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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Kähler differential} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea_and_definition}{Idea and definition}\dotfill \pageref*{idea_and_definition} \linebreak \noindent\hyperlink{the_ordinary_definition_and_its_insufficiency}{The ordinary definition and its insufficiency}\dotfill \pageref*{the_ordinary_definition_and_its_insufficiency} \linebreak \noindent\hyperlink{the_correct_definition_of_the_notion_of_module_}{The correct definition of the notion of \emph{module} \ldots{}}\dotfill \pageref*{the_correct_definition_of_the_notion_of_module_} \linebreak \noindent\hyperlink{AbstractDef}{\ldots{} and the correct definition of \emph{derivations} and \emph{K\"a{}hler modules}}\dotfill \pageref*{AbstractDef} \linebreak \noindent\hyperlink{the_fully_general_definition}{The fully general definition}\dotfill \pageref*{the_fully_general_definition} \linebreak \noindent\hyperlink{specific_definitions}{Specific definitions}\dotfill \pageref*{specific_definitions} \linebreak \noindent\hyperlink{OrdinaryRings}{Over ordinary rings}\dotfill \pageref*{OrdinaryRings} \linebreak \noindent\hyperlink{relative_version}{Relative version}\dotfill \pageref*{relative_version} \linebreak \noindent\hyperlink{higher_degree_khler_forms}{Higher degree K\"a{}hler forms}\dotfill \pageref*{higher_degree_khler_forms} \linebreak \noindent\hyperlink{relation_to_hochschild_homology}{Relation to Hochschild homology}\dotfill \pageref*{relation_to_hochschild_homology} \linebreak \noindent\hyperlink{SmoothOrPlain}{Over smooth rings regarded as ordinary rings}\dotfill \pageref*{SmoothOrPlain} \linebreak \noindent\hyperlink{the_problem_and_its_solution}{The problem and its solution}\dotfill \pageref*{the_problem_and_its_solution} \linebreak \noindent\hyperlink{detailed_comparison}{Detailed comparison}\dotfill \pageref*{detailed_comparison} \linebreak \noindent\hyperlink{CooCase}{Over smooth rings}\dotfill \pageref*{CooCase} \linebreak \noindent\hyperlink{over_general_monoids}{Over general monoids}\dotfill \pageref*{over_general_monoids} \linebreak \noindent\hyperlink{over_simplicial_rings}{Over simplicial rings}\dotfill \pageref*{over_simplicial_rings} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{on_smooth_rings_regarded_as_ordinary_rings}{On smooth rings regarded as ordinary rings}\dotfill \pageref*{on_smooth_rings_regarded_as_ordinary_rings} \linebreak \noindent\hyperlink{on_the_fully_general_case}{On the fully general case}\dotfill \pageref*{on_the_fully_general_case} \linebreak \noindent\hyperlink{for_a_categorical_approach}{For a categorical approach,}\dotfill \pageref*{for_a_categorical_approach} \linebreak \noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak \hypertarget{idea_and_definition}{}\subsection*{{Idea and definition}}\label{idea_and_definition} The notion of \emph{K\"a{}hler differential} is a very general way to encode a notion of [[differential form]]: something that is dual to a [[derivation]] or [[vector field]]. Conceptually, in dual language of algebras, a symmetry of a commutative algebra $A$ is an [[automorphism]] $g\colon A\to A$, i.e., $g(a b)=g(a)g(b)$. The `infinitesimal' symmetries are the [[derivations]] $X\colon A\to A$, with $X(a b)=X(a)b+X(b)a$. The [[module]] of K\"a{}hler differentials $\Omega^1_K(A)$ parametrizes derivations, in the sense that every derivation $X$ corresponds uniquely to a morphism of $A$-modules $\mu_X: \Omega_K^1 (A)\to A$. \hypertarget{the_ordinary_definition_and_its_insufficiency}{}\subsubsection*{{The ordinary definition and its insufficiency}}\label{the_ordinary_definition_and_its_insufficiency} K\"a{}hler differentials are traditionally conceived in terms of an algebraic construction of a certain [[module]] $\Omega_K^1(Spec R)$ on a given ordinary [[ring]] $R$. On [[space]]s modeled (in the sense