differences between CompLF v0 and v1

#Contents#
* table of contents
{:toc}

## Idea

This article documents the differences between versions [[CompLF v0|zero]] and [[CompLF v1|one]] of [[CompLF]]. It also provides pointers to explanations of some of the theory that enables some of these differences, which is perhaps surprisingly (disappointingly?) subtle and deep. It gets into the connection between meaning explanations and partial logic, and the practical advantage of intensionality at the judgment level of dependent type theory.

## Admissibility vs Soundness

One of the most important differences between v0 and v1 is that many of the rules that are merely *admissible* in v0 are actually primitive or *derivable* in v1. In this regard, v1 is closer to Nuprl, and does not continue v0's "experiment" with admissible rules [[CompLF#AdmissSem|discussed previously]].

The most conspicuous changes are the addition of some new rules in v1: the "sanity" or "presuppositionality" rule, and "inversion" rules for the type formation rules. Sanity is this one:

$$\frac{\Gamma \vdash a \Vdash A}{\Gamma \vdash A\,type}$$

There are (unfortunately) many inversion rules, but they all have the effect of saying merely that most of the type formation rules are invertible. For example, here are the inversions of $\Pi$ formation:

$$\begin{gathered}
\frac{\Gamma \vdash \Pi x:A.B\,type}{\Gamma \vdash A\,type} \\
\\
\frac{\Gamma \vdash \Pi x:A.B\,type \qquad \Gamma \vdash a \Vdash A}
{\Gamma \vdash B[a/x]\,type}
\end{gathered}$$

Note that there's a substitution baked into the $\Pi$ codomain rule. This is because substitution is still an admissible rule. Not baking in a substitution here would probably mess that up.

Less conspicuous, but just as important, are other changes throughout the rules of v0 that are possible because we no longer have to prove sanity and inversion by induction on formal derivations. Indeed, having sanity and inversion as primitive has the effect that many other rules become stronger: If you can derive $a \Vdash A$, then you also know that $A$ and many of its subexpressions are all valid types. In v0, this was by carefully designing the rules so that all those extra consequences were already derivable. In v1, it's actually revealing additional facts about the interpretation of the judgments. (Examples below.)

Due to the use of PER semantics, to make the inversion rules sound, semantic equality of types is now intensional, like in Nuprl (and intensional type theory!), and unlike in v0. (TODO: Write an explanation of this.) As for sanity, that was actually sound all along, but including it as a primitive rule would ruin admissibility of inversion, were that not also primitive.

Here is a list of changes to existing rules of v0 that are possible because sanity and inversion are primitive:

* The direct computation rule, which lets you beta convert in a type, (and which previously used the auxiliary judgment ($\equiv_\beta$) but now uses ($\longrightarrow_\beta$), which is discussed [separately](#BetaConv)) no longer requires the new type to be proven valid as an extra premise. Indeed, the new type *must* be valid, since it's semantically the same as the old one, which is valid by assumption.

* As a consequence of the previous, the other old direct computation rule, which lets you beta convert in the subject, has become redundant and has been removed. To derive it, just switch the judgment to express membership with the equality type, use direct computation in that type, and switch back.

* The "select right" rule of equality, which goes from ($a = a' \in A$) to ($a' \Vdash A$) has become redundant and has been removed. Prove symmetry in the usual way, and combine with select left. (Maybe it was already redundant before, who knows?)

* The subset elimination rule had a premise in v0 showing that the motive is a valid type family. In v1, this premise is removed. Semantically, it's redundant because it follows from the typing premise for the sole case (and the additional free variable constraint).

* Similarly, the PER elimination rule had the motive validity premise removed.

* Yet another, the $Bool$ elimination rule had the motive validity premise removed. This one is a bit more interesting, because this rule has *two* cases, which only show validity of a substitution instance of the motive. Luckily, a family over the booleans only *has* two substitution instances, semantically.

* As a consequence of the previous, the rule saying booleans are computations has become redundant and has been removed. It follows from case analysis on the boolean, and the fact that both canonical booleans are obviously computations, by the computation formation rules. The reason this argument didn't work in v0 is that you'd have trouble showing the motive is valid.

* Another consequence of the strengthened $Bool$ elimination which is perhaps much more surprising: The $\bot$ elimination rule, which effectively said that the two booleans aren't equal, has become redundant and has been removed. Like in type theory with universes, you can prove this by interpreting the booleans as true and false *types*. Why this is possible without universes is [[CompLF/HOAS#CombElim|explained elsewhere]], but it boils down to types syntactically being terms, having primitive sanity and inversion, and the strengthened $Bool$ elimination rule. (As icing on the cake, the derived $\bot$ elimination rule doesn't need a motive validity premise either. But this seems to require [identity](#IdType), a different v1 addition.)

## Relaxed Equality

Another important change is the new equality formation rule. Here is the v0 rule:

$$\frac{\Gamma \vdash p \Vdash A \prec C \qquad
\Gamma \vdash q \Vdash B \prec C \qquad
\Gamma \vdash a \Vdash A \qquad \Gamma \vdash b \Vdash B}
{\Gamma \vdash a = b \in C\,type}$$

Here is the v1 rule:

$$\frac{\Gamma \vdash a1 \Vdash Relax(A) \qquad \Gamma \vdash a2 \Vdash Relax(A)}
{\Gamma \vdash a1 = a2 \in A\,type}$$

The new rule uses the new type constructor $Relax$, instead of the defined notion of respect ($\prec$). Since respect is defined in terms of equality, respect-based equality was a rather annoying circularity in the rules, which made the soundness proof a nuisance. But that is *not* the main improvement. One could just make respect into a primitive type constructor to avoid the circularity.

The real improvement is that "relaxed equality" avoids the implicit quantification over types in the v0 formation rule. Notice that the v0 rule involves the types $A$ and $B$ which are not mentioned in the conclusion. Because of this, it was not clear at all how to give a strong inversion rule for equality formation. With relaxed equality, the inversion rules are rather obvious.

Despite this major organizational improvement, the relaxed equality formation rule is saying essentially the same thing as the respect-based equality formation rule. This is because of a characterization of $Relax$ in terms of subtyping: ($A \lt\!\!:\;Relax(B)$) if and only if ($A \prec B$). So some miscellaneous v0 rules about respect:

$$\frac{\Gamma \vdash A\,type}{\Gamma \vdash p \Vdash A \prec A}
\qquad
\frac{\Gamma \vdash A\,type}{\Gamma \vdash p \Vdash Comp \prec A}$$

are replaced by v1 rules about $Relax$:

$$\frac{\Gamma \vdash a \Vdash A}{\Gamma \vdash a \Vdash Relax(A)}
\qquad
\frac{\Gamma \vdash A\,type \qquad \Gamma \vdash t \Vdash Comp}
{\Gamma \vdash t \Vdash Relax(A)}$$

This allows the bootstrapping of general reasoning about respect to proceed in essentially the same way.

### Greatest type respected

To understand how $Relax$ avoids quantification in the equality formation rule, it's helpful to note that $Relax(A)$ is the greatest type (ordered by subtyping) that's respected by $A$. This is a corollary of that subtyping characterization of $Relax$. So starting from the v0 rule, to know that $a$ and $b$ are elements of *some* types $A$ and $B$ that are respected by $C$, it's necessary and sufficient to check that $a$ and $b$ are elements of $Relax(C)$.

### Greatest partition that's a superset

There's another way to think about $Relax$. Every type is a [[subquotient]] of $Comp$. A subquotient can be decomposed as a subset followed by a quotient, or as a quotient followed by a subset. So what happens if you decompose some arbitrary type $A$ in these ways?

It turns out there's a unique subset of $Comp$ of which $A$ is a quotient. It's ($Comp \cap A$), the type of computations that are also elements of $A$.

It turns out that the other decomposition is *not* unique. But $Relax(A)$ is the greatest (ordered by subtyping) quotient of $Comp$ of which $A$ is a subset.

Now here's the cool part: Nuprl uses an equality formation rule that essentially uses the type ($A \cup Comp$) in place of our $Relax(A)$. (Except Nuprl calls it $Base$ instead of $Comp$.) And *that* type is the *least* quotient of $Comp$ of which $A$ is a subset!

Indeed, ($A \cup Comp$) can be understood as the type you get from $A$ by adding all computations, that *aren't* elements of $A$, as additional elements. Meanwhile, $Relax(A)$ instead adds *at most one* additional element, which is implemented by all computations that aren't elements of $A$. That is, ($A \cup Comp$) and $Relax(A)$ both add all the remaining computations, but ($A \cup Comp$) uses the finest possible equality, while $Relax(A)$ uses the coarsest.

### Kleene equality

Thinking of $Relax$ as adding at most one garbage element gives yet another way to think about it. The additional element can be regarded as an undefined element. It's the element of $Relax(A)$ whose realizers fail to denote an element of the desired type $A$. So $Relax$ is sort of like a lifted domain, and equality in a lifted domain is [[Kleene equality]]. Indeed, this gives the actual PER of $Relax$ used in the semantics:

$(a = a' \in Relax(A))\;\coloneqq\;(a \in A \vee a' \in A) \to (a = a' \in A)$

From that definition, the general intro and elim rules for the $Relax$ type constructor should be relatively clear.

What this means for the equality type is that while it's *strict* equality, when regarded as a relation on partial elements, it's *non-strict* when regarded as a type-valued operation. That is, ($a = a' \in A$) might be a valid type even if $a$ and/or $a'$ are not elements of $A$ (non-strict), but it cannot be *true* unless both $a$ and $a'$ are elements of $A$ (strict).

Meanwhile, ($a = a' \in Relax(A)$) is non-strict in both senses, in terms of potential elements of $A$. But for elements of $Relax(A)$, it's strict in both senses. If ($a = a' \in Relax(A)$) is a valid type, then $a$ and $a'$ are elements of $Relax(Relax(A))$. But that means they're elements of $Relax(A)$, because $Relax$ is idempotent. (You can't add computations to a type that already has all of them.)

All of this is provable internally, using the rules for $Relax$ and equality.

## Shortcut Beta Conversion {#BetaConv}

To take a break from the big, important changes, here's a minor, unimportant one. In v0, beta conversion was formulated as a judgment form ($\equiv_\beta$) which was defined to be the congruence closure of beta reduction. It turns out that that's kind of a minor nuisance to spell out in a HOAS presentation of the rules. But because $\lambda$ is the only binding form in the syntax, there's a shortcut for getting full congruence. (In general, this shortcut does not avoid congruence rules for binding forms.)

In v1, "partially-compatible beta reduction" ($\longrightarrow_\beta$) has only two congruence rules: one for $\lambda$, and one for the argument of an application. Plus it's not reflexive, symmetric, or transitive. But because the v1 direct computation rules explicitly allow reduction in either direction, and can implicitly be applied repeatedly, this is enough.

Obviously, consecutive applications of the direct computation rules allow rewriting with the reflexive, symmetric, transitive closure (equivalence closure) of ($\longrightarrow_\beta$). Let's call that ($\leftrightarrow_\beta$). It turns out to be the same as ($\equiv_\beta$).

What we seem to be missing is most of the congruence rules. But we have a trick: if ($a \leftrightarrow_\beta a'$), then

$b[a/x] \longleftarrow_\beta
(\lambda x.b)\,a \leftrightarrow_\beta
(\lambda x.b)\,a' \longrightarrow_\beta
b[a'/x]$

by application argument congruence (extended to the equivalence closure) and the beta rule.

So ($b[a/x] \leftrightarrow_\beta b[a'/x]$). This lets us rewrite in any context *if* the hole does not appear under binders, *or* the thing we're rewriting at least does not mention a bound variable. This actually seems to cover nearly all the cases of rewriting in practice, and then you don't even need to bother with congruence rules, which is nice.

But for the proof, what we need is that it means we effectively have congruence rules for all non-binding forms. And we really have the $\lambda$ congruence rule for the only binding form in the language. So we effectively have all the congruence rules.

