit of finite-dimensional coalgebras of the form

$$(k[x]/I)^\ast$$

where $k[x]/I$ is a (finite-dimensional) quotient of the polynomial algebra $k[x]$ modulo a non-zero ideal $I$. The adjoint of the quotient map $k[x] \to k[x]/I$ is an embedding $(k[x]/I)^\ast \hookrightarrow k[x]^\ast \cong \prod_n k \cdot x^n$ into the space of formal power series in $x$. As a set, $T(k)^\circ$ is simply the union of the subspaces $(k[x]/I)^\ast \hookrightarrow \prod_n k \cdot x^n$. In other words, every $\xi \in T(k)^\circ$ belongs to some $(k[x]/I)^\ast$, and it suffices to consider first the structure of a typical such $\xi$ as a formal power series, and then how the comultiplication behaves on it.

Now $I$ is a principal ideal $(g(x))$ for some polynomial $g(x) = b_0 + \ldots + b_n x^n$, where the top coefficient is of course assumed non-zero. We are trying to understand when a formal power series $\xi = \sum_{n \geq 0} a_n x^n \in \prod_n k \cdot x^n$ vanishes on $I$, in the sense that

$$\langle p(x), \sum_n a_n x^n \rangle = 0$$

for all $p(x) \in (g(x))$, where the pairing $\langle , \rangle \colon k[x] \times \prod_n k \cdot x^n \to k$ is the canonical one:

$$\langle x^j, \sum_n a_n x^n \rangle = a_j.$$

Taking the case $p(x) = x^j g(x)$, we obtain recurrence relations

$$\langle b_0 x^j + \ldots b_n x^{n+j}, \sum_n a_n x^n \rangle = b_0 a_j + b_1 a_{j+1} + \ldots + b_n a_{n+j} = 0,$$

one for every $j$. We may pick $a_0, a_1, \ldots, a_{n-1}$ at random, but from there on the remainder of the sequence $a_n, a_{n+1}, \ldots$ is determined by the recurrence relations.

Now let $g^{rev}(x)$ be the polynomial obtained by reversing the order of the coefficients $b_0, \ldots, b_n$ of $g$,

$$g^{rev}(x) = b_n + \ldots + b_0 x^n,$$

and consider the problem of multiplicatively inverting $g^{rev}(x)$ (which we may do since the constant coefficient of $g^{rev}$ is non-zero). If $\sum_j a_j x^j$ is the reciprocal, we have

$$1 = g^{rev}(x)\sum_j a_j x^j = (b_n + \ldots + b_0 x^n)(\sum_j a_j x^j)$$

which yields a second set of recurrence relations by examining coefficients of powers of $x$. We get the first set of recurrence relations by restricting attention to the coefficients of $x^n$, $x^{n+1}$, etc.

Thus the power series of $g^{rev}(x)^{-1}$ yields a $\xi$ which vanishes on $I = (g(x))$, and similarly the power series of

$$\frac{x}{g^{rev}(x)}, \ldots, \frac{x^{n-1}}{g^{rev}(x)}$$

also vanish on $I$. They in fact form a basis for the subspace $(k[x]/(g(x)))^\ast \hookrightarrow \prod_n k \cdot x^n$.

We conclude that an arbitrary element of $T(k)^\circ$ is of the form $\frac{p(x)}{h(x)}$ for some $h$ with non-zero constant coefficient:

+-- {: .num_theorem}
###### Theorem
As subspaces of the space of formal power series $\prod_n k \cdot x^n$, there is an identification

$$T(k)^\circ = k[x]_{(x)}$$

where the right side is the localization of $k[x]$ at the prime ideal $(x)$, whose elements are considered as formal power series.
=--

It remains to identify the coalgebra structure on $k[x]_{(x)}$. This extends the standard coalgebra structure on the tensor algebra $k[x]$, where the comultiplication is given by deconcatenation:

$$\delta(x^N) = \sum_{m+n = N} x^m \otimes x^n.$$

We may as well focus on a typical subcoalgebra $(k[x]/(g(x)))^\ast$, with basis elements $\frac{x^i}{g^{rev}(x)}$. By duality, the definition of $\delta(\frac{x^i}{g^{rev}(x)})$ can be extracted from the multiplication table for residue classes $x^j \pmod g(x)$.

The calculations work out cleanest if we assume if $k$ is algebraically closed. For in that case, we can take advantage of partial fraction decompositions of the form

$$\frac{x^i}{g^{rev}(x)} = \sum_k \frac{A_k}{(1- r_k x)^{n_k}}$$

so that it suffices to give $\delta(\frac1{(1 - r x)^n})$, although the algebra works out better with a slightly different basis.

With this in mind, put $e_n(r) = \frac{(r x)^n}{(1 - r x)^{n+1}}$ for $n \geq 0$. The power series expansion is

$$e_n(r) = \sum_{k \geq 0} \binom{k}{n} r^k x^k$$

and we _formally_ calculate

$$\array{
\delta(e_n(r)) & = & \sum_k \binom{k}{n} r^k \delta(x^k) \\
 & = & \sum_k \binom{k}{n} r^k \sum_{p+q=k} x^p \otimes x^q \\
 & = & \sum_{p, q \geq 0} \binom{p+q}{n} r^p x^p \otimes r^q x^q \\
 & = & \sum_{p, q} \sum_{i+j=n} \binom{p}{i} \binom{q}{j} r^p x^p \otimes r^q x^q \\
 & = & \sum_{i+j=n} (\sum_p \binom{p}{i} r^p x^p) \otimes (\sum_q \binom{q}{j} r^q x^q) \\
 & = & \sum_{i+j=n} e_i(r) \otimes e_j(r)
}$$

which exhibits comultiplication as deconcatenation, familiar from the usual coalgebra structure on $k[x]$. In particular, we have $\delta(e_0(r)) = e_0(r) \otimes e_0(r)$: the elements $e_0(r)$ are _grouplike_.

A more respectable-looking calculation (but really with the exact same content!) shows that the comultiplication thus defined on $k[x]_{(x)}$ is indeed adjoint to multiplication on $k[x]$:

$$\array{
\langle x^p x^q, e_n(r) \rangle & = & \langle x^{p+q}, \sum_k \binom{k}{n} r^k x^k \rangle \\
 & = & \binom{p+q}{n} r^{p+q} \\
 & = & \sum_{i+j=n} \binom{p}{i} \binom{q}{j} r^p r^q \\
 & = & \sum_{i+j=n} \langle x^p, e_i(r) \rangle \langle x^q, e_j(r) \rangle \\
 & = & \sum_{i+j=n} \langle x^p \otimes x^q, e_i(r) \otimes e_j(r) \rangle \\
 & = & \langle x^p \otimes x^q, \delta(e_n(r)) \rangle.
}$$

An analogous calculation shows that the counit $\varepsilon \colon k[x]_{(x)} \to k$ is defined by $\varepsilon(e_n(r)) = \delta_{0 n}$ (Kronecker delta).

We summarize the calculations above as follows.

+-- {: .num_theorem}
###### Theorem
Let $k$ be an algebraically closed field. For each nonzero scalar $r \in k$, let $L(r)$ denote the $k$-linear span of the elements $e_n(r) = \frac{(r x)^n}{(1 - r x)^{n+1}}$ in $k[x]_{(x)}$, with coalgebra structure given in each case by

$$\delta(e_n(r)) = \sum_{i+j=n} e_i(r) \otimes e_j(r), \qquad \varepsilon(e_n(r)) = \delta_{0 n}$$

Let $L(0) = k[x]$ as a subspace of $k[x]_{(x)}$. The cofree coalgebra over a 1-dimensional space $k$, identified with $k[x]_{(x)}$, decomposes as a direct sum of coalgebras

$$k[x]_{(x)} \cong \bigoplus_{r \in k} L(r)$$

according to the partial fraction decomposition of rational functions. Each of the coalgebras $L(r)$ is isomorphic to $k[x]$ with its standard coalgebra structure.
=--

## Related concepts

* The [[duality|dual concepts]] are those of _[[free monoid]]_, and _[[tensor algebra]]_,

## References

* {#Sweedler69} [[Moss Sweedler]], _Hopf algebras_, 1969

* {#WM}  [[Walter Michaelis]], _Coassociative Coalgebras_, Handbook of Algebra Volume 3, Elsevier (2003).

See also

* Tom Fox, _The construction of cofree coalgebras_. JPAA 84 (1993) 191-198. [texfile here](http://www.math.mcgill.ca/fox/papers.html)

but note that this contains an error which is corrected in

* [[Michiel Hazewinkel ]] (2003), "Cofree coalgebras and multivariable recursiveness", [Journal of Pure and Applied Algebra 183 (1): 61–103](http://www.sciencedirect.com/science/article/pii/S0022404903000136)

[[!redirects cofree coalgebras]]
[[!redirects free coalgebra]]
[[!redirects free coalgebras]]


cogenerator

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#### Category theory
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[[!include category theory - contents]]
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#Contents#
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## Definition

A __cogenerator__ in a [[category]] $C$ is an [[object]] $S$ such that the [[functor]] $h_S = C(-,S) : C^{\mathrm{op}} \to \mathrm{Set}$ is [[faithful functor|faithful]]. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $S$ in the sense that
$$ \forall (\theta: Y \to S),\; \theta \circ g_1 = \theta \circ g_2 ,$$
then $g_1 = g_2$.

One often extends this notion to a __cogenerating family__ of objects, which is a (usually [[small category|small]]) set $\mathcal{S} = \lbrace S_a, a\in A\rbrace$ of objects in $C$ such that the family $C(-,S_a)$ is jointly faithful. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $\mathcal{S}$ in the sense that
$$ \forall (a: A),\; \forall (\theta: Y \to S_a),\; \theta \circ g_1 = \theta \circ g_2 ,$$
then $g_1 = g_2$.

## Examples

In [[Set]], the set of [[truth value]]s is a cogenerator.  More generally, in any [[well-pointed topos]], the [[subobject classifier]] is a cogenerator.

Much more generally, we have

+-- {: .num_prop}
###### Proposition
Every [[topos]] with a [[size issues|small set]] of [[separator|generators]] (e.g., a well-pointed topos, or a Grothendieck topos), and that has [[products]] of objects indexed over sets no larger in [[cardinality]] than the generating set, admits an [[injective object|injective]] cogenerator.
=--

+-- {: .proof}
###### Proof
Let $C$ be a set of generators for the topos; as usual, let $\Omega$ be the subobject classifier. We claim that a product

$$\prod_{c \in C} \Omega^c$$

is a cogenerator. For suppose $f, g \colon X \stackrel{\to}{\to} Y$ are distinct morphisms. The contravariant power object functor $\Omega^-$ is faithful (a familiar fact, since it is [[monadic functor|monadic]]), so that $\Omega^f, \Omega^g: \Omega^Y \stackrel{\to}{\to} \Omega^X$ are distinct. Since the objects $c$ form a [[separator|generating set]], there is some $h \colon c \to \Omega^Y$ such that the composites

$$c \stackrel{h}{\to} \Omega^Y \stackrel{\overset{\Omega^f}{\to}}{\underset{\Omega^g}{\to}} \Omega^X$$

are distinct. The map $h$ may be transformed to a map $\tilde{h}: Y \to \Omega^c$, and it follows that the two composites

$$X \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y \stackrel{\tilde{h}}{\to} \Omega^c$$

are distinct. For any other $c' \in C$, we may uniformly define $Y \to \Omega^{c'}$ to be the map classifying the maximal [[subobject]] of $c' \times Y$, so that these maps together with $\tilde{h}$ collectively induce a map

$$Y \to \prod_{c \in C} \Omega^c$$

that yields distinct results when composed with $f$ and $g$. This proves the claim.

