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Questions tagged [adjoint-functors]

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

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Epimorphisms preserved by pullback functor with right adjoint

I am working through some notes, and am on a section about locally cartesian closed categories. One of the results is if $\mathcal{C}$ is local cartesian closed and has coequalisers of reflexive pairs,...
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$F(f)=0$, then $f=0$ where $0$ is the zero morphism for equivalent functors.

For an additive category $\mathcal{C}$ and let $F:\mathcal{C} \to \mathcal{C}$ be a equivalent additive and covariant functor. Prove that for every morphsim $f:A \to B$ such $F(f)=0$, then $f=0$ ...
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If $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint then $G$ preserves kernels.

The theorem 7.7 in Course an homologic algebra of Peter Hilton is If $G: \mathcal{D} \to \mathcal{C}$ has a left adjoint then G preserves products, pull-backs and kernels. Hilton prove that $G$ ...
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1answer
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Prove that adjunction restricts to an equivalence between full subcategories

Given problem 2.2.11 (a) from T. Leinster's "Basic Category Theory" (I modified the question a bit since it's difficult to draw an adjunction here, but the logic is the same): Let a pair of ...
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Left and right adjoints of restriction in modules.

Given a map $f:A\rightarrow B$ of rings, we get a functor $Res_f:\text{B-Mod}\rightarrow \text{A-Mod}$, induced by our map $f$. This has left adjoint $B\otimes_A \text{_}$ where the $B$ module ...
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1answer
40 views

Currying in a locally small category with coproducts

While studying for category theory course I stumbled upon the following question taken from a previous exam: Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every ...
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1answer
47 views

Is the transpose of the projection under the exponentiation adjunction a constant morphism?

Consider a cartesian closed category $\mathbf{C}$ and fix an object $B \in \mathbf{C}$. For any $X$, we have the product $X \times B$ and a projection $\pi_B : X \times B \rightarrow B$. Under the ...
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1answer
58 views

Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
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1answer
36 views

Symmetric monoidal category which is not closed?

A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We ...
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30 views

“Adjunction” between natural transformations?

An adjunction is a relation between two functors. But I have come across something that feels like a “adjunction between natural transformations”. Does this concept exist?
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28 views

Composition of morphisms under adjunction

Suppose we have a pair of adjoint founctors $S:\mathcal{H} \to \mathcal{G}$ and $T:\mathcal{G} \to \mathcal{H}$ where $\mathcal{G}$ and $\mathcal{H}$ are both additive categories, and $S$ is left ...
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55 views

Why do the unit and counit of an adjunction work in opposite directions?

This is probably a very basic question, because it seems to me that this is literally the point of an adjunction, but I don't really get it still. (note, I don't really get adjunctions in general). ...
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1answer
48 views

Why is an adjunction a “weak form of equivalence”?

Wikipedia says: An adjunction between categories $C$ and $D$ is somewhat akin to a "weak form" of an equivalence between $C$ and $D$. I have heard this idea before, e.g. from Qiaochu. Can you ...
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1answer
27 views

Equivalent characterizations of the essential image of reflective subcategories

I'm currently trying to prove the third item of the following exercise from Category Theory in Context, Exercise 4.5.vii. Consider a reflective subcategory inclusion $D \hookrightarrow C$ with ...
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1answer
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Left Adjoint to the Forgetful Functor from Rings to Semirings.

Is there a left adjoint to the forgetful functor from the category of commutative rings to the category of commutative semirings? A semiring is like a ring, except for the existence of an additive ...
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Limits and colimits preservation under adjoint equivalence of categories

If $F\colon\mathcal{A}\to\mathcal{B}$ is left adjoint to $U\colon\mathcal{B}\to\mathcal{A}$, then $U$ preserves limits and $F$ preserves colimits. Can we say something more if $F$ is left adjoint to $...
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1answer
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Regarding the pointwise left adjoint of the $Hom_*$ functor for cartesian closed categories

This question is motivated by a lack of understanding in the following (part of an) exercise of Emily Riehl's Category Theory in Context. More specifically, I do not quite see how to make my solution ...
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What is the intuitive concept captured by “adjunction”

I just watched an introductory lecture to the concept of "adjunctions" in category theory, and read an introductory text on it. They explained the definition, some examples, and how to work with them. ...
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Is there a definition of morphism of gluing data of schemes that makes gluing an adjoint functor?

