# Cohomology

of an odd-party conversational simplex.

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Related KMR pages:

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Other related sources of information:

• Svensson & Naeve (2002): Combinatorial Aspects of Clifford Algebra.

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A conversational complex is a formal sum of conversational $\, n$-simplexes, where each $\, n$-simplex represents a conversation with $\, n \,$ participants (also referred to as an $\, n$-party conversation.

A conversational complex:

Cohomology with respect to an odd-party conversational simplex:

NOTATION: Let $\, P \,$ be a proposition, i.e., a logical statement which is either true or false.
Then $\, (P) \, \stackrel {\mathrm{def}}{=} \, 1 \,$ if $\, P \,$ is true, and $\, (P) \, \stackrel {\mathrm{def}}{=} \, 0 \,$ if $\, P \,$ is false.

Let $\, C_l \,$ denote the Clifford algebra on $\, n \,$ symbols $\, e_1, e_2, ..., e_n \,$ over $\, \mathbb{Z} \,$ and let $\, C_l^- \,$ denote the subspace of odd multivectors of $\, C_l$. If we include the empty set $\, \emptyset \,$ in the basis for $\, C_l \,$, then $\, C_l \,$ has $\, 2^n \,$ dimensions as a module over $\, \mathbb{Z}$.

For example, if $\, n = 3 \,$ we have the $\, 8 \,$ basis elements $\, \emptyset \, , e_1 \, , e_2 \, , e_3 \, , e_1 e_2 \, , e_2 e_3 \, , e_3 e_1 \, , e_1 e_2 e_3 \,$ for the Clifford algebra $\, C_l(e_1, e_2, e_3)$.

On the basis elements of $\, C_l \,$ the left inner product is defined as

$\, X \, \llcorner \, Y \, \stackrel {\mathrm{def}}{=} \, (X \subseteq Y) X Y$.

It is extended to all multivectors of $\, C_l \,$ by linearity.

Here $\, X Y \,$ denotes the geometric product of the multivectors $\, X \,$ and $\, Y$ in $\, C_l$.

On the basis elements of $\, C_l \,$ the geometric product is defined as
$\, A B \, \stackrel {\mathrm{def}}{=} \, \epsilon \, A \, \Delta \, B \,$ where $\epsilon = \pm 1$ and $\, \Delta \,$ denotes symmetric difference.
The geometric product is extended to all multivectors of $\, C_l \,$ by linearity.

On the basis elements of $\, C_l \,$ the outer product is defined as
$\, X \wedge \, Y \, \stackrel {\mathrm{def}}{=} \, ( X \cap Y = \emptyset ) X Y$. It is extended to all multivectors of $\, C_l \,$ by linearity.

Lemma 1: If $\, X \in C_l^- \,$ then we have $\, X \wedge X \, = \, 0$.

Proof: Exercise

Let $\, S_n \,$ be an $\, n$-simplex in $\, C_l \,$
and let $\, E = e_1 + e_2 + ... + e_n \,$.

We define the boundary operator $\, \partial \,$ by its action on each simplex $\, S_n \,$
of a complex of multivectors of $\, C_l$:

Definition: $\, \partial S_n \, \stackrel {\mathrm{def}}{=} \, E \, \llcorner \, S_n$.

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David Hestenes
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Making use of the following simple but important lemma
(whose proof is left as an exercise):

Lemma 2: $\, A \, \llcorner \, (B \, \llcorner \, C) \, \equiv \, (A \wedge B) \, \, \llcorner \, \, C \,$

we have

$\, \partial \, ( \partial \, X ) \, = \, E \, \llcorner \, (E \, \llcorner \, X) \, = \, (E \wedge E) \, \llcorner \, X \, = \, 0$.

Since this holds for each multivector $\, X \in C_l \,$ we can conclude that

$\, \partial \, \circ \, \partial \, = \, 0$.

Moreover, $\, \partial \,$ is a derivation and we have seen that $\, {\partial}^2 = 0$. Therefore, for each odd simplex $\, S_n$, there is a cohomology on the corresponding odd-party conversational simplex.

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