The object $\prod \Omega^c$ is injective because already $\Omega$ is injective (see Mac Lane-Moerdijk, IV.10), and it is a general fact that in a [[cartesian closed category]] (or more generally a [[closed monoidal category]]), an exponential (or internal Hom) $X^Y$ whose base $X$ is injective is also injective, and products of injective objects are injective.
=--

Notice also that the existence of a small (co)generating family is one of the conditions in one version of the [[adjoint functor theorem]]. We may conclude, for example, that [[Grothendieck toposes]] are cototal (q.v.).

##Terminology

The concept of _cogenerator_ is dual to that of [[separator]], so it can also be referred to as a _coseparator_.

## Related concepts

* [[generator]]

* [[separator]]

* **cogenerator**

[[!redirects cogenerators]]

[[!redirects cogenerating set]]
[[!redirects coseparator]]


cogerm differential form

# Cogerm differential forms
* table of contents
{: toc}

## Idea

A cogerm differential form is a vast generalization of the usual exterior [[differential forms]], which includes also [[absolute differential forms]], as well as "higher" differential forms such as $d^2x$ and a commutative differential operator (at least in the case of 1-forms).

## Definition

Let $X$ be a [[set]].  Usually, $X$ will be a [[manifold]] or a [[generalized smooth space]], but the definition does not require this.

By a *[[curve]]* in $X$ we mean a function $c:(-\epsilon,\epsilon)\to X$ for some real number $\epsilon\gt 0$.  Two curves $c:(-\epsilon,\epsilon)\to X$ and $c':(-\epsilon',\epsilon')\to X$ have the same *[[germ]]* (at 0) if there exists $\epsilon'' \le min(\epsilon,\epsilon')$ such that $c$ and $c'$ agree when restricted to $(-\epsilon'',\epsilon'')$.  Let $C X$ denote the set of germs of curves in $X$, i.e. the [[quotient]] of the set of curves in $X$ by the [[equivalence relation]] "has the same germ as".

+-- {: .un_defn}
###### Definition
A (partial) **cogerm differential 1-form** on $X$ is a [[partial function]] $\omega : C X \to \mathbb{R}$.  We write the action of $\omega$ on a curve $c$ (or germ thereof) as $\langle \omega | c \rangle$.
=--

## Examples

* If $f: X\to \mathbb{R}$ is a function, then it defines a cogerm differential 1-form, also denoted $f$, by evaluation at $0$:
  $$ \langle f | c \rangle = (f\circ c)(0).$$
  This is defined on *all* germs.

* We also have the differential of $f$, denoted $\mathrm{d}f$, defined by
  $$ \langle \mathrm{d}f | c \rangle = (f\circ c)'(0)$$
  which is defined on all germs $c$ having the property that $f\circ c$ is differentiable at $0$.  In particular, if $f$ is smooth (for $X$ a [[smooth space]]), then $\mathrm{d}f$ is defined on all smooth germs.

  When $X$ is a [[differentiable manifold]], $ \langle \mathrm{d}f | c \rangle$ depends only on the [[tangent vector]] of $c$ at $0$.

* More generally, any real-valued function on the [[tangent bundle]] of $X$ can be regarded as a cogerm differential 1-form; this includes in particular all [[differential form|exterior differential 1-forms]].

* We also have the second differential $\mathrm{d}^2f$, defined by
  $$ \langle \mathrm{d}^2f | c \rangle = (f\circ c)''(0),$$
  which depends only on the 2-[[jet]] of $c$ at $0$.  We can similarly consider higher differentials which depend on higher jets.  A cogerm differential 1-form which depends only on the jet of $c$ may be called a *cojet differential 1-form*.

* For an example of a cogerm differential form that is not a cojet differential form, let $\langle{\omega|c}\rangle$ be $0$ or $1$ according as $c$ is or is not [[analytic function|analytic]] (say for $X$ the real line).  We do not know any more interesting examples.

* We can also apply arbitrary real functions: if $\phi:\mathbb{R}^n\to \mathbb{R}$ is a function (or even a partial function) and $\omega_1,\dots,\omega_n$ are cogerm differential 1-forms, then we have $\phi(\omega_1,\dots,\omega_n)$ defined by
  $$ \langle \phi(\omega_1,\dots,\omega_n) | c \rangle = \phi(\langle\omega_1|c\rangle,\dots,\langle\omega_n|c\rangle) $$

* For instance, any $\omega$ has an absolute value ${|\omega|}$.

* And if $X=\mathbb{R}^2$, then we have the "length element" $đ s = \sqrt{\mathrm{d}x^2 + \mathrm{d}y^2}$.

* Generalizing $đs$, any [[absolute differential form|absolute differential 1-form]] can be regarded as a cogerm differential 1-form.

* Any [[symmetric bilinear form]] on tangent vectors can also be regarded as a [[quadratic form|quadratic]] covector form and therefore a quadratic cogerm form.  In particular, this applies to the [[Riemannian metric|metric]] $g$ on any (pseudo)-[[Riemannian manifold]].  We then have $g = đs^2$ in the algebra of cogerm forms on a Riemannian manifold.

* For any function $f$, we have a cogerm differential form $\Delta f$, defined by $f(x+\mathrm{d}x)-f(x)$.  When $f$ is the [[Heaviside function]], $\Delta f$ is a candidate for the [[Dirac delta function]].

* A somewhat better way to represent the [[Dirac delta function]] as a cogerm differential form is
  $$ \delta =
  \begin{cases}
    \frac{1}{|\mathrm{d}x|} &\qquad x=0 \;and\; \mathrm{d}x\neq 0\\
    0 &\qquad otherwise.
  \end{cases}
  $$
  In the section on integration below we will see why this deserves the name "delta function".

## The cogerm differential

For any curve $c$ and real number $h$, let $c_h$ be the curve defined by $c_h(t) = c(t+h)$.  If $\omega$ is any cogerm differential 1-form, then we define its **differential** $\mathrm{d}\omega$ by

$$ \langle \mathrm{d}\omega | c \rangle = \lim_{h\to 0} \frac{\langle \omega | c_h \rangle - \langle \omega | c\rangle}{h}. $$

Note that while the fraction on the right-hand side only makes sense for a specific representative curve $c$ and a sufficiently small $h$, the limit depends only on the germ of $c$.

Of course, in general, $\mathrm{d}\omega$ may not be defined on as many curves as $\omega$ is.

+-- {: .un_defn}
###### Definition
If $X$ is a smooth space, so that we have a notion of *smooth curve*, then we say that a cogerm differential 1-form $\omega$ is **smooth** if

1. $\omega$ is defined on all germs of smooth curves, and
2. $\mathrm{d}\omega$ is smooth ([[coinductive definition|coinductively]]).
=--

For example, if $f:X\to \mathbb{R}$ is a [[smooth function]], then it is also a smooth cogerm differential 1-form, and its differentials $\mathrm{d}^n f$ are those mentioned above.  By stopping the coinduction at an appropriate point and requiring only [[continuous function|continuity]], we obtain an analogous definition of $k$-times [[continuously differentiable function]] $C^k$; for example, $\omega$ is $C^2$ if $\omega$, $\mathrm{d}\omega$, and $\mathrm{d}^2\omega$ are defined on all smooth curves and the last of these is continuous.  (But the obvious definition of [[differentiable function]] without continuity is too weak, even if we require $\mathrm{d}\omega$ to be defined on all differentiable germs, as the classic example of $x y^2/(x^2 + y^2)$ shows; extended continuously to the origin, this and its differential are defined on all smooth curves, although the differential is not continuous, but this function is not differentiable, which we know because its differential is not linear as a covector form.)

Note that the cogerm differential $\mathrm{d}$ is *not* the same as the exterior differential, except when applied to $0$-forms.  In particular, $\mathrm{d}^2 \neq 0$.

We have the expected multivariable chain rule:

+-- {: .un_theorem}
###### Theorem
If $\phi:\mathbb{R}^n\to \mathbb{R}$ is a [[differentiable function]] and $\omega_1,\dots,\omega_n$ are cogerm differential 1-forms, then
$$ \mathrm{d}(\phi(\omega_1,\dots,\omega_n)) = \partial_1\phi(\omega_1,\dots,\omega_n) \mathrm{d}\omega_1 + \cdots +  \partial_n\phi(\omega_1,\dots,\omega_n) \mathrm{d}\omega_n $$
where $\partial_i\phi$ denotes the partial derivative with respect to the $i^{th}$ variable.
=--

In particular, taking $\phi(x,y) = x y$, we have the product rule:

$$ \mathrm{d}(\omega\eta) = \omega \mathrm{d}\eta + \eta \mathrm{d}\omega. $$

This also enables us to calculate the iterated cogerm differentials of functions.  If $x:X\to \mathbb{R}$ is a "coordinate" and $f:\mathbb{R}\to \mathbb{R}$ is a function, so that $f(x) : X\to \mathbb{R}$ is a function of that coordinate, then by the theorem we have

$$\mathrm{d}f(x) = f'(x) \mathrm{d}x.$$

Thus, by the product rule, we have

$$ \mathrm{d}^2 f(x) = f''(x) \mathrm{d}x^2 + f'(x) \mathrm{d}^2x $$

and so on.  Note that the first formula justifies the common notation $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ for the derivative $f'(x)$, while the second *almost* justifies the common notation $\frac{\mathrm{d}^2f(x)}{\mathrm{d}x^2}$ for the second derivative $f''(x)$ --- it would be correct only if $\mathrm{d}^2x=0$, which is not generally the case.  Instead it would be better to write $f''(x) = \frac{\partial^2f(x)}{\partial x^2}$, indicating that $f''(x)$ is the coefficient of $\mathrm{d}x^2$ in a canonical expansion of $\mathrm{d}^2 f(x)$.

## Integration

Let $c:[a,b] \to X$ be a curve and $\omega$ a cogerm differential 1-form; we would like to integrate $\omega$ over $c$.  There are at least two possible definitions.

### Naive integration

The **naive integral** of $\omega$ over $c$ is defined to be

$$\int_c \omega = \int_{t=a}^b \langle \omega | c_t \rangle \mathrm{d}t. $$

if this exists.  If $\omega$ is an exterior differential $1$-form or an absolute differential $1$-form, then this agrees with its usual line integral over $c$.

### Genuine integration

The **(genuine) integral** of $\omega$ over $c$ is defined as follows.  Given a [[tagged partition]] $a = t_0 \lt t_1 \cdots \lt t_{n-1} \lt t_n = b$ with tags $t^*_i \in [t_{i-1},t_i]$, we define the corresponding [[Riemann sum]] to be

$$ \sum_{i=1}^n \langle \omega | \Delta t_i \cdot c_{t^*_i} \rangle $$

Here $\Delta t_i = t_i - t_{i-1}$, and for a curve $c$ and a number $h$ the curve $h\cdot c$ is defined by $(h\cdot c)(t) = c(h t)$.  Now we take the limit as the tagged partitions shrink.

It is convenient to do this in the manner of the [[Henstock integral]].  That is, we consider *gauges* $\delta : [a,b] \to \mathbb{R}_{\gt 0}$ and say that a tagged partition is *$\delta$-fine* if $[t_{i-1},t_i] \subset (t^*_i - \delta(t^*_i), t^*_i + \delta(t^*_i))$ for all $i$.  Then we say that a number $I$ is the integral $\int_c \omega$ if for all $\epsilon\gt 0$ there exists a gauge $\delta$ such that the Riemann sum of any $\delta$-fine tagged partition is within $\epsilon$ of $I$.