## Let-Comp

Measured by utility, the most important change has got to be the addition of the "letcomp" rule:

$$\frac{\begin{array}{l}\Gamma \vdash a \Vdash A \qquad
\Gamma,y:A \vdash C\,type \\
\Gamma,x:Comp,x':Comp,h:(x = x' \in A) \vdash
p \Vdash c[x/y] = c'[x'/y] \in C[x/y] \\
x,x',h \notin FV(c,c')\end{array}}
{\Gamma \vdash q \Vdash c[a/y] = c'[a/y] \in C[a/y]}$$

In v0, quotienting was almost completely broken. It was [[CompLF#PERtheory|explained]] how it was *supposed* to work, with parenthetical remarks that it didn't yet seem to work. In v1, letcomp is what makes it work.

Another solution presumably would've been to add a stronger elimination rule to PER comprehension. But this would not be taking full advantage of PER semantics, where quotienting is intuitively part of the very meaning of the judgments, and all types are implicitly subquotients.

Letcomp is a subquotient elimination rule, formulated to work on an arbitrary type by regarding it as a subquotient of $Comp$. The name "letcomp" is based on the observation that it binds an element as a general pair of related computations.

Letcomp can also be understood as internalizing the meaning of the semantic hypothetical judgment form:

$(x:A \gg c = c' \in C)\;\leftrightarrow\;((A\,type) \to ((x:A \gg C\,type) \wedge \forall a,a'.(a = a' \in A) \to (c[a/x] = c'[a'/x] \in C[a/x])))$

With this grand purpose as an attempt to complete the internalization of the PER semantics, it may not be surprising that letcomp is very useful for deriving rules for type constructors defined by PER comprehension. But it's not perfect. Rules that involve metavariables without an associated typing premise tend to cause trouble. Alas, it seems very difficult to do much about that. ("Pullback" types à la [Melliès & Zeilberger](#MZFuncRefine) might address the most serious gaps.) Notice however that intrinsic dependent type systems would not need rules with metavariables without typing premises. So the ability to derive type constructors and their rules seems quite substantial already.

There was actually an early version of letcomp for v0. But the soundness of letcomp was not established until after changing the semantics to support primitive inversions.

## Natural Numbers Type

CompLF v1 adds a type $Nat$ of [[Dana Scott|Scott]]-encoded natural numbers. The Scott nats themselves are not new, of course. What's new is having them collected into an inductive type, which actually gives CompLF some serious proof-theoretic strength. It's expected to have the same strength as (first-order) [[Peano arithmetic]], but the upper bound has not been checked.

The motive premise cannot be avoided in the $Nat$ elimination rule. One probable reason is that that would be too strong, since it would allow type-level recursion. But another reason has to do with technical details of PER semantics. (TODO: Explain.) Nuprl's version of induction actually *does* avoid the motive premise. (Nuprl's logic is much stronger than Peano arithmetic, by the way.) For some, this was a major advantage of Nuprl, since it's a way to get nontrivial proofs of totality for recursive functions. But CompLF must pursue alternatives. (TODO: Explain!)

## Identity Types {#IdType}

Identity types provide another kind of equality type that also provides subsumptive rewriting. Unlike the main equality, which is type-directed, and only rewrites in well-typed goals, identity is untyped, and rewrites in any goal at all: rewriting has no motive premise. This identity should be thought of as a relation that's true only for things that are identical, not a type of "[[identity type|identifications]]" of things that generally aren't already identical.

The restriction that prevents the two notions from collapsing is that identity types are usually not valid unless they're true. Only when dealing with computations is it safe to assume identity. But in that case, it coincides with equality. In fact, identity *is* equality of computations, in the semantics. The different rules are due entirely to the difference in when the types are valid. This is [[CompLF/HOAS#IdType|explained elsewhere]], in terms of how presuppositions affect rules and other judgment-level reasoning. Because of the sanity rule, it's implicit that for any term $t$, ($t \equiv t$) is a valid type because it's true. This is why identity has no inversion rules. Meanwhile, ($t = t \in Comp$) is not true or even meaningful unless $t \in Comp$.

When dealing with computations, where equality and identity coincide, the rules for identity types do indeed expose a stronger (derived) rewrite rule for equality. The motive premise is avoided:

$$\frac{\Gamma \vdash p \Vdash t = t' \in Comp \qquad
\Gamma \vdash c \Vdash C[t/x]}
{\Gamma \vdash c \Vdash C[t'/x]}$$

The identity type constructor [comes from Nuprl](#CompEqvITP13), where it's called "computational equivalence". That's exactly what it is, semantically. But internally, it seems weird to think of it as a mere equivalence, since it's the most intensional equality that can be expressed as a type. Nuprl has additional features that facilitate internal reasoning about computational equivalence. But in v1, identity seems scarcely different from beta conversion in practice.

## Miscellaneous

Certain rules in v0 (equality formation (via respect) and PER formation) used non-dependent function types ($\to$) to express implications. In v1, non-dependent family intersection types ($\supset$) are used (in $Relax$ introduction and PER formation) instead. In some cases, the propositions involved are guaranteed to be proof-irrelevant, so it doesn't matter. In PER formation, there's no guarantee that $R$ is proof-irrelevant, but there's no good reason for the user to use a proof-relevant relation, since the proof is ignored. In both versions of PER formation, you can run into trouble trying to use a proof-relevant relation. (So the change doesn't help or hurt.)

In v0, there was a reflexivity rule for equality, in addition to the uniqueness and selectivity rules. That was redundant. So now reflexivity is derived from uniqueness and select left.

There's been some reordering of rules, premises, and renaming of metavariables.

## References

* {#CompEqvITP13} Vincent Rahli, Mark Bickford, Abhishek Anand, _Formal Program Optimization in Nuprl Using Computational Equivalence and Partial Types_, Interactive Theorem Proving (ITP) 2013 ([web](http://www.nuprl.org/KB/show.php?ShowPub=RBA13), [pdf](http://www.nuprl.org/documents/Rahli/optimization_ITP_2013_04_22))

* {#MZFuncRefine} Paul-André Melliès, Noam Zeilberger, _Functors are Type Refinement Systems_, POPL 2015 ([pdf](https://www.irif.fr/~mellies/papers/functors-are-type-refinement-systems.pdf))


differentiable (infinity,1)-category

[[!redirects differentiable (infinity,1)-category]]

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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
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## Idea

In the context of [[Goodwillie calculus]], an [[(∞,1)-category]] is called _Goodwillie-differentiable_ if [[(∞,1)-functors]] to it admit "derivatives" in the form of [[n-excisive approximations]]. Note that this concept is not related to that of [[Lie ∞-groupoids]].

## Definition

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

An [[(∞,1)-category]] $\mathcal{C}$ is **Goodwillie-differentiable** if

1. it has [[finite (∞,1)-limits]];

1. it has [[sequential limit|sequential]] [[(∞,1)-colimits]];

1. the [[(∞,1)-colimit]] [[(∞,1)-functor]] $\underset{\longrightarrow}{\lim} Func(\mathbb{N}, \mathcal{C}) \longrightarrow \mathcal{C}$ is a [[left exact (∞,1)-functor]], hence commutes with [[finite (∞,1)-limits]].

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## Examples

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###### Example

Every [[(∞,1)-topos]] is a Goodwillie-differentiable $(\infty,1)$-category.

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([Lurie, example 7.1.1.8](#Lurie))

## Properties

### $n$-Excisive reflection / Taylor tower

By [[Goodwillie calculus]], [[(∞,1)-functors]] to Goodwillie-differentiable $(\infty,1)$-categories have [[n-excisive approximations]]/[[Taylor towers]].

## Related concepts

* [[tangent (∞,1)-category]]

* [[jet (∞,1)-category]]

* [[Goodwillie calculus]]

## References

* [[Jacob Lurie]], _[[Higher Algebra]]_
  {#Lurie}

[[!redirects differentiable (infinity,1)-categories]]

[[!redirects differentiable (∞,1)-category]]
[[!redirects differentiable (∞,1)-categories]]


differentiable manifold

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#### Differential geometry
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#Contents#
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## Idea

A _differentiable manifold_ is a [[topological space]] which is _locally_ [[homeomorphism|homeomorphic]] to a [[Euclidean space]] (a [[topological manifold]]) and such that the [[gluing functions]] which relate these Euclidean [[local charts]] to each other are [[differentiable functions]], for a fixed degree of differentiability. If one considers arbitrary differentiablity then one speaks of _[[smooth manifolds]]_ and if one demands [[analytic function|analytic]] gluing functions then one speaks of _[[analytic manifolds]]_. For a general discussion see at _[[manifold]]_.

Accordingly, a differentiable manifold is a space to which the tools of ([[infinitesimal analysis|infinitesimal]]) [[analysis]] may be applied _locally_.
Notably we may ask whether a [[continuous function]] between differentiable manifolds is [[differentiation|differentiable]]
by computing its [[derivatives]] pointwise in any of the Euclidean [[coordinate charts]].

In particular one may consider smooth functions from the [[real line]] into any smooth manifold $X$, "smooth curves" in $X$. The [[equivalence classes]] of these that have the same first [[derivative]] at a given point capture the idea of "infinitsimal smooth paths" through that point in the manifold, called its _[[tangent vectors]]_. All the tangent vectors at one point $x \in X$ constitute the [[tangent space]] $T_x X$, and the collection of all these tangent spaces yields another differentiable manifold, called the _[[tangent bundle]]_ $T X$. This happens to be a [[vector bundle]] which is [[associated bundle|associated]] to a [[principal bundle]], called the _[[frame bundle]]_ $Fr(X)$.

By equipping the [[tangent bundle]] or [[frame bundle]] of a differentiable manifold with [[extra properties]] or [[extra structure]] one encodes _[[geometry]]_ on the manifold. For example equipping them with [[orthogonal structure]] encodes [[Riemannian geometry]] on manifolds. More generally one may consider any _[[G-structure]]_ on the frame bundle and thereby equip the manifold with the corresponding _[[Cartan geometry]]_, for instance _[[complex geometry]]_, _[[conformal geometry]]_ etc.

This way differential and smooth manifolds are the basis for [[differential geometry]]. They are the analogs in differential geometry of what [[schemes]] are in [[algebraic geometry]]. In fact both of these concepts are unified within _[[synthetic differential geometry]]_.

If one relaxes the condition on differentiable manifold from it being locally isomorphic to a Euclidean space to it just admitting local smooth maps from a Euclidean space, then one obtains the more general concept of [[diffeological spaces]] or even [[smooth sets]], see at _[[generalized smooth space]]_ for more on this. If one generalizes here [[differentiable functions]] to [[simplicial algebra|simplicial]] differentiable functions one obtains concepts of _[[derived smooth manifold]]_.

The generalization of differentiable manifolds to [[higher differential geometry]] are [[orbifolds]] and more generally [[differentiable stacks]]. If one combines this with the generalization to [[smooth sets]] then one obtains the concept of [[smooth stacks]] and eventually [[smooth infinity-stacks]].

## Definition

For convenience, we first recall the basic definition of

* [Topological manifolds](#TopologicalManifolds)

and of

* [Differentiable functions between Euclidean spaces](#DifferentiableFunctionsBetweenCartesianSpaces)

and then turn to the actual definition of

* [Differentiable manifolds](#DifferentiableManifolds).

### Topological manifolds
 {#TopologicalManifolds}

For convenience, we first recall here some background on [[topological manifolds]]:

+-- {: .num_defn #TopologicalManifold}
###### Definition
**(topological manifold)**

A _topological manifold is a [[topological space]] which is

1. [[locally Euclidean topological space|locally Euclidean]],

1. [[paracompact Hausdorff topological space|paracompact Hausdorff]].

If the local [[Euclidean spaces|Euclidean]] neighbourhoods $\mathbb{R}^n \overset{\simeq}{\to} U \subset X$ are all of [[dimension]] $n$
for a fixed $n \in \mathbb{N}$, then the topological manifold is said to be a _$n$-dimensional manifold_ or _$n$-fold_. This is usually assumed to be the case.

