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Stream: learning: questions

Topic: Understanding the Adjoint Functor Theorems

Ruby Khondaker (she/her) (Dec 07 2025 at 18:35):

I've recently attempted to understand the statements of the adjoint functor theorems more clearly. As far as I can tell, they're both using the underlying idea of constructing a left adjoint using the right kan extension formula $\int_y y^{\text{Hom}(a, Gy)}$ and cutting down the size of the limit needed.

In that respect, the solution-set condition (used in the general adjoint functor theorem) makes sense to me - essentially, the inclusion of the subcategory of objects in the solution-set is an initial functor, so you can compute limits using it.

My intuition for the special adjoint functor theorem is a little more hazy, but here's what I understand so far. We assume that the domain of $G$ has a coseparating set - together with completeness, this implies that the unit of the codensity monad is monic. In other words, you can "embed" any object of the domain into a product of objects in the coseparating set. This lets you cut down the weighted limit for the Kan extension in a slightly different way, reducing it to an expression involving only the cogenerators.

Is that about right? More generally I'm trying to understand the significance of these theorems - it feels like we can separate the proofs into the right kan extension formula, and then various strategies to compute these weighted limits in categories of interest.

Nathanael Arkor (Dec 08 2025 at 09:24):

Regarding the general philosophy of adjoint functor theorems, you may find Adjoint functor theorems for lax-idempotent pseudomonads, or at least the introduction, of interest, as it explains how adjoint functor theorems may be seen as ways to decompose adjointness into cocontinuity and relative adjointness (which acts as a way to control size).

Ruby Khondaker (she/her) (Dec 08 2025 at 09:41):

Ok, I’ll take a look at this! Perhaps what I need to focus on is understanding how “size” is dealt with in category theory more carefully? Since that seems to be somewhat of a common theme:

• The solution set condition controls size through the use of an initial functor from a small category
• The small-admissibility condition controls size through a formulation with relative adjoints
• The special adjoint functor theorem controls size through embedding objects into products of cogenerators

Admittedly this is an aspect of category theory I mostly ignored on my first pass through, since it appeared that size issues weren’t that relevant. But perhaps now is the time to understand it more properly!

Nathanael Arkor (Dec 08 2025 at 09:52):

The solution set condition can also be expressed in terms of relative adjointness (see footnote 3 ibid.). (I never thought about how the SAFT fits into this perspective, though.)

Ruby Khondaker (she/her) (Dec 08 2025 at 09:58):

Nathanael Arkor said:

The solution set condition can also be expressed in terms of relative adjointness (see footnote 3 ibid.). (I never thought about how the SAFT fits into this perspective, though.)

Ah of course, you can encode solution-set in terms of weak multirepresentability, so you just take the full subcategory of those