We adopt the framework of $\infty$-groupoids (which are the higher counterpart of $Set$). But first consider an ordinary groupoid $G$. The most basic kind of 'unity' (for a structured collection) is defined in terms of objects: for any two objects $a,b$ there exists a (iso)morphism $f: a \rightarrow b$. Let us call this property $\mathbb{U}^1$. Then given a groupoid $G$ either $\mathbb{U}^1(G)$ or $G$ admits a disjoint decomposition $\bigsqcup_{i\in I} G_i$ in which $\mathbb{U}^1(G_i)$ for $i \in I$.
But we can go further. Given $A$ such that $\mathbb{U}^1$ we can require that for objects $a,b$ and morphisms $f,g: a \rightarrow b$ there exists a 2-(iso)morphism $F: f \Rightarrow g$. We call this property $\mathbb{U}^2$. And analogously we define $\mathbb{U}^i$ for $i\geq 3$. We define $\mathbb{U}^\infty(X) \equiv \forall n\geq 1. \mathbb{U}^n(X)$.
Given $G$ we can decompose it as $\bigsqcup_{i\in I} G_i$ where for each $i$ either $\mathbb{U}^\infty(G_i)$ or $\mathbb{U}^n(G_i)$ for some $n$. This is one possible interpretation of Proclus' proposition 6 : every manifold is composed either of unified groups or henads.
But what about aetiology, i.e. causality ? It seems that we should view causality in Proclus as essentially a paradigm of relation (order) between substances, qualities, processes, events. For topological spaces $Y$ and $X$ a causal relationship would be just a continuous map $h: Y \rightarrow X$. For instance if $Y$ is a fiber bundle with fiber $F$ and base space $X$ then it does make some sense to speak of the projection to $X$ expressing that $Y$ is the 'cause' of $X$. Recall that neoplatonic causality is closer to that of ancient philosophy (in particular Aristotle) rather than our modern notion.
Thus if we view causality as a higher functor between $\infty$- groupoids, $F: G_1 \rightarrow G_2$ then Proclus' proposition 7 could be interpreted as saying something about the $\mathbb{U}^i(G_1)$ and $\mathbb{U}^i(G_2)$. However, there is a first glance a problem. If we restrict ourselves to topological spaces then given a continuous map $f: X \rightarrow Y$ we have that $Y$ may have a greater level of connectivity than $X$. The image of a connected set is connected; but the image of a non-connected set can also be connected. Thus the 'cause' $X$ does not necessarily have a higher level of connectivity relative to $Y$.
Consider the two situations. In in the first we have a simpler object, a free object, $F$ and an epi $f : F \rightarrow G$ which can be seen as introducing relations on $F$ (typically given by the kernel of $f$). In this case $F$ is like a material cause (for example a quotient topological space). In the second situation we have an embedding $j : G \rightarrow H$. In some sense $H$ has more information than $G$ and allows certainly potentialities in $G$ to unfold. For example, deformations, "unfoldings" in the technical sense. The varying of the base ring in scheme theory, etc. In this case we could see $H$ as a kind of universal "form". Also in this situation $H$ can manifest (rather hylomorphically) as a "completion" (see our previous post on Hegel's Logic) or cover (for instance an algebraic closure, a normal closure or the Galois group of the algebraic closure of $\mathbb{Q}$.
Proclus' elements is all about ontological hierarchies and in such a way that we can say that : a Lie group is more 'perfect' than a mere manifold, an algebraic group than a mere variety. This is because there is an effective unity and relationship between the parts and the whole.
A very interesting illustration of Proclus' causal scheme, in particular for the 'monad' which integrates, unifies and determines each 'level' is given by the concept of classifying space. The idea is that a more limited, internal variation and aspect of an object $C$ determines a larger, more exterior, more substantial spectrum of objects $O$ : $C$ determines the $O$. For instance, for topological groups $G$, homotopy classes of continuous maps $X \rightarrow BG$ (where $BG$ is the classifying space) determine isomorphism classes of principal $G$-bundles over $X$. This is also called a 'representation', a plurality is mirrored and represented within the central classifying object.
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