The next paper in Landry’s collection is a reprint of a short 2014 *Phil. Math.* paper by Steve Awodey, ‘Structuralism, invariance, and univalence’. You can download it here.

Awodey’s question concerns the content we can give to what he calls the Principle of Structuralism, that isomorphic objects are identical, when a naive reading makes that Principle simply false. And as the title suggests, Awodey wants to argue that Voevodsky’s Univalence Axiom is the way to make sense of the Principle. This does involve skating at great speed over some ideas from type theory, such as Propositions-as-Types. My sense is that the typical logic-and-phil-maths-minded philosopher is still pretty hazy when it comes to such type-theoretic ideas, at least once we step beyond Simple Type Theory. However, Awodey’s vigorous advocacy here and elsewhere should perhaps prompt more of us to put in the effort to get up to speed (though sadly, as I’ve complained before, the literature isn’t exactly over-supplied with good introductions to type theory!). So: short but thought-provoking.

The fifth paper is ‘Category theory and foundations’ by Michael Ernst. This might rather naturally have been the first paper in the collection, as it is a lucid state-of-play assessment of debates – some long-running — about the foundational role that category theory may or may not have.

In a previous paper, Ernst himself proved a technical result which in fact changed the state-of-play: he showed that some plausible conditions on what could count as an unlimited all-encompassing category (which all categories belong to) can’t be jointly satisfied. This means that e.g. the complaint that ZFC (plus, for example, some large cardinal axioms) fails to cope with an all-encompassing category is no strike against standard set-theoretic foundations: no consistent theory meets that impossible goal. There is a nice short presentation of Ernst’s result here.

But equally, Adrian Matthias’s well-known arguments that categorial foundations in the shape of ETCS are inadequate for mathematical purposes fail to strike a fatal blow, given the fact that we can extend ETCS in categorially motivated ways to recover a theory which can do everything that can be done by ZFC (or indeed by ZFC plus some large cardinal axioms). Also see again McLarty’s piece in this collection and references there.

So, arguably, both a conventional set theoretic framework and a deviant categorial framework starting from ETCS can be developed into adequate technical foundations for mathematical practice (in some useful sense of ‘foundations’).

But what about the complaint that category theory is not really autonomous but makes use of notions of operation/morphism and collection/objects-combined-into-a-structure which need ultimately to be elucidated in set-theoretic terms? The potentially most interesting part of Ernst’s paper, his §5, discusses this autonomy issue: but to be frank I found the discussion a bit thin. E.g. he touches on Lawvere’s supposedly Cantorian “bag-of-dots” conception of sets which I have always found opaque: and I can’t say I was really helped here. However, that said, this piece is overall definitely worth reading, with useful pointers to relevant debates.

The sixth paper is on ‘Canonical maps’ by Jean-Pierre Marquis. The idea is that there is, out there in mathematical practice, a notion of canonical map which has a reasonably definite use and which can be illuminatingly analysed in categorial terms. There’s a great deal of initial arm-waving but nowhere near enough examples for me (or you?) to latch onto the supposed notion under examination. So I found this entirely unsatisfying. But you can download the paper here, and see if you can get more out of it.