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+-- {: .num_remark}
###### Remark
**(varying terminology)**

Often a topological manifold (def. \ref{TopologicalManifold}) is required to be [[sigma-compact]]. But by [this prop.](topological#manifold#RegularityConditionsForTopologicalManifoldsComparison) this is not an extra condition as long as there is a [[countable set]] of [[connected components]].
Moreover, manifolds with uncountably many connected components are rarely considered in practice.

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+-- {: .num_defn #Charts}
###### Definition
**([[local chart]], [[atlas]] and [[gluing function]])

Given an $n$-dimensional topological manifold $X$ (def. \ref{TopologicalManifold}), then

1. an [[open subset]] $U \subset X$ and a [[homeomorphism]] $\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U$ is also called a _[[local coordinate chart]]_ of $X$.

1. an [[open cover]] of $X$ by local charts $\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I}$ is called an _[[atlas]]_ of the topological manifold.

1. denoting for each $i,j \in I$ the [[intersection]] of the $i$th chart with the $j$th chart in such an atlas by

   $$
     U_{i j} \coloneqq U_i \cap U_j
   $$

   then the induced homeomorphism

   $$
     \mathbb{R}^n
       \supset
        \phantom{AA}
     \phi_i^{-1}(U_{i j})
       \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow}
     U_{i j}
       \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow}
     \phi_j^{-1}(U_{i j})
       \phantom{AA}
       \subset
     \mathbb{R}^n
   $$

   is called the _[[gluing function]]_ from chart $i$ to chart $j$.

> graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]

=--

### Differentiable functions between Cartesian spaces
 {#DifferentiableFunctionsBetweenCartesianSpaces}

For convenience we recall the definition of [[differentiable functions]]
between [[Euclidean spaces]].

+-- {: .num_defn #DifferentiableFunctionBetweenCartesianSpaces}
###### Definition
**([[differentiable functions]] between [[Euclidean spaces]])**

Let $n \in \mathbb{N}$ and let $U \subset \mathbb{R}^n$ be an [[open subset]].

Then a [[function]] $f \;\colon\; U \longrightarrow \mathbb{R}$ is called  **differentiable** at $x\in U$
if there exists a [[linear map]] $d f_x : \mathbb{R}^n \to \mathbb{R}$ such that the following [[limit of a sequence|limit]]
exists as $h$ approaches zero "from all directions at once":

$$
  \lim_{h\to 0} \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} = 0.
$$

This means that for all $\epsilon \in (0,\infty)$ there exists an [[open subset]] $V\subseteq U$ containing $x$ such that whenever $x+h\in V$ we have $\frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} \lt \epsilon$.

We say that $f$ is _differentiable on_ a [[subset]] $I$ of $U$ if $f$ is differentiable at every $x\in I$, and _differentiable_ if $f$ is differentiable on all of $U$. We say that $f$ is *continuously differentiable* if it is differentiable and  $d f$ is a [[continuous function]].

The map $d f_x$ is called the **derivative** or **differential of $f$ at $x$**.

More generally, let $n_1, n_2 \in \mathbb{N}$ and let $U\subseteq \mathbb{R}^{n_1}$ be an [[open subset]].

Then a [[function]] $f \;\colon\; U \longrightarrow \mathbb{R}^{n_2}$ is _differentiable_  if for all $i \in \{1, \cdots, n_2\}$ the component function

$$
  f_i \;\colon\; U \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{pr_u}{\longrightarrow}  \mathbb{R}
$$

is differentiable in the previous sense

In this case, the derivatives $d f_i \colon \mathbb{R}^n \to \mathbb{R}$ of the $f_i$ assemble into a linear map  of the form

$$
  d f_x \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
  \,.
$$

If the derivative exists at each $x \in U$, then it defines itself a function

$$
  d f \;\colon\; U \longrightarrow Hom_{\mathbb{R}}(\mathbb{R}^{n_1} , \mathbb{R}^{n_2}) \simeq \mathbb{R}^{n_1 \cdot n_2}
$$

to the space of [[linear maps]] from $\mathbb{R}^{n_1}$ to $\mathbb{R}^{n_2}$, which is canonically itself a [[Euclidean space]]. We say that $f$ is _twice continuously differentiable_ if $d f$ is continuously differentiable.

Generally then, for $k \in \mathbb{N}$ the function $f$ is called _$k$-fold continuously differentiable_ or _of class $C^k$_ if the $k$-fold differential $d^k f$ exists and is a continuous function.

Finally, if $f$ is $k$-fold continuously differentiable for all $k \in \mathbb{N}$ then it is called a _[[smooth function]]_
or _of class $C^\infty$_.

=--

Of the various properties satisfied by [[differentiation]], the following plays a special role in the theory of
differentiable manifolds (notably in the discussion of their [[tangent bundles]]):

+-- {: .num_prop #ChainRuleOnEuclideanSpace}
###### Proposition
**([[chain rule]] for [[differentiable functions]] between [[Euclidean spaces]])**

Let $n_1, n_2, n_3 \in \mathbb{N}$ and let

$$
  \mathbb{R}^{n_1}
    \overset{f}{\longrightarrow}
  \mathbb{R}^{n_2}
    \overset{g}{\longrightarrow}
  \mathbb{R}^{n_3}
$$

be two [[differentiable functions]] (def. \ref{DifferentiableFunctionBetweenCartesianSpaces}). Then the
[[derivative]] of their [[composition|composite]] is the composite of their derivatives:

$$
  d(g \circ f) = (d g) \circ (d f)
$$

hence for all $x \in \mathbb{R}^{n_1}$ we have

$$
  d(g \circ f)_x = d g_{f(x)} \circ d f_x
  \,.
$$

=--

### Differentiable manifolds
 {#DifferentiableManifolds}

+-- {: .num_defn #DifferentiableManifold}
###### Definition
**([[differentiable manifold]] and [[smooth manifold]])**

For $p \in \mathbb{N} \cup \{\infty\}$ then a $p$-fold _[[differentiable manifold]]_ or _$C^p$-manifold_ for short is

1. a [[topological manifold]] $X$ (def. \ref{TopologicalManifold});

1. an [[atlas]] $\{\mathbb{R}^n \overset{\phi_i}{\to} X\}$ (def. \ref{Charts}) all whose [[gluing functions]]  are $p$ times continuously [[differentiable function|differentiable]].

A $p$-fold [[differentiable function]] between $p$-fold differentiable manifolds

$$
  \left(X,\, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I} \right)
   \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow}
  \left(Y,\, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J} \right)
$$

is

* a [[continuous function]] $f \colon X \to Y$

such that

* for all $i \in I$ and $j \in J$ then

  $$
     \mathbb{R}^n
     \supset
      \phantom{AA}
     (f\circ \phi_i)^{-1}(V_j)
       \overset{\phi_i}{\longrightarrow}
     f^{-1}(V_j)
      \overset{f}{\longrightarrow}
     V_j
       \overset{\psi_j^{-1}}{\longrightarrow}
     \mathbb{R}^{n'}
  $$

  is a $p$-fold [[differentiable function]] between open subsets of [[Euclidean space]].

(Notice that this in in general  a non-trivial condition even if $X = Y$ and $f$ is the identity function. In this case the above exhibits a passage to a different, but equivalent, differentiable atlas.)

If a manifold is $C^p$ differentiable for all $p$, then it is called a _[[smooth manifold]]_.
Accordingly a continuous function between differentiable manifolds which is $p$-fold differentiable
for all $p$ is called a _[[smooth function]]_,

=--

+-- {: .num_remark #CategoryDiff}
###### Remark
**([[category]] [[Diff]] of [[differentiable manifolds]])**

In analogy to remark \ref{TopCategory} there is a [[category]] called [[Diff]]${}_p$ (or similar) whose [[objects]] are $C^p$-[[differentiable manifolds]] and whose [[morphisms]] are $C^p$-[[differentiable functions]], for given $p \in \mathbb{N} \cup \{\infty\}$.

=--

The analog of the concept of [[homeomorphism]] (def. \ref{Homeomorphism}) is now this:

+-- {: .num_defn #Diffeomorphism}
###### Definition
**([[diffeomorphism]])**

Given [[smooth manifolds]] $X$ and $Y$ (def. \ref{DifferentiableManifold}), then a [[smooth function]]

$$
  f \;\colon\; X \longrightarrow Y
$$

is called a _[[diffeomorphism]]_, if there is an [[inverse function]]

$$
  X \longleftarrow Y \;\colon\; g
$$

which is also a [[smooth function]] (hence if $f$ is an [[isomorphism]] in the [[category]] [[Diff]]${}_\infty$
from remark \ref{CategoryDiff}).

=--

Here it is important to note that while being a [[topological manifold]] is just a [[property]]
of a [[topological space]], a [[differentiable manifold]] carries [[extra structure]] encoded in the
[[atlas]]:

+-- {: .num_defn #SmoothStructure}
###### Definition
**([[smooth structure]])**

Let $X$ be a [[topological manifold]] (def. \ref{TopologicalManifold}) and let

$$
  \left(
    \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X
  \right)_{i \in I}
  \phantom{AAA}
   \text{and}
  \phantom{AAA}
  \left(
    \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X
  \right)_{j \in J}
$$

be two [[atlases]] (def. \ref{Charts}), both making $X$ into a [[smooth manifold]] (def. \ref{DifferentiableManifold}).

Then there is a [[diffeomorphism]] (def. \ref{Diffeomorphism}) of the form

$$
  f
  \;\colon\;
  \left(
    X
      \;,\;
    \left(
      \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X
    \right)_{i \in I}
  \right)
    \overset{\simeq}{\longrightarrow}
  \left(
    X\;,\;
    \left(
      \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X
    \right)_{j \in J}
  \right)
$$

precisely if the [[identity function]] on the underlying set of $X$ constitutes such a diffeomorphism.
(Because if $f$ is a diffeomorphism, then also $f^{-1}\circ f = id_X$ is a diffeomorphism.)

That the identity function is a diffeomorphism between $X$ equipped with these two atlases means,
by definition \ref{DifferentiableManifold}, that

$$
  \underset{{i \in I} \atop {j \in J}}{\forall}
  \left(
    \phi_i^{-1}(V_j) \overset{\phi_i}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^n
    \phantom{AA}
    \text{is smooth}
  \right)
  \,.
$$

Notice that the functions on the right may equivalently be written as

$$
  \mathbb{R}^n \supset
   \,
   \phi_i^{-1}(U_i \cap U_j)
     \overset{\phi_i}{\longrightarrow}
    U_i \cap V_j \overset{\psi_j^{-1}}{\longrightarrow} \psi_j^{-1}(U_i \cap V_j) \; \subset \mathbb{R}^n
$$

showing their analogy to the glueing functions within a single atlas spring.

Hence diffeomorphism induces an [[equivalence relation]] on the set of
smooth atlases that exist on a given [[topological manifold]] $X$. An [[equivalence class]]
with respect to this equivalence relation is called a _[[smooth structure]]_ on $X$.

=--

## As Étale Spaces

Differentiable manifolds can be viewed as akin to [[étale spaces]], very much in alignment with the language of [[algebraic geometry]]. For [[smooth manifolds]] and [[complex manifolds]] the requisite atlas conditions can be stated in terms of a particular structure sheaf.  More precisely, this definition yields an [[equivalence of categories]] between the category of smooth manifolds as in \ref{Charts} and the category of locally ringed spaces modeled on domains of $\mathbf{R}^n$ with a sheaf of germs of smooth functions. The same goes for topological and complex manifolds, where the local models are domains with sheafs of germs of continuous and holomorphic functions respectively. Here we illustrate the differentiable case. For any $A \subset {\mathbf R}^n$, we write $\mathcal{D}_A$ for the sheaf of germs of functions $f \in C^{\infty}(A, \mathbf{R} )$. That is, $\mathcal{D}_A$ is the functor $ \mathcal{D}_A :  \text{Open} (A) \rightarrow {\mathbf Ring}^{\text{op}}$ s.t. $ \mathcal{D}_A  : B \mapsto C^{\infty}(B, {\mathbf R})$ and $ \mathcal{D}_A  : B \subset B' \mapsto  C^{\infty}(B, {\mathbf R}) \hookrightarrow C^{\infty}(B', {\mathbf R})$. This turns $ (A, \mathcal{D}_A)$ into a ringed space, and any ringed space modeled on one of this kind can be regarded as a differentiable (by which is always meant "real") manifold. More precisely, given a ringed space $(X, D_X)$, there must an open set $A \subset {\mathbf R}^n$ and local isomorphism $(f, f_*)$  where $f: A \rightarrow X$ is a continuous function with sheaf pushfoward $f_* : (\mathcal{O}_A \rightarrow \mathcal{O}_X )$. The following property, which holds at the continuous level, is essential.

