### Chasing around the Triangle

#### Posted by David Corfield

While looking about for material for a new $n$Lab entry on foundations and philosophy, I came across Steve Awodey’s From Sets to Types to Categories to Sets. As the title of the paper suggests, Awodey chases around a triangle of foundational styles. In the process he demonstrates their equivalence.

The newest part of the triangle takes us from categories, in the form of toposes, to set theories. This involves the construction known as ‘ideal completion’ for categories ($n$Lab stub). Awodey explains how ideals are like classes, where principal ideals are the “sets”. The ideal completion of a topos is going to allow for the right kind of summing of types/sets into a universal object, into which there is a monomorphism from its own powerobject. Here powerobjects are formed by gathering subsets rather than subclasses.

After a fourth section which looks at the results of taking two steps around the triangle, the final section takes stock of what the whole exercise teaches us.

…the objects of type theory and set theory are structured by the operations of their respective systems in certain ways that are not mathematically salient. That additional information is essentially what is lost by our comparisons, e.g. distinctions between basic data and derived objects, between types of different complexity, ordinal rank of a set, membership chains within a set, etc. Categorical structure is closer to the mathematical content, and it is not lost in translation. Equivalence of categories preserves categorical properties and structures, because these are determined only up to isomorphism in the first place.

The structural approach implemented by category theory is thus more stable, more robust, more invariant than type or set theoretic constructions.

The latter two constructions are then likened to introducing coordinate systems on a manifold:

…no one today regards a manifold as involving specific coordinate charts, and one generally works with coordinate free methods so that the results obtained will apply directly – this is the modern, structural approach. But at times it can still be useful to introduce coordinates for some purpose, and this is unobjectionable, as long as the results are invariant. So it is with categorical versus logical foundations: category theory implements the structural approach directly. It admits interpretations of the conventional logical systems, without being tied to them. Category theory presents the invariant content of logical foundations.

But perhaps you’re wondering why category theory as a foundation should be restricted to toposes. What would happen if you looked to form an analogue triangle with $(\infty, 1)$-toposes in place of toposes as the category theoretic vertex of the triangle? Can we glimpse a parallel type theory and set theory? No doubt the former is related to other work by Steve on Homotopy theoretic models of identity types. And the set theory? Perhaps it ‘just’ needs ideal completion of a quasi-category.

Then a distant target of the parallel triangle for $\infty$-toposes presents itself.

## Re: Chasing around the Triangle

If you think Steve Awodey’s paper on Homotopy theoretic models of identity types is cool, wait ‘til you see:

We are seeing the merging of homotopy theory and logic before our very eyes! I’ll say more about this in the next Week’s Finds.