Tag 01JA of the Stacks Project is about gluing schemes. There, a gluing data is defined as a collection of schemes, and isomorphisms between open subsets of those schemes satisfying the cocycle ...
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Equivalent definitions of adjunction morphisms

I am struggling with the following exercise from Emily Riehl's Category Theory in Context, regarding adjunction morphisms: paraphrasing, let $F : C \to D, G: D \to C$ and $F' : C' \to D', G' : D' \to ...
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Adjunction Signature: abuse of notation or actual functor?

In "Relational Algebra by Way of Adjunctions," found at author's page (doi), section 2.4, an adjunction is described using the signature: $$ L \dashv R:\mathscr{D}\to \mathscr{C}.$$ Based purely on ...
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Stuck with Category Theory notation. What is the meaning of 'Corner brackets' 「 」?

While reading an article, I encountered this expression. Expression I was wondering if anyone knows what does the corner brackets 「(upper) and 」(down) in this expression do? Thank you.
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Adjoint functors for Lie Algebras

Let's restrict to finite dimensional case. Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. ...
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Composition of monadic functors isn't monadic

Disclaimer: this question already has a solution here: Composition of monadic functors may not be monadic . However, I would like to understand how to solve this using another characterisation of ...
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1answer
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How is it obvious that $\times : C \times C \to C$ is right adjoint to the diagonal functor?

This is from "Sheaves in Geometry & Logic". $\times : C \times C \to C$ is the cartesian product of two objects. So assume that finite products exist in $C$ the above is a functor. To say ...
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Left adjoint to the tensor functor

Let $V$ be a vector space over a field $k$. Define $V \otimes - :$ Vect $\to$ Vect the tensor endofunctor on the category of $k$ vector spaces. Assume that $V \otimes - $ preserves limits. It can be ...
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Proving adjoint equivalence from natural isomorphic unit

Let $U:\mathscr{A}\rightarrow\mathscr{B}$ and $F:\mathscr{B}\rightarrow\mathscr{A}$ be two functors. Suppose there is a natural transformation $\eta:\mathbf{1}_\mathscr{B} \rightarrow UF$ such that ...
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Right (bi)adjoint of the inclusion of $\mathbf{Grpd}$ in $\mathbf{Cat}$

Let $\mathbf{Grpd}$ and $\mathbf{Cat}$ be respectively the 2-categories of small groupoids and of small categories. At the 1-categorical level, the inclusion $\mathbf{Grpd}\rightarrow\mathbf{Cat}$ has ...
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Commuting right adjoints implies commuting left adjoints.

Say that I have a diagram of right adjoints $$ \require{AMScd} \begin{CD} A @<{R_1}<< B \\ @V{R_3}VV @VV{R_2}V \\ D @<<{R_4}< C \end{CD} $$ does it follow that the diagram of left ...
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Sandwich natural transformation between two functors

In the Kleisli adjunction we have: $G\varepsilon F = \mu$ where $\varepsilon$ is a natural transformation called the counit. How exactly is $G\varepsilon F$ defined? I understand $G\varepsilon$ and $...
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Left and Right adjoint of forgetful functor

I am studying category theory and I came across two questions that I could not answer about the adjoint of two forgetful functors. Let $\mathbf{TAb}$ be the category of abelian groups such that all ...
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The Powerset Monad

I am struggling to prove commutativity of the diagrams for the powerset monad in Category $\mathbb{Set}$. To show that $\mu:\mathcal{P}^{2}\longrightarrow \mathcal{P}$ given by $\mu_{X}:\mathcal{P}^{...
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1answer
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Does sheafification of bundles have a right adjoint?

Given a topological space $X$, let $\mathbf{Bundle}(X)=\mathbf{Top}/X$ be the category of bundles over $X$, and let $\mathbf{Sh}(X)$ be the category of sheaves over $X$. Then there's a sheafification ...
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Coends and adjunctions

I was reading Fosco Loregian's paper This is the co/end, my only co/friend, and here's something that I don't understand in an exercise. The exercise is to prove that given $F: C\to D, U: D\to C$ ...
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Reading string diagram for counit-unit triangle identities?