described at [[space]]) on the [[site]] [[CRing]]$^{op}$, such as [[varieties]], [[scheme]]s, [[algebraic space]]s, [[Deligne-Mumford stack]]s, this produces the correct notion of [[differential form]] in this context. This is the case discussed in the section \begin{itemize}% \item \hyperlink{OrdinaryRings}{Over ordinary rings} \end{itemize} below. The definition, concrete as it is, applies of course also to function rings on spaces not modeled on $CRing^{op}$, such as rings $C^\infty(X)$ of smooth functions on a [[smooth manifold]]. One might expect that the module of K\"a{}hler differentials $\Omega_K(Spec C^\infty(X))$ of $C^\infty(X)$, regarded as an ordinary [[ring]], does reproduce the familiar notion of smooth [[differential form]]s on a [[manifold]]. But it does not. This is discussed in the section \begin{itemize}% \item \hyperlink{SmoothOrPlain}{Over smooth rings regarded as ordinary rings}. \end{itemize} This shows that the concrete algebraic construction of K\"a{}hler differential forms over plain rings, traditionally thought of as their very definition, does in fact not correctly capture their nature. There is another definition -- obtained from the [[nPOV]] -- which does capture the situation correctly: \hypertarget{the_correct_definition_of_the_notion_of_module_}{}\subsubsection*{{The correct definition of the notion of \emph{module} \ldots{}}}\label{the_correct_definition_of_the_notion_of_module_} In fact, already the definition of [[module]] has to be freed from it concrete realization in the context of ordinary rings, to exhibit its true nature. What this is has been established long ago in \begin{itemize}% \item Dan Quillen, \emph{Homotopical algebra} \end{itemize} and in Jon Beck's thesis, and is discussed in more detail in the entries \emph{[[module]]} and \emph{[[Beck module]]} : Beck and Quillen noticed that the [[category]] $R Mod$ of modules over an ordinary commutative ring $R$ is canonically [[equivalence of categories|equivalent]] to the category $Ab(CRing/R)$ of abelian [[group object]]s in the [[overcategory]] $CRing/R$ of all rings, over the given ring $R$: \begin{displaymath} R Mod \simeq Ab(CRing/R) \,. \end{displaymath} Under this equivalence an $R$-module $N$ is sent to the square-0-extension ring $R \oplus N$ that is canonically equipped with a ring homomorphism $R \oplus N \to R$ and with a unital and associative and commutative product operation \begin{displaymath} (R \oplus N) \times_R (R \oplus N) \to (R \oplus N) \end{displaymath} \begin{displaymath} ((r, n_1), (r,n_2)) \mapsto (r, n_1 + n_2) \end{displaymath} that makes it first an object in the [[overcategory]] $CRing/R$ and in there an abelian group object, hence an object in $Ab(CRing/R)$. Conversely, every module arises this way, up to [[isomorphism]]. So this gives an equivalent way of defining modules over rings. And this is the right definition. Notably, this definition does not assume anything about the ring $R$. It does not even assume that $R$ is a ring at all! It could be anything. Concretely: for $C$ \emph{any} category of test objects -- so that we may think of objects in the [[opposite category]] $C^{op}$ as function rings on the test objects $C$ -- we may define the category of \emph{module} s over an object $R \in C^{op}$ by the above equation: \begin{displaymath} R Mod := Ab(C^{op}/R) \,. \end{displaymath} Notice that this is now a definition. And that $R$ could be anything, and the definition still makes sense. The category of all modules over \emph{all} possible objects $R$ is then nothing but the [[codomain fibration]] \begin{displaymath} Mod_C := Ab([I,C^{op}]) \stackrel{p_C}{\to} C^{op} \end{displaymath} where $I$ is the [[interval category]] and fiberwise (over $C^{op}$) we form abelian group objects. This turns out to be the correct [[category theory|category theoretic]] definition of [[module]] (as discussed there). In