The genuine integral also agrees with the usual integral for exterior 1-forms, and probably also for absolute differential forms.  However, in other cases it disagrees with the naive integral.  In particular, it "detects only the degree-1 part" of a form, in a way that we can make precise as follows.

### Integration of negligible forms

Let us say that $\omega$ is **$o(dx)$** if $\lim_{h\to 0} \frac{\langle \omega | h\cdot c \rangle}{h} = 0$ for any curve $c$.  Some examples of forms that are $o(dx)$ are $\mathrm{d}x^2$ and $\mathrm{d}^2x$.

+-- {: .un_theorem}
###### Theorem
If $\omega$ is $o(dx)$, then $\int_c \omega = 0$ for any curve $c$.
=--
+-- {: .proof}
###### Proof
Suppose $\epsilon\gt 0$; then for any $t\in [a,b]$ there is a $\delta(t)$ such that ${\langle \omega {|} h \cdot c_t\rangle} \lt \frac{\epsilon}{b-a} h$ for any $0\lt h\lt \delta(t)$. This defines a gauge $\delta$ on $[a,b]$. Now suppose we have a $\delta$-fine tagged partition $a = x_0 \lt \cdots \lt x_n = b$ with tags $t_i \in [x_{i-1},x_i]$, so that $\Delta x_i = x_i - x_{i-1} \lt \delta(t_i)$. Then the corresponding Riemann sum is, by definition, $\sum_{i} \langle \omega {|} \Delta x_i \cdot c_{t_i}\rangle $. Since $\Delta x_i \lt \delta(t_i)$, each $\langle \omega {|} \Delta x_i \cdot c_{t_i}\rangle \lt \frac{\epsilon}{b-a} \Delta x_i$. Thus, when we sum them up, we get something less than $\frac{\epsilon}{b-a} \sum_i \Delta x_i = \frac{\epsilon}{b-a} (b-a) = \epsilon$. Thus, for any $\epsilon$ there is a gauge $\delta$ such that the Riemann sum over any $\delta$-fine tagged partition is $\lt\epsilon$; so the integral is zero.
=--

Thus, for instance, $\int_c \mathrm{d}^2 x = 0$ according to the genuine integral, while according to the naive integral it would be (by the fundamental theorem of calculus) $\langle \mathrm{d}x | c_b \rangle - \langle \mathrm{d}x|c_a \rangle = (x\circ c)'(b) - (x\circ c)'(a)$.

The answer $0$ given by the genuine integral is preferable if we regard $\mathrm{d}x$ as a "first-order change in $x$", so that $\mathrm{d}x^2$ or $\mathrm{d}^2x$ are "second-order" quantities and hence ought to be negligible.  For instance, we usually compute the area under a curve $y=f(x)$ between $x=a$ and $x=b$ by dividing the interval $[a,b]$ into subintervals of width $\Delta x$ and approximating the area for each subinterval using a rectangle, obtaining $f(x) \Delta x$.  This leads us to integrate the differential form $f(x) \mathrm{d}x$.

However, a *better* approximation would be to replace the rectangle with a trapezoid, having an area of $\frac{1}{2}(f(x+\Delta x) + f(x))\Delta x$.  At least if $f$ is differentiable, we can approximate $f(x+\Delta x)$ by $f(x) + f'(x) \Delta x$, and thus approximate the area by
$$ \frac{1}{2}(f(x) + f'(x)\Delta x + f(x))\Delta x = f(x) \Delta x + \frac{1}{2} f'(x) (\Delta x)^2. $$
This would lead us to integrate the form $f(x) \mathrm{d}x + \frac{1}{2} f'(x) \mathrm{d}x^2$.  Since this is based on a better approximation, we should expect it to give at least as good an answer.  And with the genuine integral it does give the same answer, since the additional term $\frac{1}{2} f'(x) \mathrm{d}x^2$ is $o(dx)$ and hence has vanishing integral.  But with the naive integral, it does not.

### The fundamental theorem of calculus

The naive integral satisfies a fundamental theorem of calculus for all cogerm forms: for any $\omega$ we have

$$\int_c \mathrm{d}\omega = \langle \omega| c_b\rangle - \langle \omega | c_a \rangle. $$

This follows directly from the definition of the naive integral and the cogerm differential $\mathrm{d}$.

The genuine integral does not satisfy as general a fundamental theorem of calculus as the naive integral.  It does, however, satisfy FTC for differentials of *functions*:
$$ \int_c \mathrm{d}f = f(c(b)) - f(c(a))$$
This can be proven in exactly the same way as the usual FTC for (line) integrals.

The restriction of FTC to differentials of functions is fairly natural if we recognize that FTC is a special case of the generalized [[Stokes' theorem]], which is about *exterior* differentials of forms, not the commutative cogerm differential.  It just so happens that if $f$ is a function, then the exterior differential of $f$ regarded as a 0-form agrees with the cogerm differential of $f$ regarded as a cogerm 1-form.

### Existence of integrals

The existence of naive integrals is fairly easy to verify, since they are just defined in terms of ordinary 1-variable integrals.  But the existence of genuine integrals is rather less obvious.  Here we prove that they exist for at least one reasonably general class of forms.

Recall that a *1-cojet differential 1-form* is a cogerm 1-form that depends only on the 1-jet of a curve, i.e. its value and its tangent vector.  A 1-cojet form is equivalently just a function (not necessarily linear) on the [[tangent bundle]] $T X$; we may write it as $\omega(\mathbf{x},\mathrm{d}\mathbf{x})$, where $\mathbf{x}$ is a point of $X$ (with coordinates $x$, $y$, $z$, ...) and $\mathrm{d}\mathbf{x}\in T_{\mathbf{x}}X$ a tangent vector at $\mathbf{x}$ (with coordinates $\mathrm{d}x$, $\mathrm{d}y$, $\mathrm{d}z$, ...).

+-- {: .un_theorem}
###### Theorem
Suppose $\omega$ is a 1-cojet differential form and that

1. $\omega(\mathbf{x},0) = 0$ for all $\mathbf{x}$.

2. For each $\mathbf{x}\in X$ and $\mathrm{d}\mathbf{x}\in T_{\mathbf{x}}X$ the function $\lambda h.\omega(\mathbf{x},h \, \mathrm{d}\mathbf{x}):\mathbb{R} \to \mathbb{R}$ is differentiable from the right at $h=0$, and the derivative of the above function is continuous as a function on $T X$.

Then the genuine integral of $\omega$ over any differentiable curve exists.
=--
+-- {: .proof}
###### Proof
The Riemann sums in the definition of the genuine integral simplify in the case of a 1-cojet form to
$$ \sum_{n=1}^\infty \omega(c(t^*_i), \Delta t_i \cdot c'(t^*_i)). $$
Note that $\Delta t_i = t_i - t_{i-1}$ is always positive.
By the assumptions, for any positive $h$ we can write
$$\omega(\mathbf{x}, h\cdot \mathrm{d}\mathbf{x}) = f(\mathbf{x}, \mathrm{d}\mathbf{x})\, h + g(\mathbf{x}, \mathrm{d}\mathbf{x},h)\,h  $$
where $f$ is continuous and $\lim_{h\to 0} g(\mathbf{x},\mathrm{d}\mathbf{x},h) = 0$.  Therefore the Riemann sum is
$$ \sum_{n=1}^\infty f(c(t^*_i), c'(t^*_i))\,\Delta t_i + g(c(t^*_i), c'(t^*_i),\Delta t_i)\,\Delta t_i. $$
Now, essentially the same argument as in the proof that $o(dx)$ forms have zero integral shows that the second term can be neglected; thus we may as well consider only
$$ \sum_{n=1}^\infty f(c(t^*_i), c'(t^*_i))\,\Delta t_i $$
However, this is just an ordinary Riemann sum for computing the integral
$$ \int_{t=a}^b f(c(t),c'(t)) \, \mathrm{d}t. $$
Since $f$ is assumed continuous, this integral exists.
=--

The hypotheses of this theorem are a bit restrictive, but they include both the usual exterior 1-forms and [[absolute differential forms]] such as ${|\mathrm{d}x|}$ and $đl = \sqrt{\mathrm{d}x^2+\mathrm{d}y^2}$.  The latter are not differentiable at $\mathrm{d}\mathbf{x}=0$, but they do have one-sided derivatives along any line approaching the origin.  It is also necessary to *exclude* forms such as the following:

* A function on $X$, regarded as a cogerm differential form.  If such a function is nonzero on any region, then its genuine integral will diverge.  These are excluded by requirement (1).

* Forms such as $\sqrt{\mathrm{d}x}$, whose genuine integral also diverges.  These are excluded by requirement (2).

There are, however, some cogerm forms that are not 1-cojet forms, nor are they $o(dx)$, yet their geniune integrals exist.  An example is $\sqrt{\mathrm{d}^2x}$, whose genuine integral over a curve $c:[a,b]\to X$ is essentially $\int_{t=a}^b \sqrt{(x\circ c)''(t)} \,\mathrm{d}t$.  There is probably a generalization of the above existence theorem to $k$-cojet forms, involving somewhat more complicated vanishing and differentiability conditions.

### Parametrization invariance

Since both naive and genuine integrals agree with ordinary line integrals when restricted to exterior 1-forms, such integrals are invariant under orientation-preserving reparametrization.  However, for general cogerm 1-forms, neither integral is so invariant.  Nevertheless, we may ask for conditions under which they are.

The naive integral is very much not invariant under reparametrization.  This is most obvious when $\omega$ is just a function, in which case its integral is just its ordinary integral.  Similarly, a naive integral like $\int_c \mathrm{d}^2 x$, which equals $(x\circ c)'(b) - (x\circ c)'(a)$ by FTC, is not invariant under reparametrization since it involves the derivative of the coordinate function of $c$.

The genuine integral is somewhat more invariant under reparametrization.  For instance,

* Since the genuine integral $\int_c \omega = 0$ whenever $\omega$ is $o(dx)$, independently of whatever $c$ might be, any such integral is parametrization-independent.

* The genuine integral of any form along any curve is invariant under orientation-preserving *affine* reparametrization.  This follows fairly directly from its definition.

There are forms whose genuine integral exists, but is not parametrization-invariant, such as $\sqrt{\mathrm{d}^2x}$.  However, we can isolate a useful class of forms, including both exterior forms and absolute forms, whose genuine integral is so invariant.  This is most easily done by reformulating the integral, as follows.

### Affine integration

For 1-cojet differential 1-forms satisfying a weak sublinearity condition, there is an equivalent definition of the genuine integral that is more obviously parametrization-invariant.

A 1-cojet form can be considered as a function on tangent vectors $(x,v)$ where $v\in T_x X$.  Let us say that such a form $\omega$ is **tangent-Lipschitz** if for each $x\in X$ there is a constant $L\gt 0$ such that
$$ {|\omega(x,v) - \omega(x,w)|} \lt L \,\Vert v-w\Vert $$
for all $v,w\in V$.  Here we are using a [[norm]] on the tangent space $T_x X$.  Since this vector space is finite-dimensional, the notion of tangent-Lipschitz is independent of the norm chosen (although the numerical value of $L$ will vary with the norm).

Note that this says nothing at all about the continuity of $\omega$ as a function of $x$.  In particular, if each $\omega(x,-)$ is separately either linear or absolute, then $\omega$ is tangent-Lipschitz.  Thus, tangent-Lipschitz is a weak replacement for linearity in the tangent variable.