On nLab, there are some string diagrams for the triangle identities for the unit and counit of an adjoint pair $(L,R,\eta,\epsilon)$. and with minimal notation, I'm confused about how to read these, ...
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How do you arrive at $\eta : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$ from $\eta : LR \to \text{id}_{C'}$?

On page 28 of "Categories and Sheaves" it says: $$ \eta : L R \to \text{id}_{C'} $$ is a functor but then they have in a commutative diagram right below that: $$ \text{Hom}_{C'}(Y, Y') \xrightarrow{...
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If $F$ and $G$ are adjoint functors, why are $\operatorname{Nat}(G,G)\simeq\operatorname{Nat}(F,F)$ as algebras? [duplicate]

Suppose $F\colon\mathcal{A}\to\mathcal{B}$ and $G\colon\mathcal{B}\to\mathcal{A}$ are adjoint functors between some $R$-linear abelian categories ($R$ a ring), via a fixed counit-unit adjunction $\...
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How to prove naturality in $c$ of $\text{Hom}_D(Fc, d) \simeq \text{Hom}_C(c, Ud)$ given adjunction $(F, U)$?

We're given an adjunction $(F, U)$ where $F: C \to D, U : D \to C$ are the left and right adjoint functors. We use the definition that the adjunction comes with a "unit of adjunction", or a natural ...
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Find a left adjoint to the forgetful functor from the category of monoids to the category of sets

Find a left adjoint to the forgetful functor from the category of monoids to the category of sets. I am not sure what to do here. Clearly we must have something akin to free product... I'm sorry I ...
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1answer
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Show that $\lim$ is a right adjoint to the constant functor

A problen in category theory: Let $I$ be a small category and $\mathcal C$ be a comolete category, and consider the functor $\lim : Func(I, \mathcal C) \to \mathcal C$. Show that $\lim$ is right ...
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Let $t: I \to \mathcal C$ be a funtor with $I$ small, and suppose that $t$ has a colimit. Prove that $\lim_{\to} (F \circ t)$ exists

A problem in category theory: Suppose $F : \mathcal C \to \mathcal D$ has a right adjoint. Fix a functor $t: I \to \mathcal C$ with $I$ small, and suppose that $t$ has a colimit. Prove that $\lim_{\...
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Looking for a pair of adjoint functors

I'm struggling with finding the left adjoint to a functor, and the right adjoint to another one. Here's some context. Given any ring $R$, we can associate two categories to it. The first one is a ...
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Operations that have adjoints rather than inverses

Is there a name for the operation ⊖ which is the left adjoint of natural number addition in the following way? c ⊖ d ≤ a ≡ c ≤ a + d Is there a name for algebraic structures like this, whose ...
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Can a field be canonically reconstructed from its multiplicative group?

I understand that there are a lot of restrictions on which abelian groups arise as the multiplicative groups of fields. My question is kind of the adjoint: Suppose I pick a field $k$, but I do not ...
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1answer
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Is the meaning of the sentence appearing in page 44 of “The Rising Sea” the following commutative diagram?

In Vakil's book "The Rising Sea", page 43-44. Two covariant functors $F:\mathscr A\to \mathscr B$ and $G: \mathscr B\to \mathscr A$ are adjoint if there is a natural bijection for all $A\in \...
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Not sure whether I understand certain adjoint impilcation correctly.

My textbook: The naturality axiom implies that from each array of maps $A_0 \rightarrow... \rightarrow A_n$, $F(A_n) \rightarrow B_0$, $B_0 \rightarrow... \rightarrow B_m$ it is possible to ...
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Groups are cocomplete by adjoint functor theorem without explicit use of free products

$$ \newcommand{GRP}{\mathsf{GRP}} \newcommand{I}{\mathcal{I}} \newcommand{im}{\mathrm{Im} \;} $$ This is a follow up for the question Royal Road to Free Groups and Free Products. I was exploiring ...
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Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
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1answer
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Adjoint functors, inclusion functor, reflective subcategory

Suppose that a category $A$ is reflective in a category $B$ and that the inclusion functor $K:A\to B$ has a left adjoint $F:B\to A.$ Now what does it technically mean that this bijection of sets $$A(...
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1answer
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Free frame generated by a poset

Suppose I have a poset $P$, is there a "best" frame for $P$; that is a frame $L$ with a monotone map $P\to L$ that is universal ? What if I add some nice conditions on $P$: the $P$'s I'm interested ...