fact, this is is the special case of the [[higher category theory|higher categorical]] definition that works for $C$ any [[(∞,1)-category]]. In that case the construction $Ab(C^{op}/R)$ of abelian group objects in the [[overcategory]] is generalized (straightforwardly! and in fact even more elegantly) to the notion of [[tangent (∞,1)-category]]. \hypertarget{AbstractDef}{}\subsubsection*{{\ldots{} and the correct definition of \emph{derivations} and \emph{K\"a{}hler modules}}}\label{AbstractDef} With the above correct notion of [[module]] in hand, all the other concepts of [[deformation theory]], notably those of [[derivation]]s and of K\"a{}hler differentials follow straightforwardly \begin{enumerate}% \item given an $R$-module $N$ regarded as an object $p_N : R \oplus N \to R$; the \textbf{[[derivation]]s} on $R$ with coefficients in $N$ are precisely the [[section]]s $\delta : R \to R \oplus N$ of $p_N$. \item The assignment $Spec R \mapsto \Omega_K(Spec R)$ of modules of \textbf{K\"a{}hler differential}s is the assignment universal with respect to derivations, which means that \begin{displaymath} \Omega_K : C^{op} \to Mod_C \end{displaymath} is the [[left adjoint]] to the above projection $p_C : Mod_C \to C^{op}$: this means that every [[derivation]] $\delta : R \to \mathcal{N}$ (being a [[section]] in $C$ of the module which is the overcategory element $\mathcal{N} \to R$) is identified conversely with a morphism $\Omega_K(R) \to \mathcal{N}$ in the category of abelian [[group object]]s in the [[overcategory]] $C^{op}/R$: \begin{displaymath} Hom_{Mod_C}(\Omega_K(R), \mathcal{N}) \simeq Hom_{C^{op}}(R, \mathcal{N}) \,. \end{displaymath} \end{enumerate} Notice that in all of the above now, $C$ is still a completely arbitrary [[category]]. \hypertarget{the_fully_general_definition}{}\subsubsection*{{The fully general definition}}\label{the_fully_general_definition} By allowing $C$ -- the collection of test spaces -- to be a general [[(∞,1)-category]], the above story gives the following completely general [[nPOV]] on the nature of K\"a{}hler differentials: For $C$ any [[(∞,1)-category]] of test spaces, write $p_{C^{op}} : Mod := T_{C^{op}} \to C^{op}$ for the [[tangent (∞,1)-category]] of its [[opposite category]]. \begin{enumerate}% \item the fiber of $Mod \to C^{op}$ over $R \in C^{op}$ is the [[(∞,1)-category]] $R Mod$ of [[module]]s over $R$; \item for $(p_{\mathcal{N}} : \mathcal{N} \to R) \in R Mod$, a [[derivation]] on $R$ with coefficients in $\mathcal{N}$ is a [[section]] $\delta : R \to \mathcal{N}$ of $p_{\mathcal{N}}$. \item The assignment of \textbf{modules of K\"a{}hler differentials} or \textbf{cotangent complexes} is the [[left adjoint]] \begin{displaymath} \Omega_K : C^{op} \to Mod \end{displaymath} of the [[tangent (∞,1)-category]] projection $p_{C^{op}}$. Its value $\Omega_R(R)$ on an object $R \in C^{op}$ is the module of K\"a{}hler differentials on $Spec R \in C$. \end{enumerate} \hypertarget{specific_definitions}{}\subsection*{{Specific definitions}}\label{specific_definitions} We spell out very concretely definitions of K\"a{}hler differentials for special concrete choices of base category $C$ as special cases of the above general story. We start with the familiar cases and then work our way up to more general or richer cases. \hypertarget{OrdinaryRings}{}\subsubsection*{{Over ordinary rings}}\label{OrdinaryRings} In terms of the above discussion, we now take $C = CRing^{op}$ to be the [[opposite category]] of the category of ordinary (commutative unital) [[ring]]. In fact without changing anything of the discusson we may assume that the ring $R$ in question is equipped with a ring homomorphism $k \to R$ from a [[ring]] or [[field]] $k$. This makes $R$ a $k$-[[algebra]], and we shall often speak of algebras in the following, where we could just as well speak of rings. Suppose $A$ is a [[commutative algebra]] over