Enriched Naturality Revisited
Let’s look back at the enriched versions of representable functors. If we fix an object we have a
-natural transformation
. This corresponds under the closure adjunction to
. There’s a similar transformation for composition on the other side.
Now remember that the closest thing we have to a “morphism” in an enriched category is an element of the underlying set of a hom-object. That is, we can talk about an arrow . We often abuse the language and say that this is a morphism from
to
, which in fact it’s a morphism in the underlying category.
Now, even though this isn’t really a morphism in our enriched category, we can still come up with a morphism sensibly called
. Here’s how it goes:
- We start with
and use the left unit isomotphism to move to
.
- We now hit
with our morphism
to land in
.
- Finally, we compose to end up in
.
We can similarly take and construct a morphism
.
These composites should look familiar from the definition of enriched naturality for a transformation . In fact, we have a more compact diagram to replace that big hexagon:
Notice here that the right and bottom arrows in this square expand out to become the top and bottom of a hexagon, and we slip the functors and
into place.
[…] Now that we’ve gone back and rewritten the definition of naturality, let’s push it a […]
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[…] awful lot of natural maps There are a bunch of natural (in both senses) maps we can consider now. Some look all but tautological over , and we may have used them […]
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