+-- {: .num_lemma}
###### Lemma

Let $F : X \rightarrow Y$ be a continuous surjection. Let $P$ be a sheaf of $R$-valued functions on $X$ (here we let $R$ be any ring). Consider an open $U \subset Y$ and define $Q(U) \subset R^U$ to be $\{ f : U \rightarrow R  | F^{*} f = f \circ F|_{f^{-1}(U)} \in P(F^{-1}(U)) \}$ i.e. those $f$ s.t. the pullback of each $f$ by $F$ belongs to $P(F^{-1}(U))$. Then $Q$ is a sheaf on $Y$.
=--

We can phrase the example of $\mathbf{R}P^n$ in this language.

+-- {: .num_example #DifferentiableManifoldCartesianSpace}
###### Example
**([[real projective space]] as a [[smooth manifold]] in this [[étale map]] sense)

We need only show that the functions $f : V \subset {\mathbf R}P^n \rightarrow {\mathbf R}^n$ which are smooth when composed with the canonical projection $\pi : {\mathbf R}^{n+1}/\{ 0 \} \rightarrow {\mathbf R }P^n$ form a sheaf on ${\mathbf R}^n$. Denote these $\mathcal{O}_{{\mathbf R}P^n}$. Certainly they form a presheaf, and checking the pullback as in the lemma confirms that they are a sheaf. So far this only confirms that ${\mathbf R}P^n$ is a real topological manifold. To establish its differentiable structure we take for each chart $(U, \varphi)$ of ${\mathbf R}P^n$ the ringed space $(\varphi(U), \varphi_* \mathcal{O}_{{\mathbf R}P^n}|_U )$ obtained from restriction and then pushforward, a local isomorphism of locally ringed spaces by construction. Hence $({\mathbf R}P^n, \mathcal{O}_{{\mathbf R}P^n})$ is a smooth manifold.

=--

## Examples
 {#Examples}

+-- {: .num_example #DifferentiableManifoldCartesianSpace}
###### Example
**([[Cartesian space]] as a [[smooth manifold]])

For $n \in \mathbb{N}$ then the [[Cartesian space]] $\mathbb{R}^n$ equipped with the [[atlas]] consisting of the single [[chart]] $\mathbb{R}^n \overset{id}{\to} \mathbb{R}^n$ is a [[smooth manifold]], in particularly a $p$-fold differentiable manifold for every $p \in \mathbb{N}$ according to def. \ref{DifferentiableManifold}.

Similarly the [[open disk]] $D^n$ becomes a [[smooth  manifold]] when equipped with the atlas whose single chart is the [[homeomorphism]] $\mathbb{R}^n \to D^n$.

This defines a [[smooth structure]] (def. \ref{SmoothStructure}) on $\mathbb{R}^n$ and $D^n$.
Strikingly, precisely for $n = 4$ there are _other_ smooth structures on $\mathbb{R}^4$,
hence called _[[exotic smooth structures]]_.

=--

+-- {: .num_example}
###### Example
**([[n-spheres]] as [[smooth manifolds]])**

For all $n \in \mathbb{N}$, the [[n-sphere]] $S^n$ becomes a smooth manfold, with [[atlas]] consisting of the two [[local charts]] that are given by the [[inverse functions]] of the [[stereographic projection]] from the two poles of the sphere onto the [[equator|equatorial]] hyperplane

$$
  \left\{
     \mathbb{R}^n
       \underoverset{\simeq}{\sigma^{-1}_i}{\longrightarrow}
     S^n
  \right\}_{i \in \{+,-\}}
  \,.
$$

By the formulas given in [this prop.](stereographic+projection#StandardStereographicProjection) the induced [[gluing function]] $\mathbb{R}^n \backslash \{0\} \to \mathbb{R}^n \backslash \{0\}$ is a [[rational function]], and hence a [[smooth function]].

Finally the $n$-sphere is a [[paracompact Hausdorff topological space]]. Ways to see this include:

1. $S^n \subset \mathbb{R}^{n+1}$ is a [[compact subspace]] by the [[Heine-Borel theorem]]. Compact spaces are evidently also paracompact. Moreover, [[Euclidean space]], like any [[metric space]], is Hausdorff, and [[subspaces]] of Hausdorff spaces are Hausdorff;

1. The $n$-sphere has an evident structure of a [[CW-complex]] and [[CW-complexes are paracompact Hausdorff spaces]].

=--

+-- {: .num_example}
###### Example
**([[real projective space]] and [[complex projective space]])**

For $n \in \mathbb{N}$

1. the [[real projective space]] $\mathbb{R}P^n$ has the structure of a [[smooth manifold]]
of [[dimension]] $n$;

1. the [[complex projective space]] $\mathbb{C}P^n$ has the structure of a [[smooth manifold]] of (real) [[dimension]] $2n$.

where in both cases the [[atlas]] is given by the standard open cover ([this def.](projective+space#TopologicalProjectiveSpaceStandardOpenCover)).

=--

By [this prop.](projective+space#SmoothManifoldRealComplexProjectiveSpace).

We now discuss some general mechanisms by which new differentiable manifolds arise from given ones:

+-- {: .num_example #ProductManifold}
###### Example
**(product manifold)**

Let $X$ any $Y$ be two differentiable manifolds with [[atlases]] $\{\mathbb{R}^n\underoverset{\simeq}{\phi_i}{\to} U_i \subset X\}$
and $\{\mathbb{R}^{n'}\underoverset{\simeq}{\psi_j}{\to} V_j \subset Y\}$. Then their [[product topological space]] $X \times Y$
becomes an differentiable manifold with respect to the atlas

$$
  \left\{
    \mathbb{R}^{n \cdot n'}
      \underoverset{\simeq}{\phi_i \times \psi_j}{\to}
    U_i \times V_j
      \subset
    X \times Y
  \right\}_{(i,j) \in I \times J}
$$

=--

+-- {: .num_example #OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds}
###### Example
**([[open subsets]] of differentiable manifolds are differentiable manifolds)**

Let $X$ be a $k$-fold differentiable manifold and let $S \subset X$
be an [[open subset]] of the underlying [[topological space]] $(X,\tau)$.

Then $S$ carries the structure of a $k$-fold differentiable manifold such that the
inclusion map $S \hookrightarrow X$ is an [[open embedding|open]]
[[embedding of differentiable manifolds]].

=--

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

Since the underlying [[topological space]] of $X$ is [[locally connected topological space|locally connected]] ([this prop.](topological+manifold#LocalPropertiesOfLocallyEuclideanSpace)) it is the [[disjoint union space]] of its [[connected components]] ([this prop.](locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity)).

Therefore we are reduced to showing the statement for the case that $X$ has a single [[connected component]]. By [this prop](topological+manifold#RegularityConditionsForTopologicalManifoldsComparison) this implies that $X$ is [[second-countable topological space]].

Now a [[subspace]] of a second-countable Hausdorff space is clearly itself second countable and Hausdorff.

Similarly it is immediate that $S$ is still [[locally Euclidean space|locally Euclidean]]: since $X$ is locally Euclidean every point $x \in S \subset X$ has a Euclidean neighbourhood in $X$ and since $S$ is open there exists an open ball in that (itself [[homeomorphism|homeomorphic]] to Euclidean space) which is a Euclidean neighbourhood of $x$ contained in $S$.

For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas.
Then the [[gluing functions]] for the Euclidean charts on $S$ are $k$-fold differentiable follows since these are restrictions of the gluing functions for the atlas of $X$.

=--

+-- {: .num_example }
###### Example
**([[general linear group]])**

For $n \in \mathbb{N}$, the [[general  linear group]] $Gl(n,\mathbb{R})$
is a smooth manifold
(as an [[open subspace]] of [[Euclidean space]] $GL(n,\mathbb{R}) \subset Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}$,
via example \ref{OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds} and example \ref{DifferentiableManifoldCartesianSpace}).

The group operations are [[smooth functions]] with respect to this smooth manifold structure, and thus $GL(n,\mathbb{R})$
is a [[Lie group]].

=--

## Related concepts

* [[differential topology]]

* [[smooth manifold]]

* [[diffeological space]], [[smooth set]]

* [[orbifold]]

* [[differentiable stack]], [[smooth stack]], [[smooth infinity-stack]]

## References

> See also the references at *[[smooth manifold]]*.

The vision of the modern notion of [[manifolds]] is attributed to:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

Monographs:

* {#Kosinski93} [[Antoni Kosinski]], _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

* [[Theodore Frankel]], chapter 1 of _[[The Geometry of Physics - An Introduction]]_, Cambridge University Press (1997, 2004, 2012)

Lecture notes include

* Claudio Gorodski, _Notes on smooth manifolds_, 2013 ([pdf](https://www.ime.usp.br/~gorodski/teaching/mat5799-2013/ch1.pdf))

* {#Schlichtkrull08} [[Henrik Schlichtkrull]], _Differentiable manifolds_, 2008 ([pdf](http://www.math.ku.dk/~jakobsen/geom2/manusgeom2.pdf))

See also

* Wikipedia, _[Differentiable manifold](https://en.wikipedia.org/wiki/Differentiable_manifold)_

[[!redirects differential manifold]]
[[!redirects differential manifolds]]
[[!redirects differentiable manifold]]
[[!redirects differentiable manifolds]]


differentiable map

> See also _[[differentiation]]_ and _[[derivative]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Differentiable maps
* table of contents
{: toc}

## Idea

A [[function]] (map) is *differentiable* at some point if it can be well approximated by a [[linear map]] near that point.  The approximating linear maps at different points together form the *derivative* of the map.

One may then ask whether the derivative itself is differentiable, and so on.  This leads to a hierarchy of ever more differentiable maps, starting with [[continuous maps]] and progressing through maps that are $n$ times (continuously) differentiable to those that are infinitely differentiable, and finally to those that are [[analytic map|analytic]].  Infinitely differentiable maps are sometimes called *smooth*.

Differentiability is first defined directly for maps between (open subsets of) a [[Cartesian space]].  These differentiable maps can then be used to define the notion of [[differentiable manifold]], and then a more general notion of differentiable map *between* differentiable manifolds, forming a [[category]] called [[Diff]].  We have a parallel hierarchy of ever more differentiable manifolds and ever more differentiable maps between them.  Since every more differentiable manifold has an [[underlying]] less differentiable manifold, we may always consider maps that are less differentiable than the manifolds between which they run.

## Definitions

In all of the following $\mathbb{R}$ denotes the [[real numbers]] and $\mathbb{R}^n$ for $n \in \mathbb{N}$
their $n$-fold [[Cartesian product]]. For $i \in \{1, \cdots, n\}$ we write

$$
  \array{
    \mathbb{R}^n &\overset{pr_i}{\longrightarrow}& \mathbb{R}
    \\
    \vec v = (v_1, \cdots, v_n) &\mapsto& v_i
  }
$$

for the [[projection]] map onto the $i$th factor.

When considering [[convergence]] of [[sequences]] of elements of these
sets we regard them as [[Euclidean space|Euclidean]] [[metric spaces]] with the [[Euclidean norm]]

$$
  {\Vert -\Vert} \;\colon\; \mathbb{R}^n \longrightarrow [0,\infty) \subset \mathbb{R}
  \,.
$$

The [[open subsets]] of the corresponding [[metric topology]] are the [[unions]] of [[open balls]] in $\mathbb{R}^n$.