To define our new version of the integral, suppose first that $X$ is a finite-dimensional real [[affine space]], regarded as a smooth manifold.  Then we can canonically identify all its tangent spaces with the same vector space $V$.

Let $c:[a,b]\to X$ be a curve in such an affine space $X$, and $\omega$ a 1-cojet differential 1-form on some open subset $U\subseteq X$ containing the image of $c$.  Then we can regard $\omega$ as a function $U\times V\to \mathbb{R}$.  Now for any [[tagged partition]] $a = t_0 \lt t_1 \cdots \lt t_{n-1} \lt t_n = b$ with tags $t^*_i \in [t_{i-1},t_i]$, we define the corresponding [[Riemann sum]] to be

$$ \sum_{i=1}^n \omega(c(t^*_i), \Delta c_i) $$

where $\Delta c_i = c(t_i) - c(t_{i-1})$, the subtraction being the usual way to subtract points in an affine space and obtain a vector.  We then take the limit of such Riemann sums in the Henstock way, as before; let us call this the **affine integral**.

Note that this Riemann sum manifestly depends only on the points $c(t_i)$ and $c(t^*_i)$ (and the order in which they occur).  Thus, any increasing bijection $\phi:[a',b'] \to [a,b]$ induces a bijection between tagged partitions which respects their Riemann sums.  It also maps intervals to intervals, so any gauge on one interval induces one on the other interval.  Thus the limits also coincide (in the strong sense that one exists if and only if the other does, and in that case they are equal), and so the affine integral is completely parametrization-independent.

We now show that the affine integral agrees with the genuine integral.

+-- {: .un_theorem}
###### Theorem
If $\omega$ is a tangent-Lipschitz 1-cojet form on an open subset of an affine space $X$, then its genuine and affine integrals over any differentiable curve agree, in the strong sense that each exists if and only if the other does and in that case they are equal.
=--
+-- {: .proof}
###### Proof
Let $c:[a,b]\to X$ be differentiable.  Then by definition of differentiability, for any $t_1 \lt t^*\lt t_2$ in $[a,b]$ we can write
$$ c(t_2) - c(t_1) - c'(t^*)(t_2-t_1) = g(t^*,t_1,t_2)(t_2-t_1) $$
where $g(t^*,t_1,t_2) \to 0$ as $t_2-t_1 \to 0$.  Thus, since $\omega$ is tangent-Lipschitz, for any tagged partition we have
$$ {|\omega(c(t^*_i), \Delta c_i) - \omega(c(t^*_i),\Delta t_i \cdot c'(t^*))|} \lt L \Vert g(t^*,t_1,t_2) \Vert \cdot \Delta t_i. $$
Since $g(t^*,t_1,t_2) \to 0$ as $t_2-t_1 \to 0$, we can choose gauges to make this difference vanish in the limit, just as we did for integrating $o(dx)$ forms.  But $\omega(c(t^*_i), \Delta c_i)$ and $\omega(c(t^*_i),\Delta t_i \cdot c'(t^*))$ are exactly the terms in the Riemann sums for the affine and genuine integrals.
=--

Finally, a similar argument shows that the affine integral can be extended to all finite-dimensional smooth manifolds by local charts.

+-- {: .un_theorem}
###### Theorem
Let $U\subseteq X$ and $V\subseteq Y$ be open subsets of affine spaces and $\phi:U\to V$ be differentiable.  Let $\omega$ be a tangent-Lipschitz 1-cojet form on $V$ and $c:[0,1]\to U$ a differentiable curve.  Then for the affine integrals we have
$$ \int_c \phi^*\omega = \int_{\phi c} \omega. $$
=--
+-- {: .proof}
###### Proof
By definition, $\phi^*\omega(x,v) = \omega(\phi(x),d_x\phi(v))$.  Thus, the LHS is a limit of Riemann sums of the form
$$ \sum_i \omega(\phi(c(t^*_i)),d_{c(t^*_i)}\phi(\Delta c_i)). $$
while the RHS is a limit of sums of the form
$$ \sum_i \omega(\phi(c(t^*_i)),\Delta (\phi c)_i). $$
Since $\phi$ is differentiable, for any point $c(t^*_i)$ we have
$$ \Delta (\phi c)_i = \phi(c(t_{i})) - \phi(c(t_{i-1})) = d_{c(t^*_i)}\phi(\Delta c_i) + E(c(t^*_i),\Delta c_i)\cdot {\Vert \Delta c_i \Vert } $$
where $E(c(t^*_i),\Delta c_i)\to 0$ as $t_i - t_{i-1} \to 0$.  Since $\omega$ is tangent-Lipschitz, this gives
$$  \Big\Vert \omega(\phi(c(t^*_i)),\Delta (\phi c)_i) - \omega(\phi(c(t^*_i)),d_{c(t^*_i)}\phi(\Delta c_i)) \Big\Vert \lt L \cdot E(c(t^*_i),\Delta c_i) \cdot {\Vert \Delta c_i \Vert }. $$
Now $L$ and $E$ depend on the point chosen, but nevertheless, we can choose gauges as before to make this term vanish in the limit.  Thus, the two integrals agree.
=--

### Stieltjes integrals, $\delta$-functions, and distributions

An alternative approach to proving FTC would be to observe that essentially by definition of $\mathrm{d}f$, the form
$$ f(x+\mathrm{d}x) - f(x) - \mathrm{d}f $$
is $o(dx)$.  Therefore, $\int_c \mathrm{d}f = \int_c (f(x+\mathrm{d}x)-f(x))$ if either exists.  If the latter integral can be calculated using only partitions tagged by their left endpoint, then it is obviously $f(c(b))-f(c(a))$ --- but it is not clear that such partitions suffice.

In the list of examples above, we denoted $f(x+\mathrm{d}x)-f(x)$ by $\Delta f$.  More generally, we might expect that integration of $\Delta f$ (perhaps multiplied by another function $g$) is a sort of [[Stieltjes integration]].  However, it is again not clear whether left endpoints suffice.

We can, at least, show that the second definition of the [[Dirac delta function]] has its expected properties.  Recall that this was
$$ \delta =
  \begin{cases}
    \frac{1}{|\mathrm{d}x|} &\qquad x=0 \;and\; \mathrm{d}x\neq 0\\
    0 &\qquad otherwise.
  \end{cases}
$$

+-- {: .un_theorem}
###### Theorem
For any function $f:\mathbb{R}\to \mathbb{R}$, if $a\le 0 \le b$ then we have $\int_a^b f(x)\, \delta \,\mathrm{d}x = f(0)$ for the affine integral.
=--
+-- {: .proof}
###### Proof
We define a gauge, denoted $\eta$ (since the letter $\delta$ is already in use) as follows:
$$\eta(x) =
\begin{cases}
  \infty &\qquad x=0\\
  {|x|} &\qquad x\neq 0.
\end{cases}
$$
Then in any $\eta$-fine tagged partition, $0$ must be the tag of the subinterval containing it.  Therefore, the Riemann sum of $f(x)\, \delta \,\mathrm{d}x$ over any such tagged partition is simply
$$ f(0) \frac{1}{|\mathrm{d}x|} \mathrm{d}x $$
where $\mathrm{d}x$ is the width of the subinterval containing $0$.  But this is positive, so the Riemann sum is simply $f(0)$.  Thus, the integral equals $f(0)$.
=--

One might instead define $\delta$ to be $1$ if $x=0$, so that we would integrate $f(x)\, \delta$ rather than $f(x) \,\delta \,\mathrm{d}x$.  Our choice matches more closely the common informal notation "$\int f(x)\, \delta(x)\,\mathrm{d}x$", although of course here $\delta$ is a function of $\mathrm{d}x$ as well as $x$.  Our choice also gives $\delta$ the usual scaling and precomposition properties, usually written as

$$ \delta(\alpha x) = \frac{\delta(x)}{|\alpha|}$$

$$ \delta(g(x)) = \sum_{g(c)=0} \frac{\delta(x-c)}{|g'(c)|}$$

In our notation, instead of $\delta(g(x))$ we would write $\delta(g(x),\mathrm{d}g)$, which is how we precompose a differential form with a function; the above formulas then follow from $\mathrm{d}g = g'(x)\,\mathrm{d}x$.

Note that $\delta$-functions are usually defined as measures or distributions, which are integrated over *unoriented* regions.  Since our integrals are oriented, we have to specify that $[a,b]$ is traversed left-to-right (as in the notation $\int_a^b$) in order to get $f(0)$ as the answer; if we integrated $\int_b^a$ instead we would get $-f(0)$.

We can, of course, avoid the minus sign by integrating against $\delta\,|\mathrm{d}x|$ instead of $\delta \,\mathrm{d}x$.  (We could also omit the absolute value in the definition of $\delta$, but this would break the usual scaling properties.)

Since we have successfully represented the $\delta$-distribution as a cogerm differential form, one may wonder whether other distributions can also be so represented.  It is unclear to me whether this is possible.  For instance, if $\phi_\epsilon$ are test functions converging to a distribution $\theta$ as $\epsilon\to 0$, we could consider a differential form such as $\theta(x,\mathrm{d}x) = \phi_{\mathrm{d}x}(x)$, but it's not clear whether this would have the right behavior under integration.

## Higher forms

One can define a **cogerm differential $k$-form** to be a function on germs of $k$-dimensional hypersurfaces in $X$.  These include in particular the usual [[differential form|exterior]] differential $k$-forms as well as [[absolute differential form|absolute]] differential $k$-forms.  And they can be integrated over $k$-dimensional hypersurfaces, by a similar formula as above.

There is a sort of "**exterior differential**" acting from cogerm $k$-forms to cogerm $(k+1)$-forms, which we denote by $(\mathrm{d}\wedge -)$ to avoid confusion with the commutative cogerm differential $\mathrm{d}$.  When $k=1$, the definition is

$$ \langle \mathrm{d}\wedge \omega {|} c \rangle = \lim_{A \to 0} \frac{1}{\Vert A \Vert} \oint_{c(\partial A)} \omega. $$

Here the limit is over some neighborhoods $A$ of $0\in\mathbb{R}^2$ whose areas $\Vert A \Vert$ shrink to zero, and the integral over the boundary is defined as above.  The definition for general $k$ is similar.

This limit can only exist if the integral $\oint_{c(\partial A)} \omega$ is invariant under oriented reparametrization of $\partial A$.  Moreover, if $\omega$ is an absolute form, then usually there will not be enough cancellation to make the limit above finite (it would be analogous to defining a 1-dimensional "derivative" as $\lim_{h\to 0} \frac{f(x+h)+f(x)}{h}$ rather than $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$).  Thus, this "exterior differential" seems barely (if at all) more general than the usual one acting on exterior $k$-forms.

It is unclear whether there is a notion of exterior differential which is significantly more general.  Similarly, it is unclear whether the wedge product of exterior forms can be sensibly extended to cogerm ones, or whether there is a sensible commutative cogerm differential $\mathrm{d}$ acting on cogerm $k$-forms for $k\gt 1$.

