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A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>
We may analyze ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'', ''v'' ∈ ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:
We may analyze ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'', ''v'' ∈ ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:
