symmetric monoidal (∞,1)-category of spectra
One usually defines cohomology with respect to some coefficient objects:
Beck modules are a simultaneous generalisation of all three types of module.
Let $\mathcal{C}$ be a category with pullbacks and let $A$ be an object in $\mathcal{C}$. A Beck module over $A$ is an abelian group object in the slice category $\mathcal{C}_{/ A}$. In particular, if $A$ is the terminal object, this reduces to the notion of an abelian group object in $\mathcal{C}$. We write $\mathbf{Ab}(\mathcal{C}_{/ A})$ for the category of Beck modules over $A$.
Let $\mathcal{C}$ be an effective regular category (resp. locally presentable category) and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is an abelian category (resp. locally presentable category).
If $\mathcal{C}$ is a effective regular category (resp. locally presentable category), then so is $\mathcal{C}_{/ A}$. Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).
Let $\mathcal{C}$ be an effective regular category with filtered colimits and let $A$ be an object in $\mathcal{C}$. If filtered colimits in $\mathcal{C}$ preserve finite limits, then $\mathbf{Ab}(\mathcal{C}_{/ A})$ (is an abelian category and) satisfies axiom AB5.
The forgetful functor $\mathcal{C}_{/ A} \to \mathcal{C}$ creates pullbacks and filtered colimits, so filtered colimits in $\mathcal{C}_{/ A}$ also preserve finite limits. The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and filtered colimits, so filtered colimits in $\mathbf{Ab}(\mathcal{C}_{/ A})$ preserve kernels. In view of the earlier proposition, it follows that $\mathbf{Ab}(\mathcal{C}_{/ A})$ satisfies axiom AB5.
Let $\mathcal{C}$ be a locally finitely presentable effective regular category and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is a Grothendieck category.
Combine the two propositions above.
Let $\mathcal{C}$ be a category with pullbacks, let $A$ be an object in $\mathcal{C}$, and let $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ be the forgetful functor. Given a Beck module $M$ over $A$, an $M$-valued derivation of $A$ is a morphism $1_A \to U M$ in $\mathcal{C}_{/ A}$, where $1_A$ is the terminal object in $\mathcal{C}_{/ A}$, and we write
for the set of $M$-valued derivations of $A$. The Beck module of differentials over $A$ is an object $\Omega_A$ in $\mathbf{Ab}(\mathcal{C}_{/ A})$ representing the functor $Der (A, -) : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathbf{Set}$.
The Beck module $\Omega_A$ is not guaranteed to exist in general. When the functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint, $\Omega_A$ is simply the value of the left adjoint at $1_A$.
Let $\mathcal{C}$ be a locally presentable category and let $A$ be an object in $\mathcal{C}$. Then the forgetful functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint.
The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and $\kappa$-filtered colimits (for some $\kappa$ large enough), so we may apply the accessible adjoint functor theorem.
Let $\mathcal{C}$ be the category of (not necessarily commutative) rings and let $A$ be a ring. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is equivalent to the category of $A$-bimodules.
Let $\epsilon : B \to A$ be ring homomorphism. To give it the structure of a Beck module over $A$, we must give ring homomorphisms $\eta : A \to B$ and $\mu : B \times_A B \to B$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (b_0, b_1)) = \epsilon (b_0) = \epsilon (b_1)$, as well as various other equations. Given elements $b_0, b_1, b_2, b_3$ of $B$ such that $\epsilon (b_0) = \epsilon (b_2)$ and $\epsilon (b_1) = \epsilon (b_3)$, we have the following interchange law:
Hence, if $a = \epsilon (b_1) = \epsilon (b_2)$,
but $\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0$, so
and we conclude that
and in particular, $\mu$ is entirely determined by $\eta$ and $\epsilon$. We also have the following interchange law,
and in particular,
hence,
so if $\epsilon (b_1) = \epsilon (b_2) = 0$, then $b_2 b_1 = 0$.
Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing abelian group structure on $\ker \epsilon$. In addition, the homomorphism $\eta : A \to B$ gives $M$ the structure of an $A$-bimodule, and we see that $B$ is naturally isomorphic to the square-0 extension ring $A \oplus M$, with componentwise addition and the multiplication given below,
regarded as a Beck module over $A$ by defining $\epsilon : A \oplus M \to A$, $\eta : A \to A \oplus M$, and $\mu : (A \oplus M) \times_A (A \oplus M) \to A \oplus M$ as follows:
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ A})$ and the category of $A$-bimodules, as claimed.
Let $A$ be a ring. Then the Beck module $\Omega_A$ is isomorphic to the $A$-bimodule of Kähler differentials (relative to $\mathbb{Z}$).
Let $M$ be an $A$-bimodule, regard $A \oplus M$ as a ring as above, and let $\epsilon : A \oplus M \to A$ be the obvious projection. A ring homomorphism $\phi : A \to A \oplus M$ satisfying $\epsilon \circ \phi = id_A$ is the same thing as an additive homomorphism $\delta : A \to M$ satisfying the following equations,
i.e. a derivation $A \to M$ (over $\mathbb{Z}$). Thus, the Beck module $\Omega_A$ has the same universal property as the $A$-bimodule of Kähler differentials.
Let $\mathcal{C}$ be the category of (not necessarily abelian) groups and let $G$ be a group. Then $\mathbf{Ab}(\mathcal{C}_{/ G})$ is equivalent to the category of left $G$-modules.
Let $\epsilon : H \to G$ be group homomorphism. To give it the structure of a Beck module over $G$, we must give group homomorphisms $\eta : G \to H$ and $\mu : H \times_G H \to H$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (h_0, h_1)) = \epsilon (h_0) = \epsilon (h_1)$, as well as various other equations. Given elements $h_0, h_1, h_2, h_3$ of $H$ such that $\epsilon (h_0) = \epsilon (h_2)$ and $\epsilon (h_1) = \epsilon (h_3)$, we have the following interchange law:
and in particular,
but on the other hand, if $g = \epsilon (h_1) = \epsilon (h_2)$, then
and writing $e$ for the unit of $G$ and $H$, we have $\epsilon (\eta (g)^{-1} h_1) = \epsilon (\eta (g)^{-1} h_2) = e$, hence
so we conclude that
and in particular, $\mu$ is entirely determined by $\eta$.
Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing group structure on $\ker \epsilon$. (In particular, $M$ is an abelian group!) We make $M$ into a left $G$-module as follows:
We can then construct the semi-direct product $M \rtimes G$, which has the following multplication:
There is a group homomorphism $M \rtimes G \to H$ defined by $(m, g) \mapsto m \eta (g)$, and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that $M \cap \operatorname{im} \eta = \{ e \}$. We may regard $M \rtimes G$ as a Beck module over $G$ by defining $\epsilon : M \rtimes G \to G$, $\eta : G \to M \rtimes G$, and $\mu : (M \rtimes G) \times_G (M \rtimes G) \to M \rtimes G$ as follows:
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ G})$ and the category of left $G$-modules, as claimed.
Let $G$ be a group and let $M$ be a left $G$-module. Under the above identification of Beck modules over $G$ with left $G$-modules, $M$-valued derivations of $G$ are precisely crossed homomorphisms $G \to M$, i.e. maps $\delta : G \to M$ satisfying the following equation:
Let $\epsilon : M \rtimes G \to G$ be the evident projection. A group homomorphism $\phi : G \to M \rtimes G$ such that $\epsilon \circ \phi = id_G$ is the same thing as a map $\delta : G \to M$ satisfying the equation below,
which is equivalent to the defining equation for crossed homomorphisms.
One may assemble the individual categories of Beck modules over the objects of $\mathcal{C}$ into a category fibred over $\mathcal{C}$, called the tangent category.
The concept is due to
and was popularized in
See also
An application to knot theory is given in
Last revised on May 6, 2019 at 19:56:55. See the history of this page for a list of all contributions to it.