## Nonsymmetrized version

More general than cojet forms (but analytic), we have the _coflare differential forms_, in which there are any number of differentials (or higher differentials) $\mathrm{d}_{0}x$, $\mathrm{d}_{1}x$, $\mathrm{d}_{1,0}x = \mathrm{d}_{1}\mathrm{d}_{0}x$, etc.  Whereas jets are based on the [[tangent bundle]] $T X$, flares (of rank, say, $3$) are based on the iterated tangent bundle $T (T (T X))$.  Both cojet forms and [[exterior differential forms]] are included in the calculus of coflare forms, as are nonlinear versions of exterior forms such as [[absolute differential forms]].  The differentials in cojet forms always have the subscript $0$ (which we may take to be the default), while the exterior form $\mathrm{d}x \wedge \mathrm{d}y$ is really $\mathrm{d}_{0}x \mathrm{d}_{1}y - \mathrm{d}_{1}x \mathrm{d}_{0}y$ (possibly divided by $2!$, depending on your conventions).

A more abstract version of coflare forms (including both cogerm forms and the coflare forms of the previous paragraph) may be based on germs of maps with domain $\mathbb{R}^p$ instead of only curves (with domain $\mathbb{R}$).  To fix notation, number the coordinates on $\mathbb{R}^p$ from $0$ to $p - 1$; if $\phi$ is a partial function from $\mathbb{R}^p$ to $\mathbb{R}$ and $U$ is the subset of $\dom \phi$ on which $\phi$ is (say) $3$-times differentiable, then write $\mathrm{D}_{i,j,k}\phi\colon U \to \mathbb{R}$ for the third partial derivative of $\phi$ with respect to variable $i$, variable $j$, and variable $k$ (in any order).  Then if $c$ is a thrice-differentiable function to $X$ from a neighbourhood of the origin in $\mathbb{R}^p$ and $f$ is a thrice-differentiable function to $\mathbb{R}$ from a neighbourhood of $c(0)$ in $X$, then we have $\langle \mathrm{d}_{i,j,k}f | c \rangle \coloneqq \mathrm{D}_{i,j,k}(f \circ c)(0)$, etc.

Even more abstractly, there is no reason to limit the domain of $c$ to $\mathbb{R}^p$, although it\'s not clear if the forms on more general classes of germs correspond to anything of independent interest.

## References

* nForum discussions: [I](http://nforum.ncatlab.org/discussion/5402/what-is-a-variable/), [II](http://nforum.ncatlab.org/discussion/5518/differentials/), [III](http://nforum.ncatlab.org/discussion/5700/cogerm-forms/) (merged into [the page discussion](https://nforum.ncatlab.org/discussion/8552/)), [IV](http://nforum.ncatlab.org/discussion/5817/cojet-differential-forms/), [V](http://nforum.ncatlab.org/discussion/5941/higher-differentials-again/), [VI](https://nforum.ncatlab.org/discussion/7636/coflare-differential-forms/).

Historical reference saved for later reading:

* Henk Bos (1973). Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus. Archive for History of Exact Sciences 14:1--90. [DOI](http://dx.doi.org/10.1007%2FBF00327456). [PDF](http://www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf) hosted by Leo Corry.

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cograph of a functor

+-- {: .rightHandSide}
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The notion of _cograph of a functor_ is dual to that of [[graph of a functor]]: for
$f : C \to D$ a functor between [[n-categories]] is the fibration
classified by the [[profunctor]] correspondence $\chi_f : C^{op}\times D \to (n-1)Cat$. But $f$ also determines a morphism $\bar f : I \to n Cat$ from the [[interval category]] $I$. The **cograph** of $f$ is the fibration classified by $\bar f$.

Recall that the graph of a function $f: A \to B$ is the [[subset]] determined by the [[monomorphism]] $\langle 1, f \rangle: A \to A \times B$. This makes sense in any category with products. Under one definition, the notion of cograph of a function $f: A \to B$ is the categorically dual notion: it is the [[quotient object|quotient]] determined by the [[epimorphism]] $(f, 1): A \sqcup B \to B$.

A more vivid presentation of cograph is given by the pictures we draw of functions $f: A \to B$: as [[directed graph]]s (in the graph-theoretic sense!) whose vertices are elements in $A \sqcup B$, with an edge drawn from $a$ to $f(a)$ for each $a$ in $A$. This can also be conceived as a [[poset]] $P_f$ with underlying set $A \sqcup B$, in which $a \leq f(a)$ and all other instances of $\leq$ are the reflexive ones. The connected components function $(P_f)_0 \to \pi_0(P_f)$ is then the cograph in the sense given above.

In this article we give a definition of cograph which generalizes this poset picture of cograph of a function, and which applies to any functor between $n$-categories.

# Examples #

## Cographs of functors between 0-categories ##

In the case that $C, D$ are [[0-category|0-categories]], i.e. [[sets]], a functor $f : C \to D$
is just a [[function]] between sets. The cograph [[2-pullback]]

$$
  \array{
    cograph(f) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& Set
  }
$$

is computed by the ordinary [[pullback]]

$$
  \array{
    cograph(f) &\to& Set_{*}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& Set
  }
$$

and identifies $cograph(f)$ with the [[category of elements]] of $\bar f$,
as described there: the objects of $cograph(f)$
are the disjoint union of $C$ and $D$: $Obj(cograph(f)) = C \coprod D$
and the nontrivial morphisms are of the form $x \to y$ whenever $x \in C$, $y \in D$
and $f(x) = y$.

What [[Bill Lawvere]] called the [[cograph of a function]] is the connected components
$\pi_0(Cograph(f))$ of this category.

## Cographs of functors between 1-categories ##

For $f : C \to D$ an ordinary  [[functor]], $cograph(f)$ is the category with $Obj(cograph(f)) = Obj(C) \coprod Obj(D)$ and with

$$
  Hom_{cograph(f)}(x,y) =
  \left\{
    \array{
      Hom_C(x,y) & if\; x,y \in C
      \\
      Hom_D(x,y) & if\; x,y \in D
      \\
      Hom_D(f(x),y) & if\; x \in C ,y \in D
      \\
      \emptyset & if\; x \in D ,y \in C
    }
  \right.
$$

with composition defined as induced from $C$, from $D$, and from the action of $f$.  This is a special case of the [[cograph of a profunctor]], specialized to the representable [[profunctor]] $D(f-,-)$.

This cograph is denoted $C \star^f D$ in section 2.3.1 (_Correspondences_) in

* [[Jacob Lurie]], [[Higher Topos Theory]]

where it is understood as a generalization of the [[join of simplicial sets]]
and where it serves as a motivation for the study of cographs of
functors between [[(∞,1)-category|(∞,1)-categories]] discussed
below.

## Adjoint functors in terms of cographs
 {#AdjointFunctorsInTermsOfCographs}

As emphasized in the beginning of section 5.2 there, cographs of functors may be used to characterize [[adjoint functors]].  This is just one way of stating the characterization of adjoints in terms of [[profunctors]] (which in turn makes sense in any [[2-category equipped with proarrows]]).

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

Two functors $L : C \to D$ and $R : D \to C$ are [[adjoint functors]] precisely if $cograph(L)$ and $cograph(R^{op})^{op}$ are isomorphic under $C$ and $D$:

$$
  (L \dashv R) \Leftrightarrow
  (cograph(L) \cong cograph(R^{op})^{op})
$$

where the isomorphism is in the [[co-slice category]] $(C\sqcup D)/Cat$.
=--

More precisely, there is a bijection between adjunctions $L\dashv R$ and isomorphisms as above.

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

The category $cograph(L)$ is the category with $Obj(cograph(L)) = Obj(C) \coprod Obj(D)$ and with

$$
  Hom_{cograph(L)}(x,y) =
  \left\{
    \array{
      Hom_C(x,y) & if\; x,y \in C
      \\
      Hom_D(x,y) & if\; x,y \in D
      \\
      Hom_D(L(x),y) & if\; x \in C ,y \in D
      \\
      \emptyset & if\; x \in D ,y \in C
    }
  \right.
$$

(This set is also called the set of [[heteromorphisms]] between objects in $C$ and $D$.)

The category $cograph(R^{op})^{op}$ accordingly is the category with $Obj(cograph(R)) = Obj(C) \coprod Obj(D)$ and with

$$
  Hom_{cograph(R^{op})^{op}}(x,y) =
  \left\{
    \array{
      Hom_C(x,y) & if x,y \in C
      \\
      Hom_D(x,y) & if x,y \in D
      \\
      Hom_C(x,R(y)) & if x \in C ,y \in D
      \\
      \emptyset & if x \in D ,y \in C
    }
  \right.
$$

Evidently these categories are isomorphic under $C$ and $D$ precisely if for all $x \in C, y \in D$ we have

$$
  Hom_D(L(x),y) \cong Hom_C(x,R(y))
  \,.
$$

naturally in $x$ and $y$.  It is natural because the isomorphism is an isomorphism of categories, and the functoriality of $Hom_D(L(-),-)$ and $Hom_C(-,R(-))$ is encoded by composition in the cograph.  Of course, such a natural isomorphism is precisely the structure of an adjunction $L\dashv R$.

=--

Note also that just as $cograph(L)$ is the cograph of the profunctor $D(L-,-)$, also $cograph(R^{op})^{op}$ is the cograph of the profunctor $C(-,R-)$.  Thus, this theorem can be viewed as one way of stating the characterization of adjunctions in terms of homsets, as can be [formulated](http://ncatlab.org/nlab/show/2-category+equipped+with+proarrows#HomsetAdjn) in terms of profunctors in any 2-category equipped with proarrows.

## Cographs of functors between $(\infty,1)$-categories ##

In the context of [[(∞,1)-category]] theory there is a good theory of [[Cartesian fibrations]] $X \to S$ and of their classification by [[(∞,1)-functors]] $S^{op} \to (\infty,1)Cat$ to the [[(∞,1)-category of (∞,1)-categories]] as described at [[universal fibration of (∞,1)-categories]].

Accordingly, the above notion of cograph of a functor has a direct generalization to [[(∞,1)-functors]]:

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

For $f : C \to D$ an [[(∞,1)-functor]], identified with a morphism

$$
  \bar f : I \to (\infty,1)Cat
$$

in the [[(∞,1)-category of (∞,1)-categories]], it **cograph** is the [[Cartesian fibration]] $cograph(f) \to I$ classified by it. In terms of the [[universal fibration of (∞,1)-categories]] this is the [[homotopy pullback]]

$$
  \array{
    cograph(f) &\to& S^{op}
    \\
    \downarrow && \downarrow
    \\
    I &\stackrel{\bar f}{\to}& (\infty,1)Cat
  }
  \,.
$$

=--

As for every [[Cartesian fibration]] the functor $f : C \to D$ is determined uniquely up to equivalence by its cograph. In general, obtaining the classifying $(\infty,1)$-functor from a given [[Cartesian fibration]] may be difficult. In the special case of cographs as Cartesian fibrations over the simple [[interval category]] it is easier. This is discussed in the following:

...

## Related concepts

* [[graph of a functor]]

* **cograph of a functor**, [[cograph of a profunctor]]

## References

The notion of cographs of $(\infty,1)$-functors and the theory of how to re-obtain $(\infty,1)$-functors from their cographs is the content of section 5.2.1, _Correspondences and associated functors_,  of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .


cograph of a profunctor

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## In category theory

### Definition

Let $H\colon A ⇸B$ be a [[profunctor]], i.e. a [[functor]] $B^{op}\times A\to Set$.  Its **cograph**, also called its **collage**, is the [[category]] $\bar{H}$ whose set of objects is the [[disjoint union]] of the sets of objects of $A$ and $B$, and where

$$ \begin{aligned}
  \bar{H}(a_1,a_2) &= A(a_1,a_2)\\
  \bar{H}(b_1,b_2) &= B(b_1,b_2)\\
  \bar{H}(b,a) &= H(b,a)\\
  \bar{H}(a,b) &= \emptyset
  \end{aligned}
$$

where composition is defined as in $A$, $B$, and according to the actions of $A$ and $B$ on $H$.

