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Stream: learning: questions

Topic: Category of functors with different domain

Pim de Haan (Aug 16 2021 at 11:06):

Hi all,
I've got a basic question about how to construct a category of functors with different domains. If we have some category of categories (like Cat or a subcategory thereof) C and some category Z, it seems like that we can make a category [-, Z] whose

• objects tuples (A, F), where A is a category in C and F is a functor A -> Z
• morphisms between (A, F) -> (A', F') consist of a functor phi: A -> A' and a natural transformation alpha: F => (phi; F')

If the natural transformations are the identity, this looks like an over category Cat/Z. Would this be the over 2-category Cat/Z?

Thanks!
Pim

Reid Barton (Aug 16 2021 at 11:15):

If the natural transformation isn't assumed to be invertible then it should have "lax" somewhere in the name, like at https://ncatlab.org/nlab/show/slice+2-category.

Pim de Haan (Aug 16 2021 at 11:46):

Ah, so it's the lax-slice 2-category C//Z, thank you!