a field $k$. We may define K\"a{}hler differentials either by an explicit construction or by a universal property. In fact there are two explicit constructions. The simplest construction, maybe, is as follows. The module of \textbf{K\"a{}hler differentials} $\Omega^1_K(A)$ over $A$ is generated by symbols $d a$ for all $a\in A$, subject to these relations: \begin{itemize}% \item $d c = 0$ when $c$ is a `constant', that is, an element of $k$ regarded as an element of $A$. \item $d(a b)=(d a)b+a(d b)$. \item $d(a+b)=d a+d b$. \item $(d a)b = b(d a)$. \end{itemize} In particular there are only finite sums in the module of K\"a{}hler differentials. Another more sophisticated construction of $\Omega^1_K(A)$ is given below. But turning to the universal property, note that we can define \textbf{derivations} from $A$ to any $A$-[[module]] $M$: they are $k$-linear maps $X : A \to M$ satisfying the product rule: \begin{displaymath} X(a b) = X(a) b + a X(b) \end{displaymath} Then $\Omega^1_K(A)$ may be defined as the [[universal property|universal]] $A$-module equipped with a derivation. In other words, there is a derivation \begin{displaymath} d : A \to \Omega^1_K(A), \end{displaymath} and if $X:A\to M$ is any [[derivation]] from $A$ to some $A$-module $M$, then there is a unique $A$-module morphism \begin{displaymath} \mu_X:\Omega_K^1(A)\to M \end{displaymath} such that the following diagram commutes: \begin{displaymath} \itexarray{ A&\overset{X}\to & M\\ & \underset{d}\searr&\uparrow \mu_X\\ & & \Omega_K^1{A} } \end{displaymath} We say that $X$ \textbf{factors through} $d$. \hypertarget{relative_version}{}\paragraph*{{Relative version}}\label{relative_version} We can replace the commutative algebra $A$ more generally by a [[morphism]] of [[commutative unital rings]] $f:R\to S$. Then the \textbf{module of K\"a{}hler differentials} is the $S$-[[module]] $\Omega^1_K(S/R)$ corepresenting the functor \begin{displaymath} Der_R(S,f_*(-)) : S Mod\to Set : M\mapsto Der_R(S,f_* M) \end{displaymath} that assigns to every $S$-[[module]] $M$ the [[set]] of [[derivations]] on $S$ with values in the (bi)module $f_* M$, where $f_*:S Mod\to R Mod$ is the restriction of scalars. In other words, $Der_R(S,f_*M)\cong Hom_S(\Omega^1_K(S/R),M)$. In a diagram: for every $R$-derivation $X\colon S \to M$ there is a unique morphism (of $S$-modules) $\mu\colon \Omega^1_K(S/R) \to M$ making the following diagram commute: \begin{displaymath} \itexarray{ S&\overset{X}\to & M\\ & \underset{d}\searr&\uparrow \mu\\ & & \Omega^1_K(S/R) } \end{displaymath} This framework also gives another construction of the module of K\"a{}hler differentials, instead of the generators and relations definition given above. Let $I$ be the [[augmentation ideal]], i.e. the [[kernel]] of the multiplication map \begin{displaymath} I := Ker(m:S\otimes_R S\to S) \end{displaymath} Then $\Omega^1_K_{S/R}= I/I^2$ and there is a canonical induced map $d: S\to \Omega^1_{S/R}$ given by $d s = [1\otimes s - s\otimes 1]$. \hypertarget{higher_degree_khler_forms}{}\paragraph*{{Higher degree K\"a{}hler forms}}\label{higher_degree_khler_forms} Furthermore, if $R$ is in characteristic zero, one may introduce \textbf{K\"a{}hler $p$-forms} , which are elements of the $p$-th [[exterior power]] $\Omega^p_K_{S/R}:=\Lambda_R^p \Omega^1_K_{S/R}$. The [[module]] of K\"a{}hler differentials readily generalizes as a [[sheaf]] of K\"a{}hler differentials for a separated morphism $f:X\to Y$ of (commutative) [[schemes]], namely it is the [[pullback]] along the embedding of the ideal sheaf of the [[diagonal subobject|diagonal subscheme]] $X\hookrightarrow X\times_Y X$. Compare the role of [[universal differential envelope]] and [[Amitsur complex]] for analogous constructions in the noncommutative case. The appropriate extension of the module of relative K\"a{}hler differentials to the derived setting is the [[cotangent