Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming

We introduce the category SSS of sample-smooth spaces over a mixed site. The test objects are the products Omegan := ℝn × Omega of a Cartesian space with the universal Hilbert cube Omega carrying all universally measurable sets, and a space is a set with a family of admissible plots Omegan → X closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because Omega absorbs its own square; its Karoubi envelope contains every ℝn; and it has mixed morphisms ω mapsto (W(ω),Phi(ω)), which turn measurability of a smooth family from an axiom into a consequence. SSS is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the C^∞ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string Pi dashv flat dashv natural dashv sharp dashv Lambda, making SSS cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of mathsfP(X) as push-forwards of X-plots at every test object, mathsfP is an unconditional strong commutative affine monad on all of SSS -- functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting -- and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.