The [[cograph of a functor]] is the special case when $H$ is a corepresentable profunctor of the form $B(f-,-)$ for some functor $f\colon A\to B$.

### Properties

The cograph of a profunctor can be given a universal property: it is the [[lax colimit]] of that profunctor, considered as a single arrow in the [[bicategory]] of profunctors.  (The word "collage" is also sometimes used more generally for any lax colimit, especially in a $Prof$-like bicategory.)  The cograph of a profunctor is also a [[cotabulator]] in the [[proarrow equipment]] of profunctors.  Furthermore, the [[cospans]] $A\to \bar{H} \leftarrow B$ which are cographs of profunctors can be characterized as the two-sided [[codiscrete cofibrations]] in the [[2-category]] [[Cat]].

Cographs of profunctors can also be characterized as categories equipped with a functor to the [[interval category]] $(0\to 1)$, where $B$ is the fiber over $0$ and $A$ is the fiber over $1$.  See [[joyalscatlab:Distributors and barrels]].

## In $(\infty,1)$-category theory

The notion of a cograph of a profunctor generalizes to [[(∞,1)-category theory]].

### Definition

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

For $C$ and $D$ two [[(∞,1)-categories]] a **correspondence** between them is an $(\infty,1)$-category $p : K \to \Delta[1]$ over the [[interval category]] $\Delta[1] = \{0 \to 1\}$ with an [[equivalence of (∞,1)-categories|equivalences]] $K_0 \simeq C$ and $K_1 \simeq D$.

=--

This appears as ([Lurie, def 2.3.1.3](#Lurie)).

### Properties

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

There is a canonical bijection between equivalence classes of correspondences between $C$ and $D$ and equivalence classes of [[(∞,1)-profunctors]], i.e., [[(∞,1)-functors]]

$$
  C^{op} \times D \to \infty Grpd
$$

from the product of $D$ with the [[opposite-(∞,1)-category]] of $C$ to [[∞Grpd]].

=--

This appears as ([Lurie, remark 2.3.1.4](#Lurie)).

Therefore the correspondence corresponding to a profunctor is its cograph/collage.

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

An $(\infty,1)$-profunctor comes from an ordinary [[(∞,1)-functor]] $F : C \to D$ precisely if its cograph $p : K \to \Delta[1]$ is not just an inner fibration but a [[coCartesian fibration]].

And it comes from a functor $G : D \to C$ precisely if it is a [[Cartesian fibration]]. And precisely if both is the case is $F$ the right [[adjoint (∞,1)-functor]] to $G$.

=--

Because by the [[(∞,1)-Grothendieck construction]]

* coCartesian fibrations $K \to \Delta[1]$ correspond to [[(∞,1)-functor]]s $\Delta[1] \to$ [[(∞,1)Cat]];

* [[Cartesian fibration]]s $K \to \Delta[1]$ correspond to [[(∞,1)-functor]]s $\Delta[1]^{op} \to$ [[(∞,1)Cat]].

* and as discussed at [[adjoint (∞,1)-functor]], an $(\infty,1)$-functor has an adjoint precisely if the coCartesian fibration corresponding to it is also Cartesian.

## Related concepts

* [[graph of a functor]]

* [[cograph of a functor]], **cograph of a profunctor**

* [[cotabulator]]

## References

For ordinary and [[enriched category theory|enriched]] categories, cographs were studied (and used to characterize profunctors) by:

* [[Ross Street]], "Fibrations in bicategories"

* Carboni and Johnson and Street and Verity, "Modulated bicategories"

The $(\infty,1)$-category theoretic notion ("correspondence") is the topic of section 2.3.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}

See Ross Street's post in category-list 2009, [Re: pasting along an adjunction](http://article.gmane.org/gmane.science.mathematics.categories/255/match=).

[[!redirects collage]]
[[!redirects collages]]


cogroup

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#Cogroup objects#
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## Idea ##

**Cogroup objects** are [[group object]] in an [[opposite category]], and often one takes the [[opposite category]] of group object in an opposite category to be the category of cogroup objects.

The defining property of a cogroup object is that morphisms *out* of it form a [[group]].  Specifically, if $C$ is a category, then $G$ is a cogroup object in $C$ if $\operatorname{Hom}(G,X)$ is a group for any object $X$ in $C$ (and the group structure must be natural in $X$).

There are many examples of cogroup objects.  Perhaps the most well-known are the [[n-sphere]] in the [[homotopy category]] of [[pointed topological spaces]], $\operatorname{hTop}_*$.  Then the fact that $S^n$ is a cogroup object in $\operatorname{hTop}$ is precisely the statement that the [[homotopy group]] $\pi_n(X)$ for $n \geq 1$ is indeed a [[group]], naturally in $X$, for all topological spaces $X$.

## Definition ##

The basic definition is as follows.

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

Let $C$ be a category.  To give an object $G$ of $C$ a **cogroup structure** in $C$ is to give the functor $\operatorname{Hom}(G,-)$ a [[lift]] from $\operatorname{Set}$ to $\operatorname{Grp}$.

A **cogroup object** in $C$ is an object $G$ together with a choice of cogroup structure.

A **morphism of cogroup objects** $G_1 \to G_2$ is a morphism in $C$ between the underlying objects of the $G_i$ such that the [[natural transformation]] $\operatorname{Hom}(G_2,-) \to \operatorname{Hom}(G_1,-)$ lifts to a natural transformation of functors into $\operatorname{Grp}$.
=--

Thus cogroup objects and their morphisms can be thought of as the category of [[representable functors]] from $C$ to $\operatorname{Grp}$.

Providing $C$ has enough coproducts of $G$ (the $0,1,2,3$th copowers to be precise), the concept of a cogroup structure on $G$ can be internalised.

+-- {: .num_theorem}
###### Theorem
To give an object $G$ of $C$ a cogroup structure is equivalent to choosing morphisms $\mu \colon G \to G \amalg G$, $\eta \colon G \to 0_C$, and $\iota \colon G \to G$ satisfying the diagrams for associativity, unit, and inverse but _the other way around_.
=--

Here, the phrase "the other way around" means: take the normal diagrams for a [[group object]] that express the properties of associativity, unit, and inverses, invert all the arrows, and replace products by coproducts.

## Relationship To Group Objects ##

A cogroup object in a category, say $C$, is nothing more than a [[group object]] in the [[opposite category]]: $C^{op}$.  However, morphisms in the cogroup category go the other way around.  That is to say, with the obvious notation:

$$
C\operatorname{coGrp} = (C^{op}\operatorname{Grp})^{op}
$$

## Relationship to Other Objects ##

Of course, there is nothing special about groups here.  The same style of definition works for any [[variety of algebras]] in the sense of universal algebra, where $C coAlg_T \coloneqq (C^{op}Alg_T)^{op}$.

Some terminological care should be taken in the case of [[comonoid]], which makes sense in any [[monoidal category]], not just cocartesian monoidal categories which is the general default environment for discussing co-$T$-algebras. Thus, check with the author to see which monoidal product is meant; in the case of comonoid it's likely that it's *not* the cocartesian notion that is intended. Whereas in the case of cogroups, confusion is not so likely: one needs the cocartesian structure (codiagonals, etc.) essentially because the axioms of a group involve duplication of variables, whereas this is not the case for axioms of a monoid. (Cf. the distinction between [[operad]] and [[Lawvere theory]], where the latter can be viewed as a kind of "cartesian operad".)

## Examples ##

1. All [[n-spheres]] for all $n$ are cogroup objects in the [[homotopy category]] of [[based topological spaces]], $\operatorname{hTop}_*$.

   This is the origin of the [[group]] structure on [[homotopy groups]]. it is also crucial in the structure of the [[Brown representability theorem]].

   The higher spheres are actually _abelian_ cogroup objects, as demonstrated by the fact that $\pi_n(X)$ is abelian for $n \ge 2$.

1. More generally, any [[suspension]] is a cogroup object with the "pinch" map as the comultiplication.  See at _[[suspensions are H-cogroup objects]]_. (Since the $0$-sphere is not a suspension in $\operatorname{hTop}_*$, but only in $\operatorname{hTop}$, it need not be a cogroup and in fact is not.)  This is dual to, and equivalent to, the statement that (based) [[loop spaces]] are group objects in $\operatorname{hTop}_*$ since there is an [[adjunction]], internal to $\operatorname{hTop}_*$:

   $$
   \operatorname{Hom}(\Sigma X,Y) \cong \operatorname{Hom}(X,\Omega Y)
   $$

1. There are examples of spaces that are cogroups in $\operatorname{hTop}_*$ that are **not** suspensions, see Bernstein & Harper _Cogroups which are not suspensions_.  Note that cogroups in $\operatorname{hTop}_*$ are the same as [[co-H-spaces]] which are additionally (co-)associative and have (co-)inverses.

1. Cogroup objects in the [[Grp|category of groups]] are [[free groups]], and to give a free group the structure of a cogroup object is the same a choosing a generating set.  This is an old result of [[Daniel Kan]].

1. On the other hand, every abelian group is again an abelian cogroup since $\operatorname{Ab}$ is self-enriched.  Indeed, in an [[abelian category]] every object is simultaneously an abelian group object and an abelian cogroup object.  In $\operatorname{Ab}$, the abelian cogroup object structure is unique, with comultiplication given by the [[diagonal morphism]].

1. In [[Set]], the only cogroup object (abelian or otherwise) is the [[empty set]].  This is because the counit map must be a morphism from $X$ to the terminal object _of the opposite category_.  In the case of $\operatorname{Set}$, this is the empty set.

1. This extends further: any category with a [[faithful functor]] to $\operatorname{Set}$ which preserves an [[initial object]] will have no non-trivial cogroup objects.  In particular, the category [[Top]] of *unbased* topological spaces has only the [[empty space]] as a cogroup object.

1. The case of cogroups, and some other co-things, in certain other varieties of algebras has been extensively studied by [Bergman & Hausknecht 1996](#BergmanHausknecht96).

   In particular, a co-group in the category of (unital) commutative rings is a commutative [[Hopf ring]] and a cogroup in the category of (unital) commutative $k$-algebras is a commutative [[Hopf algebra|Hopf]] $k$-algebra; a fact highlighted in homotopy theory by [[Haynes Miller]] (in view of his generalization to [[commutative Hopf algebroid]]s as cogroupoids in commutative algebra) in the context of discussion of [[dual Steenrod algebras]], see ([Ravenel 86, appendix A](#Ravenel86)) for review.