complex]] of Grothendieck--Illusie. \hypertarget{relation_to_hochschild_homology}{}\paragraph*{{Relation to Hochschild homology}}\label{relation_to_hochschild_homology} The module of K\"a{}hler differentials on $R$ is isomorphism to the first [[Hochschild homology]] of $R$ \begin{displaymath} \Omega_K^1(R) \simeq HH_1(R,R) \,. \end{displaymath} Under mild conditions the anaous statement is true for higher K\"a{}hler differentials and higher Hochschild homology: this is the [[Hochschild-Kostant-Rosenberg theorem]]. \hypertarget{SmoothOrPlain}{}\subsubsection*{{Over smooth rings regarded as ordinary rings}}\label{SmoothOrPlain} We have seen that we define K\"a{}hler differentials $\Omega^1_K(A)$ for any [[commutative algebra]] $A$. The following special case deserves special attention: The algebra $A = C^\infty(X)$ of smooth functions on some smooth space $X$ (a [[smooth manifold]] or a [[generalized smooth space]]) is in particular a commutative algebra. So one might think that its K\"a{}hler differentials form the ordinary [[differential form]]s on $X$ -- in analogy to the case when $A$ consists of the algebraic functions on an affine [[algebraic variety]] in which case K\"a{}hler differentials are often taken as a \emph{definition} of 1-forms. \hypertarget{the_problem_and_its_solution}{}\paragraph*{{The problem and its solution}}\label{the_problem_and_its_solution} However, when $A = C^\infty(X)$ consists of smooth functions on a manifold, the \emph{ring theoretic} K\"a{}hler differentials do \emph{not} agree with the ordinary smooth [[differential form|1-form]]s on this manifold! (Unless $X$ is, for instance, the [[point]], of course). However, there is a canonical map from the K\"a{}hler differentials to the ordinary 1-forms. But there is a solution to this, and an explanation for why something goes wrong: Smooth spaces such as [[manifold]]s are \emph{not} modeled on the category $C =$ [[CRing]]${}^{op}$, as [[varieties]] are. Instead, they are modeled on the category $C = \mathbb{L}$ of [[smooth loci]], which is $= C^\infty Ring^{op}$ the opposite of the category of [[generalized smooth algebra|C-infinity rings]]. In particular, the algebra $A = C^\infty(X)$ of smooth functions on a [[manifold]] carries naturally the structure of such a $C^\infty$-ring. This does have ``underlying'' it the ordinary commuttative ring of functions that forget the $C^\infty$-ring structure, but forgetting this structure is precisely what makes the definition of K\"a{}hler differentials fail to reproduce that of ordinary smooth 1-forms. If we do regard $C^\infty(X)$ as a [[generalized smooth algebra|C-infinity ring]], the its K\"a{}hler differentials do agree with ordinary 1-forms on $X$. \hypertarget{detailed_comparison}{}\paragraph*{{Detailed comparison}}\label{detailed_comparison} We discuss how K\"a{}hler differential forms relate to the ordinary notion of [[differential form]]s. Since there are only finite sums in the module of K\"a{}hler differentials, the usual $d f=f' d t$ works only if $f$ is a finite polynomial in $t$, say, if $A$ is $C^\infty(\mathbb{R})$ ([[smooth map|smooth maps]]) or $\mathbb{R}[\![t]\!]$ ([[power series]]). For example, let $f = t^n$ then \begin{displaymath} \begin{aligned} d f &= d(t^n) \\ &= t^{n-1} d t + t(d t^{n-1}) \\ &= 2 t^{n-1} d t + t^2 d(t^{n-2}) \\ &= r t^{n-1} d t + t^r d(t^{n-r}) \\ &= n t^{n-1} d t \\ &= f' d t. \end{aligned} \end{displaymath} However, we have \begin{displaymath} d e^t \ne e^t d t \end{displaymath} as K\"a{}hler differentials. Intuitively, the reason is that $d$ cannot pass through the infinite sum \begin{displaymath} d e^t = d\left(\sum_{n=0}^{\infty} \frac{t^n}{n!}\right) \ne \sum_{n=0}^{\infty} \frac{d(t^n)}{n!