are fibrations evil?

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

On the occasion of the discussion about "evil" I want to point out an example
where speaking about equality of objects seems to be indispensible.
If P : XX -> BB is a functor and one wants to say that it is a fibration
then one is inclined to formulate this as follows

     if u : J -> I is a map in BB and PX = I then there exists a morphism
     \phi : Y -> X with P\phi = u and \phi cartesian, i.e. ...

I don't see how to avoid reference to equality of objects in this formulation.

This already happens if XX and BB are groupoids where P : XX -> BB is a
fibration iff for all u : J -> I in BB and PX = I there is a map \phi : Y -> X
with P\phi = u.

Ironically the category of groupoids and fibrations of groupoids as families
of types was the first example of a model of type theory where equality may
be interpreted as being isomorphic.

So my conclusion is that equality of objects is sometimes absolutely
necessary. Avoiding reference to equality is also not a question of using
dependent types as some people implicitly seem to say. Even in intensional
type theory there is a notion of equality. But it is sometimes inconvenient
to use. As pointed out by Ahrens one can and should use extensional type theory
whenever convenient.
Intensional type theory allows one to interpret equality as being isomorphic,
a kind of reward for the inconvenience of using intensional identity types.

Thomas


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