### In the real numbers

#### Epsilon-delta definition

A function $f:\mathbb{R} \to \mathbb{R}$ is **differentiable** if it comes with a function $\frac{d f}{d x}:\mathbb{R} \to \mathbb{R}$ and a function $M_f:\mathbb{Q}_+ \to \mathbb{Q}_+$ in the positive rational numbers, such that

* for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M_f(\epsilon)$, and for every real number $x \in \mathbb{R}$,
$$\left|f(x + h) - \frac{d f}{d x}(x)\right| \lt \epsilon |h|$$

#### Infinitesimal definition

Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A [[partial function]] $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is **differentiable at a subset** $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d f}{d x}:S \to \mathbb{R}$ such that for all Archimedean ordered Artinian local $K$-algebras $A$ with ring homomorphism $h:K \to A$ and [[nilradical]] $D$ such that for all $\epsilon \in D$, $\epsilon^2 = 0$, for all nilpotent elements $\epsilon \in D$,
$$f_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon$$

A function $f:I \to \mathbb{R}$ is **differentiable at an element** $a \in I$ if it is differentiable at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is **differentiable** if it is differentiable at the [[improper subset]] of $I$.

### From a Cartesian space to the real numbers

+-- {: .num_defn #DerivativeOfFunctionFromRnToR}
###### Definition
**(differentiation of real-valued functions on Cartesian space)**

Let $n \in \mathbb{N}$ and let $U \subset \mathbb{R}^n$ be an [[open subset]].

Then a [[function]] $f \;\colon\; U \longrightarrow \mathbb{R}$ is called  **differentiable** at $x\in U$
if there exists a [[linear map]] $d f_x : \mathbb{R}^n \to \mathbb{R}$ such that the following [[limit of a sequence|limit]]
exists as $h$ approaches zero "from all directions at once":

$$
  \lim_{h\to 0} \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} = 0.
$$

This means that for all $\epsilon \in (0,\infty)$ there exists an [[open subset]] $V\subseteq U$ containing $x$ such that whenever $x+h\in V$ we have $\frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} \lt \epsilon$.

We say that $f$ is __differentiable on__ a [[subset]] $I$ of $U$ if $f$ is differentiable at every $x\in I$, and __differentiable__ (tout court) if $f$ is differentiable on all of $U$. We say that $f$ is **continuously differentiable** if it is differentiable and
$d f$ is a [[continuous function]].

The map $d f_x$ is called the **derivative** or **differential of $f$ at $x$**.

=--

+-- {: .num_remark}
###### Remark
**(Notation and Terminology)**

If $n=1$, as in classical one-variable calculus, then $d f_x$ in def. \ref{DerivativeOfFunctionFromRnToR} may be identified
with a [[real number]], and that number is also called the derivative of $f$ at $x$ and often written $f'(x)$.  (In that case, the notation $d f$ is generally still reserved for the corresponding linear map, with its input denoted by $d x$, so that we have $d f = f'(x) d x$.)

=--

+-- {: .num_remark}
###### Remark
**(equivalent formulation)**

An equivalent way to state def. \ref{DerivativeOfFunctionFromRnToR} is to say that
$$ f(x+h) = f(x) + d f_x(h) + E(h){\Vert h\Vert} $$
where $E$ is a function such that $\lim_{h\to 0}E(h) = 0$.  This is easy to see; just let $E(h) = \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert}$.

Another equivalent way to say it is that
$$ f(x+h) = f(x) + d f_x(h) + E_1(h) h_1 + \cdots + E_n(h) h_n $$
where $E_i$ are functions such that $\lim_{h\to 0}E_i(h) = 0$.  For if this is true, then $E(h) = \frac{1}{\Vert h\Vert}(E_1(h) h_1 + \cdots + E_n(h) h_n)$ satisfies the previous definition.  Conversely, if the previous definition holds, then defining $E_i(h) = \frac{h_i}{\Vert h \Vert} E(h)$ satisfies this definition.

=--

### Partial and directional derivatives

A weaker notion of differentiability is the following:

+-- {: .num_defn #DirectionalDerivative}
###### Definition
**([[partial derivative|directional derivative]])**

Let $n \in \mathbb{N}$ and $U \subset \mathbb{R}^n $ an [[open subset]].

Then a [[function]] $f \colon U \longrightarrow \mathbb{R}$ is said to have **directional derivative**
in the direction of $v \in \mathbb{R}^n$ at $x\in U$ if the [[limit of a sequence|limit]]

$$
  \lim_{h\to 0} \frac{f(x+h v)- f(x)}{h}
$$

exists. Here $h$ is just a real number.

=--

Historically, the term 'directional derivative' was reserved for when $v$ is a [[unit vector]] (or divide the derivative above by $\|v\|$), but the general concept involves less structure and is more important but has no other established name.  If $v$ is a standard basis vector $e_i$, then the directional derivative is called a **[[partial derivative]]** with respect to the corresponding coordinate, and often written $\frac{\partial f}{\partial x_i}$ or $f_{x_i}$.

If $f$ is differentiable at $x$ in the sense of def. \ref{DerivativeOfFunctionFromRnToR}, then $d f_x(v)$ is its directional derivative along $v$.  In particular, the coordinates of $d f_x$ are the partial derivatives of $f$.

In general, $f$ may have all partial derivatives, and even all directional derivatives, without being differentiable;
see the examples [below](#DifferentiabilityOpposedToPartialDifferentiability).  However, if $f$ has all partial derivatives and they are *continuous* as functions of $x$, then in fact $f$ is differentiable (and indeed continuously differentiable).

### Functions between Cartesian spaces

+-- {: .num_defn #FunctionBetweenCartesianSpacesDifferentiation}
###### Definition
**(differentiation of functions between Cartesian spaces)**

Let $n_1, n_2 \in \mathbb{N}$ and let $U\subseteq \mathbb{R}^{n_1}$ be an [[open subset]].

Then a [[function]] $f \;\colon\; U \longrightarrow \mathbb{R}^{n_2}$ is _differentiable_
if for all $i \in \{1, \cdots, n_2\}$ the component function

$$
  f_i \;\colon\; U \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{pr_u}{\longrightarrow}  \mathbb{R}
$$

is differentiable in the sense of def. \ref{DerivativeOfFunctionFromRnToR}.

In this case, the derivatives $d f_i \colon \mathbb{R}^n \to \mathbb{R}$ of the $f_i$ assemble into a linear map
of the form

$$
  d f_x \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
  \,.
$$

=--

### Functions between differentiable manifolds

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

For $X$ and $Y$ [[differentiable manifolds]], then a [[function]] $f \colon X \longrightarrow Y$ is
called _differentiable_ if for
$\left\{ \mathbb{R}^{n} \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\right\}$
an [[atlas]] for $X$
and
$\left\{ \mathbb{R}^{n'} \underoverset{\simeq}{\psi_j}{\to} V_j \subset X_2\right\}$
and atlas for $Y$ then for all $i \in I $ and $j \in J$ the function

  $$
     \mathbb{R}^n
     \supset
      \phantom{AA}
     (f\circ \phi_i)^{-1}(V_j)
       \overset{\phi_i}{\longrightarrow}
     f^{-1}(V_j)
      \overset{f}{\longrightarrow}
     V_j
       \overset{\psi_j^{-1}}{\longrightarrow}
     \mathbb{R}^{n'}
  $$

is differentiable in the sense of def. \ref{FunctionBetweenCartesianSpacesDifferentiation}.

=--

## Higher differentiability

### Iterated differentiability

For $U \subset \mathbb{R}^n$ an [[open subset]] and  $f \;\colon\; U\to \mathbb{R}^m$ a differentiable function (def. \ref{DerivativeOfFunctionFromRnToR}), we may regard its differential $d f$ as a function from a subset of $U$ (the points where $f$ is differentiable) to the space $L(\mathbb{R}^n,\mathbb{R}^m)$ of [[linear maps]].  Since $L(\mathbb{R}^n,\mathbb{R}^m) \cong \mathbb{R}^{n m}$ is again a Cartesian space, we may then ask whether $d f$ is differentiable.

We can then iterate, obtaining the following hierarchy of differentiability.  Because iterated differentiability by itself is not very useful, and a differentiable map is necessarily continuous, one generally includes continuity of the last assumed derivatives.

* The map $f$ is __continuous__ or __$C^0$__ if it is a [[continuous map]] between underlying [[topological spaces]].  We begin with this since a differentiable map is necessarily continuous.

* The map $f$ is __continuously differentiable__ or __$C^1$__ on $U$ if it is differentiable at all points of $U$ and the resulting map $d f$ is [[continuous map|continuous]].  (At this point, if we generalize to infinite-dimensional spaces, we get a variety of notions of when the differential is 'continuous'; see [[continuously differentiable map]] for discussion.)

* The map $f$ is __twice differentiable__ if it is differentiable and its derivative $d f$ is differentiable.  A twice differentiable map must be continuously differentiable.  Similarly, $f$ is __twice continuously differentiable__ or __$C^2$__ if it is twice differentiable and the second derivative $d d f$ is continuous.

* By [[recursion]], $f$ is __$n + 1$ times differentiable__ if $f$ is $n$ times differentiable and the $n$th derivative of $f$ is differentiable.  Similarly, $f$ is __$n$ times continuously differentiable__ or __$C^n$__ if $f$ is $n$ times differentiable and the $n$th derivative of $f$ is continuous.

* The map $f$ is __[[smooth map|smooth]]__ or __infinitely differentiable__ or __$C^\infty$__ if it is $n$ times differentiable for all $n$, or equivalently if it is $C^n$ for all $n$.  (There is no difference between infinite differentiability and infinite continuous differentiability.)  One may also define this notion [[coinductive definition|coinductively]]: $f$ is infinitely differentiable if it is differentiable and its derivative $d f$ is infinitely differentiable.

* One step higher, we may ask whether $f$ is [[analytic function|analytic]] or $C^\omega$.

### Uniform differentiability

Note that $f$ is differentiable on a set $U$ iff there is a function $f'$ on $U$ (necessarily unique, assuming that $U$ has no [[isolated points]]) such that

$$ \forall\, \epsilon \gt 0,\; \forall\, x \in U,\; \exists\, \delta \gt 0,\; \forall\, y \in U,\; {|{y - x}|} \lt \delta \;\Rightarrow\; {|{f(y) - f(x) - f'(x)(y - x)}|} \lt \epsilon \,{|{y - x}|} .$$

Reverse quantifiers, and $f$ is __uniformly differentiable__ on $U$ iff there is a function $f'$ on $U$ such that

$$ \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, x \in U,\; \forall\, y \in U,\; {|{y - x}|} \lt \delta \;\Rightarrow\; {|{f(y) - f(x) - f'(x)(y - x)}|} \lt \epsilon \,{|{y - x}|}.$$

In [[classical mathematics]], $f$ is uniformly differentiable if and only if $f$ is differentiable and its derivative $f'$ is [[uniformly continuous map|uniformly continuous]]; in other words, $f$ is uniformly differentiable iff $f$ is _uniformly-continuously differentiable_.  The same is true in [[constructive mathematics]] as long as one assumes [[dependent choice]]. In the absence of dependent choice, however, the argument only goes one way, and uniform differentiability is stronger.  Furthermore, just as pointwise continuity is not as well behaved constructively as uniform continuity, so pointwise differentiability is not as well behaved constructively as uniform differentiability.  For this reason, uniform differentiability is particularly important in constructive mathematics.

In addition, one could talk about **locally uniform differentiablilty**, which is a continuously differentiable function on a set $U$ which is uniformly differentiable on every closed and bounded subset $V \subseteq U$.

### Symmetry of higher derivatives {#symmetry}

If $f:U\to \mathbb{R}^m$ is twice differentiable with $U\subseteq \mathbb{R}^n$, its second derivative
$$d(d f) : U \to L(\mathbb{R}^n,L(\mathbb{R}^n,\mathbb{R}^m)) \cong Bilin(\mathbb{R}^n,\mathbb{R}^n;\mathbb{R}^m)$$
is a function from $U$ into the space of [[bilinear maps]] from $\mathbb{R}^n\times \mathbb{R}^n$ to $\mathbb{R}^m$.