} = e^t d t. \end{displaymath} However, the only proof we know that $d e^t \ne e^t d t$ is quite tricky: in fact it uses the Axiom of Choice! \begin{itemize}% \item David Speyer, K\"a{}hler differentials and ordinary differentials. (\href{http://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials/9723#9723}{Math Overflow}) \end{itemize} It would be desirable to either find a proof that avoids the Axiom of Choice, or show that axioms beyond ZF are necessary for this result. To avoid this annoying property of K\"a{}hler differentials we can proceed as follows. Given a commutative algebra $A$, let $Der(A)$ be the $A$-bimodule of derivations. Define $\Omega^1(A)$ to be the dual of $Der(A)$: \begin{displaymath} \Omega^1(A) = Der(A)^* \end{displaymath} in other words, the set of $A$-module maps $\omega : Der(A) \to A$, made into an $A$-module in the usual way. There is a derivation \begin{displaymath} d : A \to \Omega^1(A) \end{displaymath} given by \begin{displaymath} d f (X) = X(f) \end{displaymath} for $X \in Der(A)$. Now, suppose $A=C^\infty(M)$ where $M$ is any smooth manifold. Then elements of $\Omega^1(A)$ can be identified with ordinary smooth 1-forms on $M$: from the [[Hadamard lemma]] it follows that $Der(C^\infty(M)) = \Gamma(T M)$ is precisely the $C^\infty(X)$-module of [[vector field]]s on $X$, and [[differential form|1-forms]] are the $C^\infty(X)$-linear duals of vector fields, by definition. And in this case, one can show that any [[derivation]] $X: A \to M$ factors through $\Omega^1(A)$ when $M$ is free, and in particular if $M=A$. We can expand on this remark as follows. Quite generally, for any commutative algebra $A$ over a field $k$, we have \begin{displaymath} Der(A) \cong \Omega^1_K(A)^* \end{displaymath} by the universal property of K\"a{}hler differentials, which identifies derivations $A \to A$ with $A$-module morphisms $\Omega^1_K(A) \to A$. Using the definition \begin{displaymath} \Omega^1(A) = Der(A)^* \end{displaymath} (of ordinary smooth 1-forms in the case that $A = C^\infty(M)$) we have that these are the linear bidual of the K\"a{}hler differentials: \begin{displaymath} \Omega^1(A) \cong \Omega^1_K(A)^{**} \end{displaymath} There is always a homomorphism from a module to its double dual, so we have a morphism \begin{displaymath} j: \Omega^1_K(A) \to \Omega^1(A) \end{displaymath} In the case when $A = C^\infty(M)$ this map is onto but typically not one-to-one, as witnessed by the fact that $d e^t = e^t d t$ in the ordinary 1-forms $\Omega^1(A)$ but not in the K\"a{}hler differentials $\Omega^1_K(A)$. However, one can show that when $M$ is a free $A$-module, any derivation $X: A \to M$ not only factors through $\Omega^1_K(A)$ (as guaranteed by the universal property of K\"a{}hler differentials), but also $\Omega^1(A)$. More generally this is true for torsionless $M$, i.e. $A$-modules that inject into their double duals, since $\Omega^1(A)$ is torsionless. So $\Omega^1_K(A)$ is the universal differential module for all modules, while $\Omega^1(A)$ is the universal differential module for torsionless $A$-modules. [[Martin Gisser]] Couldn't resist the above uncredentialed drive-by edit\ldots{} Is it torsionlessness that ultimately discerns differential geometry from algebraic geometry? (Need to fill in more dots between torsionless modules and ``geometric modules'' in the sense of Jet Nestruev.) \vspace{.5em} \hrule \vspace{.5em} [[Eric]]: Does this universal property mean that there is some diagram in some category for which the K\"a{}hler differentials can be thought of as a (co)limit? [[John Baez]]: Yeah, take the category all $A$ modules $M$ equipped with a derivation $X : A \to M$, and take the diagram which consists of every object in this category and every morphism, and take the colimit of that, and you'll get $\Omega^1_K(A)$. But this is just a cutesy way to say that $\Omega^1_K(A)$ is the [[initial object]] of this category. And \emph{this}, in turn, is just a cutesy way to say that there is a derivation \begin{displaymath} d : A \to \Omega^1_K(A), \end{displaymath} such that if $X:A\to M$ is also a derivation, then there exists a unique $A$-module morphism \begin{displaymath} \mu_X:\Omega_K^1(A)\to M \end{displaymath} such that the following diagram commutes: \begin{displaymath} \itexarray{ A&\overset{X}\to & M\\ & \underset{d}\searr&\uparrow \mu_X\\ & & \Omega_K^1{A} } \end{displaymath} All this is general abstract nonsense, nothing special to this example! Any universal property involving maps out of an object says that object is initial in some category --- and that, in turn, is equivalent to saying that object is the colimit of the enormous diagram consisting of all objects of the same kind! There's a lot less here than meets the eye. [[Eric]]: Thank you! That actually makes a little sense to me. As trivial as it may seem, the fact that I was even able to ask this question represents tremendous progress :) [[Herman Stel]]: Dear Eric and Prof. Baez. There is a mistake in the explanation by John Baez here. The two latter properties (both being that the derivation $d : A \to \Omega^1_K(A)$ is initial) are correct. The first one is not, though. If $\Omega^1_K(A)$ were the colimit of the huge diagram then for every derivation there would be a morphism from that derivation to the universal derivation, which is not true. Instead, note that an initial object is the vertex of a colimit of the empty diagram in any category (use that $\forall x\in X P(x)$ is true if $X$ is empty). \hypertarget{CooCase}{}\subsubsection*{{Over smooth rings}}\label{CooCase} A \emph{$C^\infty$-ring} (see [[generalized smooth algebra]]) is a ring that remembers that it carries extra smooth structure akin to the smooth structure carried by a ring of smooth functions on a smooth [[manifold]]. If we take a smooth function ring $C^\infty(X)$, regard it as a $C^\infty$-ring and then determine its module of K\"a{}hler differentials with respect to the category $C = \mathbb{L} = C^\infty Ring^{op}$, we do recover the ordinary notion of smooth [[differential form]]s. For the moment, this case is described in detail in the entry on [[Fermat theory]]. \hypertarget{over_general_monoids}{}\subsubsection*{{Over general monoids}}\label{over_general_monoids} An ordinary (commutative) [[ring]] is precisely a comutative [[monoid]] in the [[category]] $Ab$ of abelian [[group]]s. The case of K\"a{}hler differentials over ordinary rings discussed \hyperlink{OrdinaryRings}{above} may therefore be thought of as the case where the category of test objects is taken to be \begin{displaymath} C = (CMon(Ab))^{op} \,. \end{displaymath} This has an evident generalization: we may replace here $Ab$ with \emph{any} category $K$ and consider \begin{displaymath} C = (CMon(K))^{op} \,. \end{displaymath} In practice $K$ is usually required to be an [[abelian category]], but our definitions so far are general enough not to be concerned about this: for any such $K$ fixed we follow the \hyperlink{AbstractDef}{general definition}, consider the [[functor]] \begin{displaymath} p : Mod := Ab([I,CMon(K)^{op}]) \to CMon(K)^{op} \end{displaymath} and define the assignment of \textbf{K\"a{}hler differentials} \begin{displaymath} \Omega_K : CMon(K)^{op} \to Ab([I,CMon(K)^{op}]) \end{displaymath} to be the [[left adjoint]] of this functor. For this to make good sense, everything here should be regarded as taking place in [[(∞,1)-categories]], typically [[model category|modeled]] by the [[simplicial ring|model structure on simplicial rings]]. Then one finds that for $C = sCRing^{op}$ the corresponding notion of module [[(∞,1)-category]] reproduces the [[derived category]] of ordinary ring modules. This is Example 8.6. in \begin{itemize}% \item [[Jacob Lurie]], \emph{Stable $\infty$-categories} . \end{itemize} \hypertarget{over_simplicial_rings}{}\paragraph*{{Over simplicial rings}}\label{over_simplicial_rings} If in the above setup we choose $K = sAb = [\Delta^{op}, Ab]$ the category of