## References

Discussion of commutative [[Hopf algebras]] as cogroups:

* {#Ravenel86} [[Doug Ravenel]], appendix A1 of _[[Complex cobordism and stable homotopy groups of spheres]]_, Academic Press 1986

Cogroups [[internalization|internal to]] the category of (graded or not) [[associative algebras]] are very rare (unlike Hopf algebras) -- in fact the underlying algebras are free; this has been clear since

* Israel Berstein, On cogroups in the category of graded algebras. Trans. Amer. Math. Soc. 115 (1965), 257–269 [jstor](http://www.jstor.org/stable/1994268)

Monograph:

* {#BergmanHausknecht96} [[George M. Bergman]], Adam O. Hausknecht, _Cogroups and co-rings in categories of associative rings_, A.M.S. Math. Surveys and Monographs __45__ (1996) [ISBN 0-8218-0495-2, [ams:surv-45](https://bookstore.ams.org/surv-45), [MR 97k:16001](http://www.ams.org/mathscinet-getitem?mr=97k:16001), [errata and updates](http://math.berkeley.edu/~gbergman/papers/updates/coalg.html)]

 > (also for [[corings]])

This fact is later observed in greater generality

* [[Benoit Fresse]], _Cogroups in algebras over an operad are free algebras_, Commen. Math. Helv. __73__:4, 1998, 637-676
[[doi](http://dx.doi.org/10.1007/s000140050072)

[[!redirects cogroups]]

[[!redirects co-H-object]]

[[!redirects cogroup object]]
[[!redirects cogroup objects]]

[[!redirects co-group object]]
[[!redirects co-group]]


Cohen-Macaulay module

## Definition

A *Cohen-Macaulay module* is a [[module]] over a [[commutative]] [[noetherian]] [[local ring]] which is [[finitely generated]], non-zero, and such that its [[depth]] is equal to its [[Krull dimension]].

## References

Related notions in $n$Lab include [[Cohen-Macaulay ring]]

* Wikipedia, _[Cohen-Macaulay ring](https://en.wikipedia.org/wiki/Cohen%E2%80%93Macaulay_ring)_
* Winfried Bruns, Jürgen Herzog, _Cohen-Macaulay Rings_, Cambridge University Press (1993)
* Rangar-Olaf Buchweitz, _On maximal Cohen-Maculay modules and Tate cohomology_, (1986) 155 pages [pdf](https://tspace.library.utoronto.ca/bitstream/1807/16682/1/maximal_cohen-macaulay_modules_1986.pdf)

category: algebra


Cohen-Macaulay ring

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## Idea

Cohen-Macaulay rings are an important class of [[commutative rings]].

## Definition

A [[commutative ring]] is Cohen-Macaulay iff it is a [[Cohen-Macaulay module]] as a module on itself.

## References

* Wikipedia, _[Cohen-Macaulay ring](https://en.wikipedia.org/wiki/Cohen%E2%80%93Macaulay_ring)_
* Winfried Bruns, Jürgen Herzog, _Cohen-Macaulay Rings_, Cambridge University Press (1993)
* Ch. 18 of [[David Eisenbud]], _Commutative algebra with a view toward algebraic geometry_, Graduate Texts in Mathematics __150__

category: algebra


coherence law

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## In the definition of $(n,r)$-categories

In the definition of [[(n,r)-categories]] in [[higher category theory]] the **coherence laws** assert that:

+-- {: .num_defn}
###### Semi-formal Definition
**(Coherence)**

While [[associativity]] and [[uniticity]] of [[composition]] of [[k-morphism]]s holds only up to choices of higher morphisms, **coherence** is the demand that the collection of these choices forms a _[[contractible]]_ [[∞-groupoid]].

A **[[coherence theorem]]** is an assertion that with a given definition of [[n-category]]-structure, coherence is satisfied.

=--

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

It is the fact that this condition makes recourse among all [[(n,r)-categories]] to [[(∞,0)-categories]]  = [[∞-groupoid]]s that it is easier to define and handle an [[(∞,n)-category|(∞,r)-category]] than an [[(n,r)-category]] for finite $n$: in the former case one just needs that certain spaces are [[contractible]] without necessarily being _equal_ to the point, while in the latter case one demands some of these to be exactly _equal_ to the point, which is a condition much harder to get under control. Accordingly, a _coherence law_ in an [[(n,n)-category]] such as a [[bicategory]] ($n = 2$) or a [[tricategory]] ($n=3$) or a [[tetracategory]] ($n = 4$) is in degree $n+1$ a complicated _equation_ that asserts that a certain contractible space of higher morphisms is exactly equal to the point.

=--

### Examples

We start the list of examples with a warning on how not to misunderstand the above definition.

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

The demand for contractible spaces of choices of [[associator]]s and [[unitor]]s is not to be confused with asserting contractible [[hom-space]]s in general (which would make the theory of $(n,r)$-categories trivial, anyway!)

For instance in a [[braided monoidal category]] there is, in general, a non-contractible space of morphisms $x \otimes y \to x \otimes y$ for any two objects, because the double braiding $B_{y \otimes x}\circ B_{x \otimes y}$ will not in general be equal to the identity morphism. But the braiding morphisms also is not the kind of structural morphism that the above definition refers to. In the definition of braided monoidal category it may look as if it is on par with the [[associator]], but this is in fact not so:

in the context of $(n,r)$-categories we may use the [[periodic table of higher categories]] to identify the braided monoidal category with a one-object-one-morphism [[3-category]]. In this, the non-trivial double braiding $B_{y \otimes x}\circ B_{x \otimes y}$ is a nontrivial [[3-morphism]] that represents a nontrivial element of the 3rd [[homotopy group]] of the 3-category. Demanding this to be trivial would be demanding the 3-category to be trivial!

But the role of the associator, which now is a 3-morphism witnessing the non-associativity of 2-morphisms under [[horizontal composition]] is quite different. This is best seen by looking at a [[geometric definition of higher categories]], such as an  [[(∞,n)-category]] for $n = 3$: here the associator is a _choice_ of [[horn]]-filler, and this choice by construction happens in a contractible space and by construction cannot contain nontrivial cells like the double braiding. This is also amplified by the following example

=--

### Monoidal categories

+-- {: .num_example}
###### Example
**(monoidal categories)**

The [[coherence theorem for monoidal categories]] asserts that with the standard definition of [[monoidal category]], there is a unique composite of [[associators]] that re-bracket any sequence of [[tensor product]]s.

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### Quasi-categories
 {#QuasiCategories}

+-- {: .num_example #CoherenceForQuasiCategories}
###### Example
**(quasi-categories)**

For $C$ a [[simplicial set]] that is a [[quasi-category]] we have (as discussed there) that the canonical morphism

$$
  sSet(\Delta[2], C) \to sSet(\Lambda^1[2], C)
$$

is an acyclic [[Kan fibration]]. This means that its [[fiber]]s are [[contractible]] [[∞-groupoid]]s. But these fibers are exactly the spaces whose points are choices of [[composition]] rules, whose morphisms are comparison maps between these, and so on.

Therefore the contractibility of these fibers is the _coherence_ of the [[associators]] for the quasi-category.

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### Trimble $\omega$-categories

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###### Example
**(Trimble $\omega$-categories)**

In a [[Trimble n-category]] the space of choices of composing a sequence of $n$ morphisms is explicitly the [[topological space]] $Top_{0,1}(I, I^{\vee n})$ of surjections from the unit interval $[0,1]$ onto the length $n$-interval $[0,n]$. The coherence law of composition in a Trimble $n$-category is the fact that these spaces are [[contractible]].

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### $n$-Groupoids

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###### Example

In an [[n-groupoid]] modeled as a $(n+1)$-[[coskeletal]] [[Kan complex]], the coherence law is the condition that every $(n+1)$-sphere $\partial \Delta^{n+1}$ has a _unique_ filler. This says that the corresponding space of choices is a point.

More generally, for a [[homotopy n-type]] modeled as a [[Kan complex]], the coherence law is just that all these sphere fillers exist, which only says that there is a contractible space of choices.

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## For other algebraic structures

One can consider coherence laws for algebraic structures other than $(n,r)$-categories. See [[coherence theorem]] for more.

## Related concepts

* [[rectification]]

## References

The coherence law of [[associativity]] is stated in

* [[Hermann Grassmann]], §3 of _[[Ausdehnungslehre]]_, 1844

[[!redirects coherence laws]]

[[!redirects coherent]]
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coherence space

> Not to be confused with the notion of *[[coherent space]]* in [[topology]].

***

#Contents#
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## Idea

The notion of _coherence space_ ([Girard 1989, §8 ](#Girard89), also sometimes "[[coherent space]]" but beware of the alternative meaning discussed there) is used for the [[semantics]] of [[linear logic]] in [[proof theory]].  They form a [[star-autonomous category]].

A coherence space is analogous to a [[locale]] defined by a [[topological base|basis]] for its [[frame]] of opens. A coherence space has a set of "tokens" we think of as "primitive observations" along with a coherence relation that says when primitive observations are "coherent" in that they can both be satisfied by the same point of the space. A point of the space can then be defined as a [[clique]] of observations: a set of observations such that every pair in the set is coherent.

## Definitions

### As a Set of Cliques

A coherence space $X$ is a collection of subsets of a set $|X|$ which verify some conditions. We call:

* $|X|$ the web of $X$
* the elements of $|X|$, the points of $X$
* the elements of $X$, the cliques of $X$

The conditions to be verified are:

* Every subset of a clique is a clique.
* If $a$ is a point, then $\{a\}$ is a clique.
* If $A$ is a family of cliques such that $(x,y \in A) \Rightarrow (x \cup y \in A)$, then $\bigcup A$ is a clique.

### As an Undirected Graph

Equivalently, a coherence space $X$ consists of

1. A set $|X|$, called the web of $X$, whose elements are called "tokens"
2. A reflexive, symmetric binary relation $~$ on $|X|$ called the _coherence_ relation

A clique is then a subset of $|X|$ such that each pair of elements is coherent. A co-clique is a subset of $|X|$ where any two *distinct* elements $x \neq y$ are *incoherent* $\neg (x ~ y)$.

### As an Arity Space

Coherence spaces can be defined as [[arity spaces]] for the "subunary" class of arities $\{0,1\}$.

## Morphisms

A linear function of coherence spaces $f : X \multimap Y$ is a relation $f \subseteq |X|\times|Y|$ that

1. preserves cliques: if $U$ is a $X$-clique then $f_*(U) = \{ y \in Y | \exists x \in U. f(x,y) \}$ is a $Y$-clique .
2. reflects co-cliques: if $V$ is a $Y$-co-clique then $f^*(U) = \{ x \in X | \exists y \in V. f(x,y)\}$ is an $X$-co-clique.

## Properties

The notion in terms of a set of cliques and a coherence relation are equivalent by the following:
\begin{proposition}
The map which send an [[undirected simple graph]] $G$ to the pair $(C(G), V(G))$ where $V(G)$ is the set of all the vertices of $G$ and $C(G)$ is the set of all the [[cliques]] of $G$ is a bijection between undirected simple graphs and coherence spaces where the coherence space associated to $G$ is $C(G)$ with $|C(G)|=V(G)$. The reciprocal map associates to a coherence space $X$ the undirected simple graph $G$ obtained by taking $V(G)=|X|$ and the set $E(G)$ of edges equal to the set of all cliques of cardinal $2$.
\end{proposition}

## References

The terminology "coherence space" seems to be due to:

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), Section 8 of: *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

Other references say instead "coherent space":

* Wikipedia, _[Coherent space](http://en.wikipedia.org/wiki/Coherent_space)_

* LLwiki, _[Coherent semantics](http://llwiki.ens-lyon.fr/mediawiki/index.php/Coherent_semantics)_

*  [[Jean-Yves Girard]], _Linear logic_,   Theoretical Computer Science 50:1, 1987.  ([pdf](http://iml.univ-mrs.fr/~girard/linear.pdf))

[[!redirects coherence space]]
[[!redirects coherence spaces]]


coherence theorem

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# Contents
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## Idea ##

Generally, a **coherence theorem** in [[category theory]] and [[higher category theory]] asserts that a [[coherence law]] is satisfied: it is a means of getting a handle on categorical structures where laws only hold up to [[isomorphism]] or higher [[k-morphism]] equivalences.  Precisely, it can mean one of several slightly different things.