+-- {: .num_theorem #symmetrythm}
###### Theorem
**(Symmetry of higher derivatives)**.
If $f:U\to \mathbb{R}^m$ is twice differentiable, then its second derivative $d(d f)$ lands in the space of *symmetric* bilinear maps, i.e. for any $x\in U$ and $v,w\in \mathbb{R}^n$ we have
$$ d(d f)_x(v,w) = d(d f)_x(w,v). $$
=--
+-- {: .proof}
###### Proof
It suffices to assume $m=1$; otherwise we just consider it componentwise.  Define a function $g:\mathbb{R}\to \mathbb{R}$ by
$$ g(\xi) = f(x+\xi v+w) - f(x+\xi v). $$
Then by the chain rule, $g$ is differentiable and
$$
\begin{aligned}
  g'(\xi) &= d f_{x+\xi v+w}(v) -  d f_{x+\xi v}(v)\\
  &= \Big(d f_{x+\xi v+w}(v) - d f_{x}(v)\Big) - \Big(d f_{x+\xi v}(v) - d f_x(v)\Big)\\
  &= d(d f)_{x}(v,\xi v + w) + E_1 |v|\,|\xi v + w| - d(d f)_x(v,\xi v) - E_2 |v|\,|\xi v|\\
  &= d(d f)_x(v,w) + E |v|\, (|v|+|w|)
\end{aligned}
$$
where $E_1, E_2, E \to 0$ as $(v,w)\to 0$.  Now the [[mean value theorem]] tells us that for some $\xi\in(0,1)$ we have
$$
\begin{aligned}
  f(x+v+w) - f(x+v) - f(x+w) + f(x)
  &= g(1) - g(0)\\
  &= g'(\xi)\\
  &= d(d f)_x(v,w) + E (|v|+|w|)^2.
\end{aligned}
$$
But the "second-order difference" $f(x+v+w) - f(x+v) - f(x+w) + f(x)$ is manifestly symmetric in $v$ and $w$, so we have
$$ d(d f)_x(v,w) - d(d f)_x(w,v) = E (|v|+|w|)^2 $$
where $\lim_{(v,w)\to 0} E = 0$.  But bilinearity of the LHS then implies that it is identically zero.
=--

The components of the bilinear map $d(d f)$ are the second-order partial derivatives $\frac{\partial^2 f}{\partial x_i \partial x_j}$ of $f$.  Thus, this theorem says that if $f$ is twice differentiable, then the *mixed partials* are equal,
$$\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.$$

The second-order partial derivatives may *exist* without the mixed partials being equal; see below for a counterexample.  However, the theorem shows that if we require $f$ to actually be twice differentiable rather than merely having second-order partial derivatives, then this cannot happen.

In particular, we have the following corollary, which is more commonly found in textbooks.

+-- {: .num_cor #symmetrycontcor}
###### Corollary
**(Symmetry of continuous higher derivatives)**.
If $f$ has first and second-order partial derivatives, and the latter are continuous in a neighborhood of $x$, then the mixed partial derivatives are equal,
$$\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.$$
=--
+-- {: .proof}
###### Proof
Continuity of the second-order partials implies that the first-order partials are differentiable, and hence so is the differential $d f$.
=--

### A direct definition of the second derivative

Note that the proof of the theorem implies that if $f$ is twice differentiable at $x$, then there exists a bilinear map $\partial^2 f_x : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$ such that

$$f(x+v+w) - f(x+v) - f(x+w) + f(x) = \partial^2 f_x(v,w) + E(v,w) ({|v|+|w|})^2 $$

where $\lim_{v,w\to 0} E(v,w) = 0$.  This is a condition that makes sense as a condition on an arbitrary $f$ without assuming differentiability. and if it holds, then the bilinear map $\partial^2 f_x$ must be symmetric.

Moreover, as explained [here](http://mathoverflow.net/a/165733/49), if $f$ is differentiable in a neighborhood of $x$ and satisfies this condition at $x$, then it is in fact twice differentiable at $x$.  To see this, it suffices to show that each coordinate of $d f$ is differentiable, so let $w=\delta e_i$, with $e_i$ a unit basis vector and $\delta$ a real number $\neq 0$.  Then we have

$$ E(v,\delta e_i) = \frac{f(x+v+\delta e_i) - f(x+v) - f(x+\delta e_i) + f(x) - \partial^2 f_x(v,\delta w)}{{|v|}{\delta}}$$

Taking the limit as $\delta\to 0$ and using differentiability of $f$ at $x$ and $x+v$ (for sufficiently small $v$), we get

$$ \lim_{\delta \to 0} E(v,\delta e_i) = \frac{d f_{x+v}(e_i) - d f_x(e_i) - \partial^2 f_x(v,e_i)}{{|v|}}. $$

Now take the limit as $v\to 0$; on the left we get $0$ by assumption, so the function $y\mapsto d f_y(e_i)$ is differentiable at $x$ with derivative $\partial^2 f_x(-,e_i)$.  Thus, $d f$ is differentiable at $x$.

On the other hand, this condition by itself does not even imply that $f$ is continuous.  For instance, if $f$ is a $\mathbb{Q}$-linear map $\mathbb{R}\to \mathbb{R}$, then the second-order difference $f(x+v+w) - f(x+v) - f(x+w) + f(x)$ is identically zero.

### Higher differentiability on tangent spaces and the chain rule

Let $f:\mathbb{R}^n\to \mathbb{R}$ be differentiable.  Instead of asking whether $d f : U \to L(\mathbb{R}^n,\mathbb{R})$ is differentiable, we can ask whether its exponential transpose $d f : U\times \mathbb{R}^n \to \mathbb{R}$ is differentiable.  (Note that $U\times \mathbb{R}^n$ is the [[tangent bundle]] of $U\subseteq \mathbb{R}^n$.)  This amounts to asking that for $x\in U$ and $v\in \mathbb{R}^n$, we have
$$ d f_{x+w}(v+h) - d f_x(v) = d^2 f_{(x,v)}(w,h) + E(|w|+|h|) $$
for a linear map $d^2 f_{(x,v)}:\mathbb{R}^{2n} \to \mathbb{R}$, where $\lim_{(w,h)\to 0} E = 0$.  Setting $h=0$, we see that this implies that $d f : U \to L(\mathbb{R}^n,\mathbb{R})$ is differentiable, with differential $d(d f)_x = d^2 f_{(x,v)}(w,0)$ for any $v$.  And setting $w=0$, we obtain $d f_x(h) = d^2 f_{(x,v)}(0,h)$ for any $v$.  Thus, we can write
$$ d^2 f_{(x,v)}(w,h) = \partial^2 f_x(v,w) + d f_x(h) $$
where $\partial^2 f_x$ is the symmetric bilinear map from the previous section.  Conversely, if $f$ is twice differentiable, then using linearity and continuity of $d f$ it is easy to see that the above condition holds.

Therefore, these two kinds of twice-differentiability of $f$ are equivalent as *conditions* on $f$, but the resulting *second differential* is different.  In the second case, we get

$$d^2f = \partial^2 f + d f = \sum_{i,j} \frac{\partial^2f}{\partial x_i \partial x_j} d x_i \, d x_j + \sum_i \frac{\partial f}{\partial x_i} d^2 x_i. $$

rather than merely the first term $\sum_{i,j} \frac{\partial^2f}{\partial x_i \partial x_j} d x_i \, d x_j$.  There are two advantages to the second approach (asking that $d f : U\times \mathbb{R}^n \to \mathbb{R}$ be differentiable).

Firstly, we can reformulate it in terms of $f$ by asking that there exists a linear map $d f_x : \mathbb{R}^n \to \mathbb{R}^m$ and a bilinear map $\partial^2 f_x : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$ such that

$$f(x+v+w+h) - f(x+v) - f(x+w) + f(x) = \partial^2 f_x(v,w) + d f_x(h) + E(v,w,h) \sqrt{{|v|}^2{|w|}^2 + {|h|}^2}.$$

where $\lim_{v,w,h \to 0} E(v,w,h) = 0$.  This holds if $f$ is twice differentiable, for then we can write
$$
\begin{aligned}
  f(x+v+w+h) &= f(x+v+w) + d f_{x+v+w}(h) + E {|h|}\\
  &= f(x+v+w) + d f_x(h) + d(d f)_x(v+w,h) + E {|v+w|}{|h|} + E{|h|}
\end{aligned}
$$
and $d(d f)_x(v+w,h)$ can also be incorporated into the error term.  Of course, setting $h=0$ we obtain the characterization of twice-differentiability from the previous section.  But setting $v=w=0$, we find that $f$ is differentiable at $x$ with derivative $d f_x$.  So here we have a direct characterization of the second derivative which also implies that the first derivative exists (although it seems that it doesn't imply differentiability in a neighborhood of $x$, so that the resulting "second derivative" may not actually be the derivative of the first derivative).

Secondly, the virtue of a second differential incorporating the first derivatives is that like the first differential $d f$, but unlike the bilinear map $\partial^2 f$, it satisfies [[Cauchy's invariant rule]].  This means that we can express the [[chain rule]] for second differentials of composite maps simply by substitution: if $y = f(u)$ and $u = g(x)$, then finding $d^2 y$ in terms of $d u$ and $d^2 u$, and $d u$ and $d^2 u$ in terms of $d x$ and $d^2 x$, and substituting, gives the correct expression for $d^2 y$ in terms of $d x$ and $d^2 x$.

In fact, this can be proven using the above direct characterization of the second differential, in essentially exactly the same way that we prove the ordinary chain rule for first derivatives.  We sketch the proof, omitting the explicit error terms.  We can write

$$g(f(x+v+w+h)) - g(f(x+v)) - g(f(x+w)) + g(f(x)) $$

as

$$ g(f(x) + v' + w' + h') - g(f(x) + v') - g(f(x) + w') + g(f(x)) $$

where $v' = f(x+v) - f(x)$ and $w' = f(x+w) - f(x)$ and $h' = f(x+v+w+h) - f(x+v) - f(x+w) + f(x)$.  Now by extra-strong twice differentiability of $g$, this is approximately equal to

$$ \partial^2 g_{f(x)}(v',w') + d g_{f(x)}(h'). $$

But by differentiability of $f$, we have $v' \approx d f_x(v)$ and $w' \approx d f_x(w)$, while by extra-strong twice differentiability of $f$ we have $h' \approx \partial^2 f_x(v,w) + d f_x(h)$.  Thus, we obtain approximately

$$ \partial^2 g_{f(x)}(d f_x(v), d f_x(w)) + d g_{f(x)}(\partial^2 f_x(v,w) + d f_x(h)) $$

which is exactly what we would get by substitution.

### For maps between manifolds

If $X$ and $Y$ are $C^k$-[[differentiable manifolds]], then we may define what it means for a map $f:X\to Y$ to be $n$ times differentiable, or $C^n$, for any $n\le k$, by asking that it yield such a map when restricted to any charts.  The most common case is when $X$ and $Y$ are smooth (infinitely differentiable) manifolds, so that we can define $C^n$ functions between them for all $n\le \infty$.  (Analytic manifolds, which are necessary in order to define analyticity of $f$, are somewhat rarer.)

A differentiable map between manifolds induces a map between their [[tangent bundles]] $d f : T X \to T Y$; this operation extends to a [[functor]] from $C^{k+1}$ manifolds and $C^{k+1}$ maps to $C^k$ manifolds and $C^k$ maps.  See [[differentiation]] for more.

