On the Algorithmic Factorization of Characteristic Morphisms in Topoi

In classical topos theory, the classification of subobjects is achieved directly via a characteristic morphism to the subobject classifier Omega. In this paper, we introduce an internal categorical framework that defers this direct logical assignment by factoring the characteristic morphism through an intermediate computational space. We define the Algorithm Space as an Eilenberg-Moore algebra over a finitary monad T. This factorization strictly separates the computational evaluation (the algorithm) from the structural logical assignment (the judgment). The framework is subsequently modeled within the topos of presheaves, strictly adhering to constructive logic and finite algorithmic paths. [email protected]