### Descriptions of free algebras

Given a [[doctrine]], such as the doctrine for monoidal categories, a coherence theorem may provide a full or partial algorithm for deciding equality in free algebras.  Usually such a doctrine is specified only by certain "generating" operations and constraint morphisms between them; the coherence question then consists in determining exactly what additional operations and morphisms are derivable from these, and in particular when two such derived morphisms are equal.  For instance, one may be interested in the coherence problem for closed categories: how does one decide equality of morphisms in freely generated closed categories, or in other words, how does one decide which diagrams (generated from the closed structure) commute?

The first, and perhaps easiest, coherence theorem is Mac Lane's, which says that in a free [[monoidal category]], *every* diagram of constraint morphisms commutes.  The especially nice form of this theorem should not lead one to expect that all coherence theorems are of the form "every diagram commutes"---in many categorical structures, it is *not* the case that all diagrams of constraints commute, and so the coherence question must address more precisely *which* diagrams of constraints commute.  For instance, in a [[braided monoidal category]] not every diagram of constraints commutes, but the coherence theorem gives a concise classification of those that do in terms of their underlying braids.

This sort of coherence theorem is often the most useful when actually working with algebras in a doctrine, since it provides a way to tell whether two particular composites of constraints which may arise in practice are actually equal.  One should beware, however, that coherence theorems only relate to the behavior of *free* algebras, or equivalently to the equality of *formally defined* composites.  Even in a structure having a coherence theorem of the "every diagram commutes" sort, it may happen in a *particular* algebra that there exist diagrams of constraints which do not commute.  For instance, one can find a monoidal category containing objects $x,y,z$ such that $x\otimes (y\otimes z) = (x\otimes y)\otimes z$ on the nose, but such that the [[associator]] isomorphism $a_{x,y,z}:x\otimes (y\otimes z) \stackrel{\simeq}{\to} (x\otimes y)\otimes z$ is not the identity.  (For example, [[skeletons]] of naturally occurring monoidal categories often have this property.  This issue also arises in the classification of [[2-groups]] via cohomology.)

### Strictification

On the other hand, given a doctrine for "weak" categorical structures of some sort, a coherence theorem may provide a means for fully or partially strictifying such structures, to the effect that every weak structure is weakly equivalent to a strictified structure.  The applicable version of Mac Lane's original coherence theorem here is that every monoidal category is (monoidally) equivalent to a *strict* monoidal category.  Similarly, every [[bicategory]] is equivalent to a [[strict 2-category]].

As before, one should not be misled by these examples into thinking that all coherence theorems are of the form "every weak thing is equivalent to a strict thing."  For example, the strictification coherence theorem for symmetric monoidal categories states that every symmetric monoidal category is symmetric-monoidally equivalent, not to a commutative monoid in $Cat$, but to a symmetric strict monoidal category, whose monoidal structure is strict but whose symmetry is not.  This may be regarded as a "partial" or "semi-"strictification result.

Naturally, it should not be thought that coherence problems are restricted to 1-category theory.  For instance, the main result in Gordon, Power and Street's _Coherence for Tricategories_ is that every [[tricategory]] is equivalent (not to a [[strict 3-category]] but) to a [[Gray-category]].  This is another example of a semi-strictification result, and is the tip of an iceberg of hoped-for further coherence theorems for algebraic notions of $n$-category.

### Relating coherence theorems

These two senses of "coherence theorem" are connected, of course.  On the one hand, an explicit description of the free algebras for a doctrine is often helpful, if not essential, in proving a strictification result, since in a strict structure where constraints become equalities, many more diagrams will turn out to commute (being composed of identities).  On the other hand, given a strictification theorem, if we have an explicit description of the strict structure (which is generally easier to come by), we can often deduce a characterization of the free algebras for the weak structure.

One thing to beware of is that even for structures whose free-algebra coherence theorem is of the form "all diagrams commute," it does not necessarily follow that all such algebras can be fully strictified.

## List of coherence theorems
 {#ListOfTheorems}

* [[coherence theorem for monoidal categories]]
* [[coherence theorem for symmetric monoidal categories]]
* [[coherence theorem for braided monoidal categories]]
* [[coherence theorem for tortile categories]]
* [[coherence theorem for closed symmetric monoidal categories]]

* [coherence theorem for bicategories](/nlab/show/bicategory#Coherence)
* [[Lack's coherence theorem]] (not a true coherence theorem)
* [coherence theorem for tricategories](/nlab/show/tricategory#Coherence)

* [[coherence theorem for bicategories with finite limits]]
* [[coherence theorem for monoidal bicategories]]
* [[coherence theorem for braided monoidal bicategories]]
* [[coherence theorem for symmetric monoidal bicategories]]

## General frameworks for coherence theorems

There exist several general frameworks for describing "doctrines" or algebraic structures on categories, which can be used to describe free algebras and state coherence theorems.  All are closely related.

### 2-monads {#2monads}

A [[2-monad]] can be regarded as the "extensional essence" of an algebraic structure on categories (or on objects of some other 2-category).  Of course, if $T$ is a 2-monad describing some structure, then $T A$ is the free such structure on $A$; thus the first sort of coherence theorem can be precisely stated as "describe $T A$ as explicitly as possible in terms of $A$."

However, the notion of monad is so general that in practice, for proving coherence theorems it is useful to have a more explicit way of "presenting" a monad in terms of generating operations and relations; this is the purpose of the structures presented below, such as [[operads]] and [[clubs]].  Not all monads can be presented in such a more explicit way, but for those that can, it is a very useful simplification.

The notion of *strict* 2-monad on a [[strict 2-category]] also provides a general way to state a strictification theorem, although one must beware that this theorem is not always true, and when it is true, it doesn't necessarily give what one was looking for.  This general "coherence theorem schema" for a 2-monad $T$ would be that every [[pseudoalgebra for a 2-monad|pseudo]] $T$-algebra is equivalent to a strict one.  A stronger statement, which is true in most cases when the weaker version is, would be that the inclusion
$$ Str T Alg \hookrightarrow Ps T Alg $$
of the 2-category of strict $T$-algebras and strict $T$-morphisms into the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms has a strict left 2-adjoint, usually written $(-)'$, and moreover for a pseudo algebra $A$, the unit $A \to A'$ is an [[equivalence]] in $Ps T Alg$.  Thus $A$ is not only equivalent to a strict $T$-algebra, it is equivalent to a canonically defined one which is characterized by a universal property.

This "coherence theorem" is true in a lot of generality, although there are 2-monads for which it fails (see ([Shulman](#Shulman))).  However, often it leaves a lot of work to be done in order to extract what is commonly thought of as a coherence theorem.  This is because the notion of pseudo $T$-algebra is always an *unbiased* weak structure, whereas it is more usual to study biased structures.  Moreover, in most cases, proving an equivalence between biased and unbiased notions requires the full strength of the coherence theorem (in the description-of-free-algebras sense) for the biased notion!

### Operads

See [[operad]].

### Clubs ##

See [[club]].

## References

[[Saunders Mac Lane]] discusses some of the history of the coherence problem in section 5 of

* [[Saunders Mac Lane]], Topology and Logic as a Source of Algebra (Retiring Presidential Address), _Bulletin of the AMS_ 82:1, January 1976. ([euclid](https://projecteuclid.org/euclid.bams/1183537593))

An account of some of the history of (proofs of) coherence theorems is at

* [linear type theory -- History of categorical semantics](linear+type+theory#HistoryCategoricalSemantics) .

See also at _[[Kelly-Mac Lane graph]]_, at _[[proof net]]_ and at _[[Trimble rewiring]]_ for more on the [[syntax|syntactic]] proofs of coherence.

For specific references see at the above sub-entries in the [List of coherence theorems](#ListOfTheorems).

See also

* [[Steve Lack]], Codescent objects and coherence, [MR](http://www.ams.org/mathscinet-getitem?mr=1935980)

* [[Mike Shulman]], "Not every pseudoalgebra is equivalent to a strict one", *Adv. Math.* 229 no. 3 (2012), 2024–2041, [arXiv](http://arxiv.org/abs/1005.1520)
{#Shulman}

[[!redirects coherence theorems]]


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## Theorem

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###### Theorem
**(Power)**
Any [[bicategory]] with finite [[2-limit|bilimits]] is [[equivalence of bicategories|equivalent]] to a [[strict 2-category]] with finite [[flexible limits]].
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+--{: .proof}
###### Proof
Let $K$ be a bicategory with finite bilimits, let $K \hookrightarrow [K^{op},Cat]$ be its [[Yoneda embedding for bicategories|Yoneda embedding]], and let $K'$ be the closure of $K$ in $[K^{op},Cat]$ under finite flexible limits.  Since $Cat$ is a strict 2-category with finite flexible limits, so is $[K^{op},Cat]$.  And since $K$ has finite bilimits, and these are preserved by its Yoneda embedding, while flexible limits are in particular bilimits, every object of $K'$ is equivalent to an object of $K$.  Thus, $K\simeq K'$.
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## References

* [[John Power]], "Coherence for bicategories with finite bilimits", in *Categories in computer science and logic*, Contemporary Mathematics **92** (1989) pp 341-347  [MR1003207](http://www.ams.org/mathscinet-getitem?mr=1003207), doi:[10.1090/conm/092](https://dx.doi.org/10.1090/conm/092), ([Google Books](https://books.google.com.au/books?id=VxAcCAAAQBAJ&lpg=PA345&ots=P8rWN00524&dq=%E2%80%9CCoherence%20for%20bicategories%20with%20finite%20bilimits%E2%80%9D&pg=PA343#v=onepage&q=%E2%80%9CCoherence%20for%20bicategories%20with%20finite%20bilimits%E2%80%9D&f=false))

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coherence theorem for braided monoidal bicategories

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## Idea

The _coherence theorem for [[braided monoidal bicategories]]_, like many [[coherence theorems]], has several forms (or, alternatively, refers to several different theorems):

1. Every diagram of constraint 2-cells in a free [[braided monoidal bicategory]] commutes; in other words, any two parallel composites of constraint 2-cells are equal.  Moreover, two parallel composites of constraint 1-cells are isomorphic if and only if they have the same underlying [[braid]], in which case they are uniquely isomorphic.

1. Every braided monoidal bicategory is equivalent to a strict braided monoidal bicategory.

## References

* [[Nick Gurski]], "Loop spaces, and coherence for monoidal and braided monoidal bicategories".  *Adv. Math* 226(5):4225--4265, 2011


coherence theorem for braided monoidal categories

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## Idea

The _coherence theorem for [[braided monoidal categories]]_ can be expressed as:

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###### Theorem
Every diagram in a free braided monoidal category made up of [[associators]] and [[unitors]] and [[braidings]], and in which both sides have the same underlying [[braid]], commutes.
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## References

* [[Andre Joyal]] and [[Ross Street]].  "Braided tensor categories".  *Adv. Math* 1993


coherence theorem for closed symmetric monoidal categories

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## Idea

A kind of _coherence theorem for [[closed symmetric monoidal categories]]_, conjectured by Saunders Mac Lane in the 60's and proved by Sergei Soloviev in the 90's, says the following:

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###### Theorem
A diagram in a free closed symmetric monoidal category is commutative if and only if all its instantiations in vector spaces are commutative.
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One may use vector spaces over any (fixed) field. In fact, Soloviev's result is more general that this and provides an axiomatic description of "test categories" for which the theorem holds.

## References

* Sergei Soloviev.  "Proof of a Conjecture of S. Mac Lane".  [BRICS 1996](http://www.brics.dk/RS/96/61/BRICS-RS-96-61.pdf)