David Corfield Heuristics

Rome conference, Category theory as a heuristic tool

• Reference to old paper.

• Danger that I could have given you this talk 50 years ago.

• That people are using category-theoretic understanding to forge concepts: twisted equivariant differential cohomology. New techniques in modalities on $\infty$-toposes.

• Place entities in categories, not just structured objects but cobordisms, propositions and entailments, etc.

• Better a good category than good objects: products, limits, coproducts, closed.

• Every functor, seek an adjoint.

• Form the arrow-based version and look for internalisation.

• Look to enrich.

• Lawvere slogans, Category Theory in Barcelona

• Universal constructions are central to category theory.

As a ‘logic’

Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial. (Peter Freyd)

The purpose of being categorical is to make that which is formal, formally formal. (Peter May)

Don’t understand ‘formal’ as in formalism or syntax, but in terms of form.

Examples

Logic

You have a proposition, and duplicate it. For any pair of propositions, there should be something, $f$, such that $(A, A) \vdash (B, C)$ or $A \vdash B$ and $A \vdash C$ if and only if $A \vdash f(B, C)$. This is $B \wedge C$.

Similarly, there should be something such that $(B, C) \vdash (A, A)$ or $B \vdash A$ and $C \vdash A$ if and only if $g(B, C) \vdash A$. This is $B \vee C$.

Then with a specific $C$, there’s the operation $h: - \mapsto - \wedge C$. For any pair of propositions, there should be something, $k$, such that $h(A) \vdash B$ if and only if $A \vdash k(B)$.

$A \wedge C \vdash B$ if and only if $A \vdash C \to B$.

Order

• Paper on categorification. Taking concepts at one level and raising them to the next. Natural numbers to finite sets.

• This already presupposes that category theory is understood. Since it probably isn’t, a talk here to establish the heuristical power of category theory.

• Could have been given 50 years ago. But tools keep being developed to forge new concepts: twisted equivariant differential cohomology. New techniques in modalities on $\infty$-toposes.

• An entity is generally understood as belonging to a collection of similar entities related by something like a function or a relation. So, a collection of spaces (manifolds, vector spaces, topological spaces, etc.), of algebraic entities (groups, rings, fields, etc.), of propositions, and so on. Then also a span.

• Sometimes we encounter an entity first and by its nature we know what are the equivalences in the collection it belongs to.

• Then in addition we may specify mappings. Manifolds and smooth ; fields and field homomorphisms; propositions and implication.

• We may expand the collection of objects to allow the category to have better properties.

• Whenever we have two of these collections, we often would like to know if there’s a mapping, $F$. This is useful, since if $F(X) \neq F(Y)$ then $X \neq Y$.

• Then if there are two, $F, G$, we want to know whether they are the ‘same’ or perhaps whether there’s a higher function between them, $\alpha: F \to G$.

• We also want to know if $F$ is invertible. It may be that it isn’t but there’s something as close as possible, and this on either side.

• Examples

• Then general theory, right adjoint preservation of limits.

• Examples

• Nucleus of adjunction produces a duality.