mathematics free of a blackboard (I. Gelfand and M. Gromov)

José Manuel Rodríguez Caballero josephcmac at
Sun Mar 14 14:48:30 EDT 2021

Vaughan Pratt wrote:

> What I have great difficulty with is Jos?'s suggestion that, unlike set
> theory, category theory somehow transcends representation.

and used the example

So unlike *Grph* [assuming that its objects are representation-free], the
> objects of *Set*^*G* are not themselves graphs but
> merely representations of them.

and asked the question:

Unless you have a representation-free conception of *Grph*, how can you
> draw the distinction between *Grph* and *Set*^*G* t

I recall my suggestion:

> Category theory is about the mathematical objects, independently of the
> representations, whereas set theory is about the representations of the
> mathematical object.

Roughly speaking (I know that there are technicalities involved), you
showed that there is no way to distinguish between an abstract graph and
the set of all its representations. In my suggestion framework, category
theory is about the abstract graph, whereas set theory is about a
particular representation of this graph that can be transformed into other

How does category theory transcend representation in this case? Simply by
considering all possible representations at the same time. Notice that I
have never claimed that if something is mentioned in category theory, it is
representation-free. My suggestion was that category theory aims to be as
representation-free as possible. It is a historical claim, considering the
activity in the community of category theory. It is not a mathematical

Kind regards,
José M.

Kind regards,
Jose M.
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