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Stream: theory: category theory

Topic: (non)-factorization of adjoints

James Deikun (Apr 06 2022 at 12:38):

In order to better understand formal (co)limits of toposes I'm looking for examples of factorizations of functors $F = GH$ where $F$ and $H$ have right adjoints and $G$ doesn't, or dually where $F$ and $G$ have left adjoints and $H$ doesn't. Is this a thing that even happens? (I can't see an obvious reason why not, though...)

Reid Barton (Apr 06 2022 at 13:07):

Sure, for example the second can easily be constructed with $H : C \to D$, $G : D \to 1$, $F = GH : C \to 1$.

Morgan Rogers (he/him) (Apr 06 2022 at 13:09):

Take your favourite example of a reflective or coreflective subcategory where there aren't further adjoints, and then the composite of the inclusion with the reflection functor is (up to natural isomorphism) the identity on the subcategory, so has itself as an adjoint on either side, but by assumption each of the functors involved doesn't have an adjoint on the other side.

Morgan Rogers (he/him) (Apr 06 2022 at 13:10):

(Take the inclusion of sheaves on the reals into the category of presheaves on the reals, say)

James Deikun (Apr 06 2022 at 18:09):

Morgan Rogers (he/him) said:

Take your favourite example of a reflective or coreflective subcategory where there aren't further adjoints, and then the composite of the inclusion with the reflection functor is (up to natural isomorphism) the identity on the subcategory, so has itself as an adjoint on either side, but by assumption each of the functors involved doesn't have an adjoint on the other side.

In the case of a reflection don't we have: $F$ and $G$ have right adjoints, $F$ and $H$ have left adjoints? For a coreflection, though, $F$ and $G$ have left adjoints and $F$ and $H$ have right adjoints, so that seems to work ...

James Deikun (Apr 06 2022 at 18:12):

Reid Barton said:

Sure, for example the second can easily be constructed with $H : C \to D$, $G : D \to 1$, $F = GH : C \to 1$.

Ah, like when C and D have initial objects and H preserves the initial object of C but little else ...

James Deikun (Apr 06 2022 at 18:21):

I guess actually H doesn't even need to preserve anything ...