abelian [[simplicial group]]s, then $Mon(K)$ is the category of [[simplicial ring]]s. The category $Mon(K)^{op}$, regarded as a higher category, is the site used in [[higher geometry]] in place of $CRing^{}$ \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[canonical bundle]] \item [[cotangent complex]], [[André-Quillen homology]], [[topological André-Quillen homology]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{on_smooth_rings_regarded_as_ordinary_rings}{}\subsubsection*{{On smooth rings regarded as ordinary rings}}\label{on_smooth_rings_regarded_as_ordinary_rings} For a proof that every derivation of $A = C^\infty(\mathbb{R})$ comes from a smooth vector field on the real line, and an extensive discussion of K\"a{}hler differentials versus ordinary 1-forms, see: \begin{itemize}% \item This Week's Finds in Mathematical Physics (\href{http://math.ucr.edu/home/baez/week287.html}{Week 287}) \end{itemize} See also the discussion at the $n$-Caf\'e{}: \begin{itemize}% \item Blog discussion of Week 287. (\href{http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_48.html#c030626}{Summary}) \item \href{http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-726Spring-2009/8C4F62C5-7AE2-482B-9643-890EE76499F5/0/MIT18_726s09_lec13_differentials.pdf}{The module of K\"a{}hler differentials}, MIT OpenCourseWare: 18.726 Algebraic Geometry, Spring 2009. \end{itemize} \hypertarget{on_the_fully_general_case}{}\subsubsection*{{On the fully general case}}\label{on_the_fully_general_case} A detailed discussion of K\"a{}hler differentials and their generalization from [[algebra]] to [[higher algebra]] is in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Deformation Theory]]} \end{itemize} \hypertarget{for_a_categorical_approach}{}\subsubsection*{{For a categorical approach,}}\label{for_a_categorical_approach} introducing a setting in which Kahler differentials live quite naturally (but not yet in as much generality as possibly one might hope), see \begin{itemize}% \item [[R. Blute]], [[J.R.B. Cockett]], [[T. Porter]], [[R.A.G.Seely]], \emph{K\"a{}hler categories}, Cahiers Top. G\'e{}om. Diff. cat., 52 (2011) 253 -- 268 (\href{http://www.math.mcgill.ca/rags/difftl/kahlercahiers.pdf}{pdf}) \end{itemize} \hypertarget{abstract}{}\paragraph*{{Abstract}}\label{abstract} This paper establishes a relation between the recently introduced notion of differential category and the more classic theory of K\"a{}hler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad, which is furthermore an algebra modality. An algebra modality for a monad T is a natural assignment of an associative algebra structure to each object of the form T(M). In a (co)differential category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the (co)Kleisli category as being infinitely differentiable. Finally, a differential category comes equipped with a differential combinator satisfying typical differentiation axioms, expressed coalgebraically. The traditional notion of K\"a{}hler differentials defines the notion of a module of A-differential forms with respect to A, where A is a commutative k-algebra. This module is equipped with a universal A-derivation. With this in mind, a K\"a{}hler category is an additive monoidal category with an algebra modality and an object of differential forms associated to every object. This object of differential forms satisfies a universal property with respect to derivations. Surprisingly, we are able to show that, under some natural conditions, codifferential categories are K\"a{}hler. [[!redirects Kahler differential]] [[!redirects Kahler differentials]] [[!redirects Kähler differentials]] [[!redirects Kaehler differential]] [[!redirects Kaehler differentials]] [[!redirects Kahler differential form]] [[!redirects Kahler differential forms]] [[!redirects Kähler differential form]] [[!redirects Kähler differential forms]] \end{document}