Geometric quotient theory (${\mathbb{T}}^{\prime }=\mathbb{T}+\text{additional geometric axioms}$)

A geometric quotient adds geometric sequents over the same signature. Quotients are compared modulo provable equivalence. In a syntactic site the extra sequents become additional covers, giving a subtopos of the original classifying topos. Stronger axioms correspond to a larger site topology and a smaller subtopos.