Questions tagged [adjoint-functors]

For questions about adjoint functors from category theory. Use in conjunction with the tag (category-theory).

In category theory, two functors $F : \mathsf{C} \to \mathsf{D}$ and $G : \mathsf{D} \to \mathsf{C}$ are said to be adjoint, denoted $F \dashv G$, if there is a natural bijection: $$\hom_{\mathsf{D}}(F(X), Y) \cong \hom_{\mathsf{C}}(X, G(Y)), \; \forall X \in \mathsf{C}, Y \in \mathsf{D}.$$ The functor $F$ is called the left adjoint, and the functor $G$ is the right adjoint. There are several reformulations of this property, in terms of universal morphisms, unit-counit adjunction, and hom-set adjunction as above.

This concept, initially introduced in the context of , is ubiquitous in mathematics, and many problems can be reformulated in terms of finding an adjoint to some functor. Use this tag if you have questions related to adjoint functors, in conjunction with the tag .

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Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global…
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A bestiary about adjunctions

What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying Mathematics. For the sake of clarity I would like you to…
fosco
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Mnemonic for the fact that a right(left) adjoint functor preserves limits(colimits)

A right adjoint functor preserves limits. Dually a left adjoint functor preserves colimits. I often forget which is which. Of course, you can look up a book on category theory or use internet. But it's nice if there is a good mnemonic method to…
Makoto Kato
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Adjoint functors as "conceptual inverses"

The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other. For example, the forgetful functor "ought to be" the "conceptual inverse" of the…
Nick Alger
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Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are isomorphic, we wish to come up with a natural…
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Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction, then if $D:I\to \mathcal{D}$ is a diagram that…
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In category theory why is a right adjoint not a left adjoint?

I'm learning basic category theory and teaching myself about adjoints. The definition I have is that an adjunction between $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ is a bijection, for each pair of an object $A \in…
Jon
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If adjunction arises everywhere, where is it in the fundamental theorems?

MacLane's slogan "adjunction arises everywhere" is widely known, and adjunction has been identified as a key concept (maybe the key concept?) in category theory, eg, in the books by Goldblatt Topoi, Awodey Category Theory, and others: The notion…
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Is an equivalence an adjunction?

Let $C$ and $D$ be categories and $F:C\to D$, $G:D\to C$ two functors. $F$ is left-adjoint to $G$, if there are natural transformations $\eta:id_C\to GF$ and $\epsilon:FG\to id_D$ such…
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Do the adjoint functor theorems usefully dualise?

The special and general adjoint functor theorems exist to construct left adjoints to particular functors given certain conditions on them. However, I've not been able to find much mention – at least, not in my lecture notes nor in a quick Google…
Ben Millwood
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How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I have a much harder time spotting "adjointness",…
kjo
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Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter $0$, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint functor, then $\mathcal{F}$ is left-exact. I am having some trouble proving this. If…
Hui Yu
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Examples of Monads and their Algebras

I'd like to get some examples of monads; specifically, I'd love a big list of different monads and a description of what their algebras are. Alternatively, online resources and especially exercices on monads and their algebras. A recent question…
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Fully faithful and essentially surjective is an equivalence

The question asks to prove the statement in the subject. So assume the functor is $F: \mathcal{C} \rightarrow \mathcal{D}$ is fully faithful and essentially surjective. We need to construct a map $G$, such that $F\circ G$ is naturally isomorphic…
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Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual $A^\ast:X^\ast\rightarrow X^\ast$ and its adjoint (or…
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