Applied Category Theory (ACT) diagrams vs mechanical drawings

Michael Stone, February 3, 2023, , (src), (all posts)


In November, I outlined an analogy between Applied Category Theory (ACT) and “precision machining”. Here, I’d like to flesh out an additional dimension of the analogy, this time by comparing and contrasting drawings in both fields.

Mechanical drawings

Analogous to electrical and hydraulic schematics in their respective fields, mechanical drawings are an important medium of communication and interchange at the interface between engineering (design, analysis), manufacturing (production), “application”), and maintenance (repair, retirement) in the lifecycle of assemblies made of parts.

In this setting, one of the key issues that mechanical drawings help manage are analyses of the issues of “fits” and “clearances”.

In machines, parts can be joined in ways that control future motion through the use of fixing techniques (welding, adhesives, friction, clamping, mechanical capture…) as well as via “fits” that create “bearings” with emergent properties such as allowing sliding, rotating, or both sliding and rotating motion while transmitting other loads through the resulting mechanical interface(s), as well as via “flexures” or “compliant mechanisms” (which I’m not going to talk about in any detail here).

Going further, managing both intensive (material) and extensive (structural) mechanical properties of parts and assemblies affords control over still other kinds of possible movement such as the non-rigid bending/deflection-based “mode shapes” probed by “modal analysis” and “modal testing”.

In this regard, insofar as mechanical drawings pragmatically describe the types of parts that, when manufactured and assembled, will fit together into assemblies with these desirable emergent mechanical properties… these drawings can then be seen both as “theories”, whose models are the parts that do actually fit and work together as drawn, as well as as type-level terms representing the types of a a “Curry-Howard”-style correspondence.

ACT diagrams

ACT also makes use of a kind of drawings or schematics, typically called diagrams, that depict assemblies of categorical parts (whose real-world models are typically databases themselves representing possible program and world states, or networks of actual physical elements like people or circuit components).

As in mechanical engineering and machining, the diagrams serve as a valuable interface and medium of communication between:

As in schematics of other kinds, annotations on the pictures (in ACT, coincident arrows, commutativity markings, unique existence markings, and so forth) communicate the required properties of material and form, analogous to “GD&T” (geometric dimensioning and tolerancing) marks, surface finish annotations, and alloy or input billet requirement details.

Limitations of the analogy

While the analogy I have sketched above will be provocative to some, it should also not be taken too far at this early stage: there clearly are important differences between these kinds of drawings including:

Future of the analogy

Notwithstanding the limitations mentioned above, there is one more, deeper, connection between the pictures that I may attempt to plumb in a future post which is as writings (in the sense of “tools for thinking”); to wit, how, in addition to their facial specificatory value, I find it intriguing how for both kinds of drawings [and certainly the CAD/CAM tools that support at-scale production and use of mechanical drawings today], the process of making the drawings provides practitioners, both individually and in groups, with valuable opportunities to reflect on their work-in-progress, both covert (e.g., their initially unconscious understandings of unresolved problems, desiderata, and future possibilities present in the unfinished design process) and overt (e.g., the actual consciously accessible state of the design process) in which they are so often participating with its varyingly explicit graph or log of initial problems, obstructions, concepts of possible solutions, resulting tradespaces and emergent subproblems, associated uncertainties, provisional choices, implications, and previously explored “dead ends”…