  We propose a layered hierarchical architecture called UCLA (Universal
Causality Layered Architecture), which combines multiple levels of categorical
abstraction for causal inference. At the top-most level, causal interventions
are modeled combinatorially using a simplicial category of ordinal numbers. At
the second layer, causal models are defined by a graph-type category. The
non-random ``surgical" operations on causal structures, such as edge deletion,
are captured using degeneracy and face operators from the simplicial layer
above. The third categorical abstraction layer corresponds to the data layer in
causal inference. The fourth homotopy layer comprises of additional structure
imposed on the instance layer above, such as a topological space, which enables
evaluating causal models on datasets. Functors map between every pair of layers
in UCLA. Each functor between layers is characterized by a universal arrow,
which defines an isomorphism between every pair of categorical layers. These
universal arrows define universal elements and representations through the
Yoneda Lemma, and in turn lead to a new category of elements based on a
construction introduced by Grothendieck. Causal inference between each pair of
layers is defined as a lifting problem, a commutative diagram whose objects are
categories, and whose morphisms are functors that are characterized as
different types of fibrations. We illustrate the UCLA architecture using a
range of examples, including integer-valued multisets that represent a
non-graphical framework for conditional independence, and causal models based
on graphs and string diagrams using symmetric monoidal categories. We define
causal effect in terms of the homotopy colimit of the nerve of the category of
elements.
