Category Theory - 2020-05-13

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user2103480

21:03

In the first order theory of category theory (is this theory actually important for "applications"?), what does an isomorphism between the "models"(categories) correspond to? An equivalence between categories or something weaker?

And how do natural isomorphisms compare to equivalence? I've gotten pretty used to everything being naturally isomorphic at first

When, two weeks ago, we started preparing for categories with families, the following (important) theorem came up:

For every presheaf X on A, there is an equivalence of (category of presheafs on the elements of X) with (slice category of presheafs on A with X)

So I'm wondering how to judge equivalence of these categories in comparison to a natural equivalence between them

Malice Vidrine

21:37

@user2103480 - I'm not familiar with the distinction "equivalence" versus "natural equivalence". There's isomorphism vs. equivalence of categories, and natural isomorphsm (I don't know of "natural equivalence" except maybe as a synonym for this) of functors.

So I suspect one of those two terms is being used in an imprecise way above.

And an isomorphism of models in the first order language of categories would be an isomorphism of categories.

An isomorphism of categories is the strong notion: there are two functors that, composed one way around give you the identity functor on one category, and composed the other way around give you the identity functor on the other.

Equivalence is when the composites are naturally isomorphic to the identity functors.

user2103480

@MaliceVidrine Yes, the natural isomorphism is imprecise. The functors would map presheafs X to the category of presheafs on the elements of X or the slice category of presheafs on A with X respectively

for that, of course, we need to define what the morphisms in the presheaf category are mapped to

@MaliceVidrine ok sounds sensible! but since equivalences seem to be more important, isomorphisms are probably too strong?

Malice Vidrine

They are usually too strong, yes.

user2103480

@MaliceVidrine So to actually compare these two, we need the extra structure given by some functors we defined

(these two = equivalence and natural iso)

Malice Vidrine

If I understand you, yes. There turn out to be some things that are preserved by isomorphism but not equivalence, like reflective subcategories if the subcategory isn't full (if I recall correctly), so there are some visible differences.

user2103480

Will be much to learn. thanks!

Malice Vidrine

21:56

np!

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