## Examples and non-examples

### Differentiability versus partial differentiability
 {#DifferentiabilityOpposedToPartialDifferentiability}

The function $f:\mathbb{R}^2 \to \mathbb{R}$ defined by

$$ f(x,y) =
\begin{cases}
  \frac{y^3}{x^2+y^2} &\quad (x,y) \neq (0,0)\\
  0 &\quad (x,y) = (0,0)
\end{cases}
$$

is continuous everywhere, and has directional derivatives (def. \ref{DirectionalDerivative}) at $(0,0)$ in all directions, but is not differentiable at $(0,0)$ in the sense of def. \ref{DerivativeOfFunctionFromRnToR}.  Note that the (unnormalized) directional derivative along the vector $(a,b)$ is $\frac{b^3}{a^2+b^2}$, which is not linear.

One may wonder whether existence of a linear map $d f_x$ is enough, but this is also not the case.  The function $f:\mathbb{R}^2 \to \mathbb{R}$ defined by

$$ f(x,y) = \begin{cases} \frac{y^3}{x} &\quad x\neq 0\\ 0 &\quad x=0 \end{cases} $$

has all directional derivatives at $(0,0)$ equaling $0$, so that in particular there is a linear map $d f_{(0,0)}$ whose values are the directional derivatives.  But it is not differentiable at $(0,0)$; in fact, it is not even continuous at $(0,0)$.  (Thus it also provides an example of a discontinuous function which has all directional derivatives.)

The discontinuity kind of gives this last one away; maybe a continuous function with all directional derivatives must be differentiable?  But no, because if $W$ is a [[Weierstrass function]] (continuous everywhere but differentiable nowhere) from $\mathbb{R}$ to $\mathbb{R}$ with [[periodic function|period]] $2\pi$, and $\operatorname{atan2}$ is a $2$-argument [[arctangent]] function from $\mathbb{R}^2$ to $\mathbb{R}$ (so that $\sqrt{x^2+y^2} \,\sin\,\operatorname{atan2}(y,x) = y$ and $\sqrt{x^2+y^2} \,\cos\,\operatorname{atan2}(y,x) = x$ for all $x$ and $y$), then set

$$ f(x,y) = (x^2 + y^2) W(\operatorname{atan2}(y,x)) .$$

Then $f$ is, like $W$, continuous everywhere and differentiable nowhere, yet the directional derivatives through the origin are all $0$.  There are further examples (including one that is analytic except at the origin) at [Math StackExchange question 1497043](https://math.stackexchange.com/questions/1497043/is-there-a-function-thats-continuous-and-has-all-directional-derivatives-as-a-l).

### Differentiability versus continuous and higher differentiability

Let $k$ be a [[natural number]], and consider the function
$$ f_k(x) \coloneqq x^k \sin(1/x) $$
from the [[real line]] to itself, with $f(0) \coloneqq 0$.  Away from $0$, $f_k$ is smooth (even analytic); but at $0$, $f_0$ is not continuous, $f_1$ is continuous but not differentiable, $f_2$ is differentiable but not continuously differentiable, and so on:

*  If $k = 2 n$ is even, then $f_k$ is differentiable $n$ times but not continuously differentiable $n$ times;
*  If $k = 2 n + 1$ is odd, then $f_k$ is continuously differentiable $n$ times but not differentiable $n + 1$ times.

(For a fractional value of $k$, $f_k$ has the same behaviour as $f_{\lceil{k}\rceil}$.)

Similarly, in two dimensions we can consider functions such as
$$ f(x,y) = (x^2+y^2)\sin(\frac{1}{\sqrt{x^2+y^2}}).$$
together with $f(0,0) = 0$.  This is smooth away from $0$, and once differentiable at $0$, even in the strong sense that it is well-approximated by a linear function near $0$.  However, its derivative is not continuous at $0$.  In particular, this shows that the converse of the theorem "if the partial derivatives exist and are continuous at a point, then the function is differentiable there" fails, even in higher dimensions.

Uniform differentiability is stronger than continuous differentiability but independent of twice differentiability.  For an example that is uniformly differentiable but not twice differentiable, use $f_3$ again.  For an example that that is twice differentiable (and hence continuously differentiable) but not uniformly differentiable, use $x \mapsto x^3$.  However, on a [[compact set|compact]] domain, any continuously differentiable function (and a fortiori any twice differentiable function) must be uniformly continuous (at least in [[classical mathematics|classical]] and [[intuitionistic mathematics]]).

### Symmetry of the second partial derivatives

The function $f:\mathbb{R}^2 \to \mathbb{R}$ defined by

$$ f(x,y) = \frac{x y (x^2-y^2)}{x^2+y^2}$$

plus $f(0,0) = 0$, has partial derivatives $\frac{\partial^2f}{\partial x \partial y}$ and $\frac{\partial^2f}{\partial y \partial x}$ that both exist but are not equal at $(0,0)$ (nor are they continuous at $(0,0)$).  Therefore, by the theorem proven above, it is not twice differentiable at $(0,0)$.

### Twice differentiability versus quadratic approximation

While differentiability means approximability by a linear function, twice differentiability does *not* mean approximability by a quadratic function.  For example, the function

$$ f(x) = x^3 \sin(1/x) $$

(with $f(0) =0$, to make it continuous there) is not twice differentiable at $x=0$, but it is well-approximated by the quadratic polynomial $p(x) = 0$ in the sense that their difference is $o(x^2)$ as $x\to 0$.  Even worse, the function

$$ f(x) = e^{-1/x^2} \sin(e^{1/x^2}) $$

is not twice differentiable at $x=0$, but it is well-approximated by a polynomial of *any* finite degree $n$ (namely, the zero polynomial), in the sense that their difference is $o(x^n)$ as $x\to 0$.

In some contexts, it is useful to say that functions such as these have "pointwise second derivatives".  More precisely, we say that a function $f$ has a **pointwise $k^{th}$ derivative** at $a$ if it is continuous at $a$ and there exists a polynomial $p$ of degree $k$ such that

$$ \lim_{x\to a} \frac{f(x) - p(x)}{(x-a)^k} = 0. $$

(We have to state continuity explicitly in order to match the usual notion of differentiability when $k = 1$, since nothing in this limit constrains the value of $f(a)$.)  In this case, the pointwise $k^{th}$ derivative is $f^{(k)}_{pt}(a) = p^{(k)}(a)$ (and it follows that all lower-order pointwise derivatives match as well).  Thus we would say that while $f(x) = x^3 \sin(1/x)$ does not have a second derivative at $0$, it does have a *pointwise* second derivative at $0$, and $f_{pt}''(0) = 0$.  Similarly, $e^{-1/x^2} \sin(e^{1/x^2})$ is pointwise smooth.  See e.g. [this MSE answer](http://math.stackexchange.com/a/160383/91608).

The polynomial $p$ may be thought of as the $k$th-order [[Taylor polynomial]] of $f$ at $a$, and the limit above is Taylor's theorem with the Peano remainder.  (Thus the subscript $pt$ can be thought of as standing for ‘Peano--Taylor’ as well as for ‘pointwise’.)  Note that while $k$-times pointwise differentiability is weaker than $k$-times differentiability for $k \gt 1$ and equivalent for $k = 1$, it is stronger for $k = 0$ (since even $0$-times pointwise differentiability requires continuity).

### The second derivative as a quadratic form

In the definition of strong twice-differentiability, we cannot replace the symmetric [[bilinear map]] $\partial^2 f_x(v,w)$ by the corresponding [[quadratic form]] $Q_x(v) = \partial^2f_x(v,v)$.  In other words, if we suppose that

$$f(x+2v) - 2 f(x+v) + f(x) = Q_x(v) + E(v) {|v|}^2 $$

where $\lim_{v\to 0} E(v) = 0$, for some quadratic form $Q_x$, it does not follow that $f$ is twice differentiable at $x$, even in one dimension.  The same old counterexample

$$ f(x) = x^3 \sin(1/x) $$

(with $f(0)=0$) works: we have

$$f(0+2v) - 2 f(0+v) + f(0)= 2v^3(4\sin(\frac{1}{2v}) - \sin(\frac{1}{v}))$$

so $E(v) = v (4\sin(\frac{1}{2v}) - \sin(\frac{1}{v})) \to 0$.

## Related concepts

* [[smooth function]]

* [[extremum]]

* [[chain rule]], [[total derivative]]

* [[continuous function]], **differentiable function**, [[continuously differentiable function]], [[smooth function]], [[analytic function]]

* [[derivative of distributions]]

* [[Hasse-Schmidt derivative]]

* [[integrable function]], [[square-integrable function]]

* [[bounded function]]

* [[compactly supported function]]

* [[measurable function]]

* [[rapidly decreasing function]]

## References

Early account, in the context of [[Cohomotopy]], [[cobordism theory]] and the [[Pontryagin-Thom construction]]:

* {#Pontrjagin55} [[Lev Pontrjagin]], Chapter I of: _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

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differentiable space

A kind of [[space]] that admits a notion of [[differentiation]] of [[functions]] on it.

For example:

* [[differentiable manifold]]

* [[microlinear space]]

* [[smooth space]]

* [[archimedean field]]

* [[algebraic limit field]]

[[!redirects differentiable spaces]]


differentiable stack

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## Idea

A _differentiable stack_ is a [[geometric stack]] on the [[site]] [[SmoothMfd]] of [[smooth manifold]]s: a stack which is represented by a [[Lie groupoid]].

This means that a differentiable stack is in a way _nothing else_ but a Lie groupoid: but the point is that equivalence of stacks captures the notion of [[Morita equivalence]] of their presenting (Lie) groupoids.

This means that looking at a Lie groupoid in terms of the stack it presents provides a direct way to recognizing the right notion of equivalence of these objects. The notion of Morita equivalence, on the other hand, proceeds via the reasoning of [[homotopy theory]].

Notice that the stack [[presented stack|presented]] by a (Lie) groupoid $G$ is really the stack which sends every test manifold to the category of $G$-principal bundles over that manifold. Such a $G$-principal bundle, also known as a [[torsor]], is analogous to a [[module]] for an algebra. This explains the terminology of "Morita morphisms", which originates in algebra:

Just as two algebras are Morita equivalent if their categories of modules are equivalent, two groupoids are Morita equivalent if their stacks of torsors are equivalent.

## Properties

### Characterization by Lie groupoids
 {#CharacterizatiobByLieGroupoids}

Sending a [[Lie groupoid]] to the [[smooth stack]] it represents constitutes an [[equivalence of (infinity,1)-categories]] between [[Lie groupoids]] with Morita morphisms/[[bibundles]] between them and differentiable stacks.

(e.g. [Blohmann 07](#Blohmann), [Carchedi 11](#Carchedi11)). For review in a broader context see also [Nuiten 13, around prop. 2.2.34](#Nuiten13)

### Other descriptions

A differentible stack is in particular an object in the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] that is

* [[1-truncated]]

* concrete.

It is however more special than that. The general 1-truncated concrete [[smooth ∞-groupoids]] are [[internal groupoids]] in [[diffeological spaces]].

### Mapping stacks

For $X$, $Y$ two differentiable stacks, there is the [[mapping stack]] $[X,Y]$, which a priory is only a [[smooth stack]]. It should be true that a sufficient condition for this itself to be a [[Frechet manifold|Frechet]]-differentiable stack is that $X$ has an [[atlas]] by a [[compact topological space|compact]] [[smooth manifold]].

(See also [[manifold structure of mapping spaces]].)

