# nLab Beck module

Contents

### Context

category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

One usually defines cohomology with respect to some coefficient objects:

• For group cohomology of a group $G$, the coefficients come from a (left) $G$-module.
• For Lie algebra cohomology of a Lie algebra $\mathfrak{g}$, the coefficients come from a (left) $\mathfrak{g}$-module.
• For Hochschild cohomology of an associative algebra $A$, the coefficients come from an $A$-bimodule.

Beck modules are a simultaneous generalisation of all three types of module.

## Definition

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 $A$.

## Properties

###### Proposition

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).

###### Proof

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).

###### Proposition

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.

###### Proof

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.

###### Corollary

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.

###### Proof

Combine the two propositions above.

## Derivations

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

$Der (A, M) = \mathcal{C}_{/ A} (1_A, U M)$

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 $1_A$.

###### Proposition

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.

###### Proof

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.

## Examples

### Beck modules over associative algebras

###### Proposition

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.

###### Proof

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:

$\mu (b_0 + b_1, b_2 + b_3) = \mu (b_0, b_2) + \mu (b_1, b_3)$

Hence, if $a = \epsilon (b_1) = \epsilon (b_2)$,

$\mu (b_1, b_2) = \mu (\eta (a) + b_1 - \eta (a), \eta (a) + b_2 - \eta (a)) = \mu (\eta (a), \eta (a)) + \mu (b_1 - \eta(a), b_2 - \eta (a)) = \eta (a) + \mu (b_1 - \eta(a), b_2 - \eta (a))$

but $\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0$, so

$\mu (b_1 - \eta(a), b_2 - \eta(a)) = \mu (0, b_2 - \eta (a)) + \mu (b_1 - \eta (a), 0) = b_2 - \eta (a) + b_1 - \eta (a)$

and we conclude that

$\mu (b_1, b_2) = b_1 - \eta (\epsilon (b_1)) + b_2 = b_1 + b_2 - \eta (\epsilon (b_2))$

and in particular, $\mu$ is entirely determined by $\eta$ and $\epsilon$. We also have the following interchange law,

$\mu (b_0 b_1, b_2 b_3) = \mu (b_0, b_2) \mu (b_1, b_3)$

and in particular,

$\mu (\eta (\epsilon (b_2)) b_1, b_2 \eta (\epsilon (b_1))) = \mu (\eta (\epsilon (b_2)), b_2) \mu (b_1, \eta (\epsilon (b_1))) = b_2 b_1$

hence,

$b_2 b_1 = \eta (\epsilon (b_2)) b_1 - \eta (\epsilon (b_2 b_1)) + b_2 \eta (\epsilon (b_1))$

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,

$(a_0, m_0) \cdot (a_1, m_1) = (a_0 a_1, a_0 m_1 + m_0 a_1)$

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:

$\epsilon (a, m) = a$
$\eta (a) = (a, 0)$
$\mu ((a, m_0), (a, m_1)) = (a, m_0 + m_1)$

Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ A})$ and the category of $A$-bimodules, as claimed.

###### Proposition

Let $A$ be a ring. Then the Beck module $\Omega_A$ is isomorphic to the $A$-bimodule of Kähler differentials (relative to $\mathbb{Z}$).

###### Proof

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,

$\delta (a_0 a_1) = \delta (a_0) a_1 + a_0 \delta (a_1)$

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.

### Beck modules over groups

###### Proposition

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.

###### Proof

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:

$\mu (h_0 h_1, h_2 h_3) = \mu (h_0, h_2) \mu (h_1, h_3)$

and in particular,

$\mu (\eta (\epsilon (h_2)) h_1, h_2 \eta (\epsilon (h_1))) = \mu (\eta (\epsilon (h_2)), h_2) \mu (h_1, \eta (\epsilon (h_1))) = h_2 h_1$

but on the other hand, if $g = \epsilon (h_1) = \epsilon (h_2)$, then

$\mu (h_1, h_2) = \mu (\eta (g) \eta (g)^{-1} h_1, \eta (g) \eta (g)^{-1} h_2) = \mu (\eta (g), \eta (g)) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \eta (g) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2)$

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

$\mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \mu (e, \eta (g)^{-1} h_2) \mu (\eta (g)^{-1} h_1, e) = \eta (g)^{-1} h_2 \eta (g)^{-1} h_1$

so we conclude that

$\mu (h_1, h_2) = h_2 \eta (g)^{-1} h_1$

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:

$g \cdot m = \eta (g) m \eta (g)^{-1}$

We can then construct the semi-direct product $M \rtimes G$, which has the following multplication:

$(m_0, g_0) \cdot (m_1, g_1) = (m_0 \eta (g_0) m_1 \eta (g_0)^{-1}, g_0 g_1)$

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:

$\epsilon (m, g) = g$
$\eta (g) = (0, g)$
$\mu ((m_0, g), (m_1, g)) = (m_1 m_0, g)$

Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ G})$ and the category of left $G$-modules, as claimed.

###### Proposition

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:

$\delta (g_0 g_1) = \delta (g_0) + g_0 \cdot \delta (g_1)$
###### Proof

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,

$(\delta (g_0), g_0) \cdot (\delta (g_1), g_1) = (\delta (g_0 g_1), g_0 g_1)$

which is equivalent to the defining equation for crossed homomorphisms.

## The tangent category

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

• Jon Beck, Triples, algebras and cohomology, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 (TAC)

and was popularized in

• Daniel G. Quillen, On the (co-)homology of commutative rings, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.