nLab biaction

Idea

A ternary function which simultaneously exhibits an action on a set from both the left and the right side.

Sets with biactions are the bimodule objects internal to Set.

Definition

Given a set $S$ and monoids $(M, e_M, \mu_M)$ and $(N, e_N, \mu_N)$, a $M$-$N$-biaction or two-sided action is a ternary function $\alpha:M \times S \times N \to S$ such that

• for all $s \in S$, $\alpha(e_M, s, e_N) = s$

• for all $s \in S$, $a \in M$, $b \in M$, $c \in N$, and $d \in N$, $\alpha\big(a, \alpha(b, s, c), d\big) = \alpha\big(\mu_M(a, b), s, \mu_N(c, d)\big)$

Left and right actions

The left $M$-action is defined as

$$\alpha_M(a, s) \coloneqq \alpha(a, s, e_N)$$

for all $a \in M$ and $s \in S$. It is a left action because

$$\alpha_M(e_M, s) = \alpha(e_M, s, e_N) = s$$

$$\alpha_M\big(a, \alpha_L(b, s)\big) = \alpha\big(a, \alpha(b, s, e_N), e_N\big) = \alpha\big(\mu_M(a, b), s, \mu_N(e_N, e_N)\big) = \alpha\big(\mu_M(a, b), s, e_N\big) = \alpha_M\big(\mu_M(a, b), s\big)$$

The right $N$-action is defined as

$$\alpha_N(s, c) \coloneqq \alpha(e_M, s, c)$$

for all $c \in N$ and $s \in S$. It is a right action because

$$\alpha_N(s, e_N) = \alpha(e_M, s, e_N) = s$$

$$\alpha_N\big(\alpha_N(s, c), d\big) = \alpha\big(e_M, \alpha(e_M, s, c), d\big) = \alpha\big(\mu_M(e_M, e_M), s, \mu_N(c, d)\big) = \alpha\big(e_M, s, \mu_N(c, d)\big) = \alpha_N\big(s, \mu_N(c, d)\big)$$

The left $M$-action and right $N$-action satisfy the following identity:

• for all $s \in S$, $a \in M$ and $c \in N$, $\alpha_M\big(a, \alpha_N(s, c)\big) = \alpha_N\big(\alpha_M(a, s), c\big)$.

This is because when expanded out, the identity becomes:

$$\alpha\big(a, \alpha(e_M, s, c), e_N\big) = \alpha\big(e_M, \alpha(a, s, e_N), c\big)$$

$$\alpha\big(\mu_M(a, e_M), s, \mu_N(c, e_N)\big) = \alpha\big(\mu_M(e_M, a), s, \mu_N(e_N, c)\big)$$

$$\alpha(a, s, c) = \alpha(a, s, c)$$

See also

• action

• bimodule