## Related concept

* [[geometric stack]]

  * [[algebraic stack]]

  * [[topological stack]]

  * **differentiable stack**

    * [[groupoid K-theory]]

* [[geometric ∞-stack]]

* [[cohesive ∞-groupoid]]

  * [[discrete ∞-groupoid]]

  * [[Euclidean-topological ∞-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

## References

A comprehensive account is in

* {#Carchedi11} [[David Carchedi]], _Categorical Properties of Topological and Diffentiable Stacks_, PhD thesis, Universiteit Utrecht, 2011 ([dspace:1874/208971](https://dspace.library.uu.nl/handle/1874/208971), [pdf](http://math.gmu.edu/~dcarched/Thesis_David_Carchedi.pdf), [[CarchediDifferentiableStacks.pdf:file]])

See also

* {#Blohmann} [[Christian Blohmann]], _Stacky Lie groups_, Int. Mat. Res. Not. (2008) Vol. 2008 rnn082 ([arXiv:math/0702399](http://arxiv.org/abs/math/0702399), [doi:10.1093/imrn/rnn082](https://doi.org/10.1093/imrn/rnn082))

* [[Kai Behrend]], [[Ping Xu]], _Differentiable Stacks and Gerbes_ J. Symplectic Geom. Volume 9, Number 3 (2011), 285-341. ([arXiv:math/0605694](http://arxiv.org/abs/arXiv:math/0605694))

* [[Grégory Ginot]], _Introduction to Differentiable Stacks (and gerbes, moduli spaces ...)_, 2013 ([pdf](https://webusers.imj-prg.fr/~gregory.ginot/papers/DiffStacksIGG2013.pdf), [[GinotDifferentiableStacks.pdf:file]])

* [[Jochen Heinloth]], *Notes on differentiable stacks* (2004) [[pdf](https://math.nyu.edu/~tschinke/WS04/pdf/heinloth.pdf), [[Heinloth-DifferentiableStacks.pdf:file]]]

* [[Richard Hepworth]], _Vector fields and flows on differentiable stacks_ ([arXiv](http://arxiv.org/abs/0810.0979)).

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, MSc thesis, Utrecht, August 2013

See also the references at [[geometric stack]] and [[topological stack]].

[[!redirects differentiable stacks]]


differentiable vector bundle

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## Idea

A _differentiable vector bundle_ is a [[vector bundle]] in the context of _[[differential geometry]]_: a _differentiably varying collection of [[vector space]] over a given [[differentiable manifold]].

All this for some specified degree of [[differentiation|differentiability]]. If one demands arbitrary differentiabiliy then one speaks of _smooth vector bundles_ over [[smooth manifolds]].

## Examples

For $X$ a [[differentiable manifold]], then its [[tangent bundle]] $T X \to X$ is a differentiable vector bundle, see [this lemma](tangent+bundle#DifferentiableTangentBundle).

## Related concepts

* [[topological vector bundle]]

* [[differentiable vector bundle]]

[[!redirects differentiable vector bundles]]

[[!redirects smooth vector bundle]]
[[!redirects smooth vector bundles]]


differential

> maybe you are looking for the (total) _[[derivative]]_ (differential) of a map

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## Definition

Abstractly, a __differential__ is the morphism of a [[differential object]] defining a [[chain complex]]: the _boundary operator_.

So in a [[category with translation]] $T : C \to C$ a differential is a morphism $d_V : V \to T(V)$ for some [[object]] $V$ such that

$$
  V \stackrel{d_V}{\to} T V \stackrel{T(d_V)}{\to}
  T T V
$$

is the [[zero morphism]].

Unwrapping this for the case where the [[category with translation]] is a [[category of chain complexes]], it reproduces the ordinary notion of a differential as a degree-$(-1)$ morphism that squares to $0$.

More concretely, the boundary operator on a chain complex is called a _differential_ if this is part of the structure of a [[dg-algebra]] on the complex.

## Examples

The archetypical example that gives the concept its name is the differential in the [[de Rham complex]] $\Omega^\bullet(X)$ of a [[smooth manifold]] $X$, which is given by actual [[differentiation]] of [[smooth function]]s and [[differential form]]s. See also [[differential calculus]].

[[!redirects differentials]]


differential 2-crossed module

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* [[crossed complex]] $\Leftarrow$ [[crossed module]] $\Rightarrow$ [[2-crossed module]]

* [[dg-Lie algebra]] $\Leftarrow$ [[differential crossed module]] $\Rightarrow$ **differential 2-crossed module**

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## Idea

A _differential 2-crossed module_ encodes a _[[semi-strict infinity-category|semistrict]] [[Lie 3-algebra]]_ in terms of a system of ordinary [[Lie algebra]]s and morphisms between them. It generalizes the way a [[differential crossed module]] encodes a strict [[Lie 2-algebra]]

## References

The definition is reproduced for instance in section 1.2 of

* [[Joao Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv:0907.2566](http://arxiv.org/abs/0907.2566))


differential algebra

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## Idea

A _differential algebra_ is an [[associative algebra]] $A$ equipped with a [[derivation]] $d \colon A \to A$, typically required to satisfy $d \circ d = 0$.

If $A$ is a [[field]] one accordingly speaks of a _differential field_.

If $A$ is a [[graded algebra]] and $d$ is of degree 1 one speaks of a _[[differential graded algebra]]_.

Etc.

## Related concepts

* [[differential ideal]]

* [[differential graded algebra]]

* [[differential graded-commutative algebra]]

* [[differential algebroid]]

## References

* [[differential Galois theory]]

[[!include oidification - table]]

[[!redirects differential algebras]]
[[!redirects differential field]]
[[!redirects differential fields]]


differential algebraic K-theory

> under construction

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## Idea

Differential algebraic K-theory is the [[differential cohomology]]-refinement of [[algebraic K-theory]].

In ([Bunke-Tamme 12](#BunkeTamme12)) this is realized effectively as the [[schreiber:differential cohomology in a cohesive topos]] of the [[tangent (∞,1)-topos]] of the [[cohesive (∞,1)-topos]]

$$
  \mathbf{H} \coloneqq Sh_\infty\left(SmthMfd, \mathbf{B} \right)
  \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  \mathbf{B}
$$

of [[∞-stacks]] on the [[site]] of [[smooth manifolds]] with values in turn in [[∞-stacks]]  over a [[site]] of [[arithmetic schemes]], hence by [[smooth ∞-groupoids]] but over a [[base (∞,1)-topos]]

$$
  \mathbf{B}
  \coloneqq
  Sh_\infty\left(Sch_{\mathbb{Z}}\right)
$$

of algebraic [[∞-stacks]].

This may be regarded as sitting inside the [[smooth E-∞-groupoids]].

It is observed in this context that the [[Beilinson regulator]] in [[algebraic K-theory]] is naturally understood as a [[Chern character]] in this perspective of [[differential cohomology]] ([Bunke-Tamme 12](#BunkeTamme12)), which helps with studying it.

## Definition

### Absolute Hodge cohomology

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

Write

$$
  \Omega^\bullet_{\mathbb{C}} \in Stab(Sh_\infty(Sch_{\mathbb{C}}))
$$

for the [[chain complex]] of [[abelian sheaves]] (regarded as a [[sheaf of spectra]] under the [[stable Dold-Kan correspondence]]) which computes [[absolute Hodge cohomology]] of [[complex varieties]].

=--

([Bunke-Tamme 12, section 3.1](#BunkeTamme12))

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

Write

$$
  \Omega^\bullet_{\mathbb{Z}} \coloneqq compl^\ast \Omega^\bullet_{\mathbb{C}} \in Stab(\mathbf{B})
$$

for the [[inverse image]] of $\Omega^\bullet_{\mathbb{C}}$ under the [[base change]] given by

$$
  compl
  \coloneqq
  (-)\times_{\mathbb{Z}}Spec(\mathbb{C})
  \;\colon\;
  Sch_{\mathbb{Z}}\longrightarrow Sch_{\mathbb{C}}
  \,.
$$

=--

([Bunke-Tamme 12, section 3.2](#BunkeTamme12))

There is a resolution of
$\Omega^\bullet_{\mathbb{Z}} \in Stab(\mathbf{B}) \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})$ by a sheaf of complexes of differential forms on smooth manifolds tensored with $\Omega^\bullet_{\mathbb{Z}}$

$$
  \Omega^\bullet \in Stab(\mathbf{H})
$$

([Bunke-Tamme 12, (47)](#BunkeTamme12)).

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

While $\Omega^\bullet \simeq \Omega^\bullet_{\mathbb{Z}}$, below we use the chain-level truncation $\Omega^{\bullet \geq 0}$ which is no longer in the image of $Disc$, hence no longer a [[flat modality]]-[[modal type]].

=--

### Algebraic K-theory sheaf of spectra
 {#AlgebraicKTheorySheafOfSpectra}

Write

$$
  \mathbf{Vect}_{lc} \in \mathbf{H}
$$

for the [[stack]] which to $X\times S \in SmthMfd \times Sch_{\mathbb{Z}}$ assigns the groupoid of locally free and locally finitely generated $pr_S^\ast \mathcal{O}_S$-[[modules]] (modules over the [[inverse image]] of the [[structure sheaf]] of $S$ under the [[projection]] map $X \times S \to S$).

This is a [[commutative monoid]] object with respect to [[direct sum]]. Write

$$
  K \coloneqq \mathcal{K}(\mathbf{Vect}_{lc}^{\oplus})
$$

for the stackification of the objectwise [[K-theory of a symmetric monoidal (∞,1)-category]]-construction.

This is the ordinary [[algebraic K-theory]] of [[schemes]], as in ([Thomason-Trobaugh 90](#ThomasonTrobaugh90)) ([Bunke-Tamme 12, section 3.3](#BunkeTamme12)), see at _[algebraic K-theory -- as the K-theory of algebraic vector bundles](algebraic%20K-theory#AsTheKTheoryOfAlgebraicVectorBundles)_.

### The refined Beilinson regulator
 {#RefinedBeilinsoRegulator}

There is a refinement of the [[Beilinson regulator]] to a smoothly parameterized version $\mathbf{K}$ of [[algebraic K-theory]]:

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

(...)

$$
  r^{Beil}
  \;\colon\;
  K
    \longrightarrow
  \Omega^\bullet_{\mathbb{Z}}
$$

=--

As a homomorphism of [[spectrum objects]] this is ([Bunke-Tamme 12, def. 4.26](#BunkeTamme12)). As a homomorphism of [[E-∞ ring]] objects, this is ([Bunke-Tamme 13, def. 2.18](#BunkeTamme13)).

### Differential algebraic K-theory

Recall the discussion at _[[differential cohomology hexagon]]_.

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

_Differential algebraic K-theory_ is the [[homotopy fiber product]]

$$
  \hat K \coloneqq K \underset{\Omega^\bullet}{\times} \Omega^{\bullet \geq 0}  \in Stab(\mathbf{H})
$$

of the inclusion of the non-negative degree truncation of the de Rham resolution of the absolute Hodge complex, def. \ref{CyclesForHodgeComplexOverComplexNumbers}, with the refined Beilinson regulator, def. \ref{TheBeilinsonRegulator}

$$
  \array{
    \hat K &\longrightarrow& \Omega^{\bullet \geq 0}
    \\
    \downarrow &(pb)& \downarrow
    \\
    K &\underset{r^{Beil}}{\longrightarrow}& \Omega^\bullet
  }
  \,.
$$

=--

([Bunke-Tamme 12, def. 5.1](#BunkeTamme12), [Bunke-Tamme 13, def. 3.1](#BunkeTamme13)).

## Applications

* differential refinement of [[Becker-Gottlieb transfer]] and its image under [[Borel regulators]]. See at _[[transfer index conjecture]]_.

## Related concepts

* [[differential K-theory]]

* [[differential form with logarithmic singularities]]

## References

Differential algebraic K-theory as above is introduced with application to [[higher regulator|higher]] [[Beilinson regulators]] in:

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_, Advances in Mathematics Volume 285, 5 November 2015, Pages 1853-1969
([arXiv:1209.6451](http://arxiv.org/abs/1209.6451), [doi:10.1016/j.aim.2015.08.004](https://doi.org/10.1016/j.aim.2015.08.004))

* [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_, Memoirs of the American Mathematical Society 2021 ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247), [ISBN:978-1-4704-4685-7](https://bookstore.ams.org/cdn-1608276327798/memo-269-1316/))

* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_,  Ann. K-Theory 1(3): 227-258 (2016)   ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421), [doi:10.2140/akt.2016.1.227](https://doi.org/10.2140/akt.2016.1.227))

Relevant references in ordinary [[algebraic K-theory]] include

* {#ThomasonTrobaugh90} [[Robert Thomason]] and Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhauser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)

[[!redirects differential algebraic K-theory]]

[[!redirects differential algebraic K-theories]]

[[!redirects differential algebraic K-theories]]