Questions tagged [adjoint-functors]

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

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38 views

Additive functors from the category of free abelian groups are right adjoints?

I am reading Lectures on algebraic topology by Albrecht Dold. Let $t$ be an additive functor from the category of free abelian groups to the category of abelian groups. In VI 7.3 he writes: For any ...
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1answer
36 views

Inituition of adjoint functors as best solution, toughest problem pairs

I'm trying to understand the use of adjoint functors and came across the interpretation of them as the optimal solution to some problem. But am trying to wrap my head around how to connect them with ...
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1answer
55 views

Smash Product as Pullback?

In considering the $\mathsf{Hom}$-functor $$ \mathsf{Hom}_*: \mathsf{Set}_*^{\text{op}} \times \mathsf{Set}_* \to \mathsf{Set}_* $$ it feels somewhat clear to me that the left adjoint is the smash ...
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1answer
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construction of adjoint of forgetful functor Set_\star to Set

I need to determine if the forgetful functor \begin{equation} U: Set_\star \longrightarrow Set \end{equation} that forgets "the base points" has left adjoint or right adjoint, but I'm ...
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1answer
53 views

New terminology for left and right adjoints [closed]

Let $C$, $D$ be categories and $F:C\to D$, $G: D\to C$ be functors. Consider the following properties : $p_1$ : "$F$ is a left adjoint of $G$" $p_2$ : "$F$ has a right adjoint" $...
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1answer
26 views

When is there an “intuitive” functor from F-coalgebras to T-coalgebras?

Suppose $F, T : Set \rightarrow Set$ are two functors on the category of sets. Let $F^{coalg}$, $T^{coalg}$ denote the categories of $F$, respectively $T$ coalgebras. Vaguely, I'm interested in when ...
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1answer
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Viewing Objects as Objects of a Reflective Subcategory with Bonus Structure

A category $\mathcal{C}$ is called a reflective subcategory of $\mathcal{D}$ whenever the inclusion $i : \mathcal{C} \to \mathcal{D}$ admits a left adjoint (confusingly called $R$, the "reflector&...
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1answer
50 views

How to use the fact that a pair of functors are adjoint?

This may be more of a heuristic question but I feel like it should have some concrete answers. I want to understand why a pair of functors being adjoint is useful. How do we use that fact to prove ...
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How to show that proper direct image is left adjoint to inverse image?

Let $f : X \rightarrow Y$ be a proper map of schemes. How do I show that $f^* \vdash f_!$? That is, how do I show that proper direct image is left adjoint to inverse image? Somewhere I need to use ...
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Coalgebras of a free-forgetful adjunction

An algebra for a monad $T : \mathcal{C} \to \mathcal{C}$ is an arrow $m : Tc \to c$ satisfying some relevant conditions regarding the unit $\eta : 1 \Rightarrow T$ and the multiplication $\mu : T^2 \...
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Showing $X \rightarrow k\{X\}$ is left adjoint to the forgetful functor

following Page 280 of Christian Kassels "Quantum Groups" Example of adjoint functors: Let $X$ be a set and $k\{X\}$ be the free k-algebra associated to $X$. Then $X \rightarrow k\{X\}$ is a ...
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1answer
38 views

Conditions for adjoints to preserve monos?

Given an adjoint pair $L \dashv R$ and a mono $f \in \text{Hom}(X, RY)$, what are some conditions which will guarantee $\tilde{f} \in \text{Hom}(LX, Y)$ is still a mono? Obviously this becomes easy if ...
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1answer
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Is the natural isomorphism in an adjunction uniquely determined by the pair of adjoint functors

An adjunction is a triple $(F, U, \zeta)$, where $F\colon C\to D$ and $U\colon D\to C$ are functors and $\zeta$ is an isomorphism between the functors $\operatorname{Hom}(-, U(-))$ and $\operatorname{...
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1answer
36 views

Equivalence between two definitions of a category having exponential objects

A category with products is said to have exponentials if for all objects $x, y$ there exists an object $y^x$ equipped with an arrow $e\colon x\times y^x\to y$ such that for all objects $z$ and all ...
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1answer
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Naturality of $\varphi : \textbf{Vct}_K(V(x), w) \xrightarrow{\sim} \textbf{Set}(x, U(w))$ in the variable $x$ (Cats for the Working Mathematician).

Consider the forgetful functor $U : \textbf{Vct}_K \to \textbf{Set}$ and the functor $V : \textbf{Set} \to \textbf{Vct}_K$ that takes an object $x$ in set to the $K$-vector space $V(x)$ with basis $x$ ...
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1answer
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Dual image map restricts to open sets?

A book I'm reading on category theory says that if $A$ and $B$ are topological spaces and $f:A\to B$ is continuous, then the "dual image" map $$f_*(U)=\{\,b\in B\mid f^{-1}(b)\subseteq U\,\}$...
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1answer
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What are some examples of self-adjoint functors? Is this an example?

I've been trying to figure out what some examples of "self-adjoint" functors are, or when this even happens, since I've never seen this before. What I mean is if $F: \mathcal{C} \to \...
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1answer
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Dual of a statement involving adjoint functors

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a functor. The following conditions are equivalent: $F$ is full and faithful and has a full and faithful left adjoint $G$. $F$ has a left adjoint $G$ and ...
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1answer
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Solution set condition and the forgetful functor $U:\textbf{Gr}\rightarrow\textbf{Set}$

Let $U:\textbf{Gr}\rightarrow\textbf{Set}$ be the forgetful functor from the category of groups to the category of sets. Let $X\in\textbf{Set}$. I want to construct the solution set $S_X\subset\text{...
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1answer
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Right adjoint to the forgetful functor $\text{Ob}$

Let $\text{Ob}:\textbf{Cat}\rightarrow\textbf{Set}$ be the forgetful functor mapping a small category to its set of objects. Consider the functor $R:\mathbf{Set}\rightarrow\textbf{Cat}$ mapping a set $...
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1answer
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Adjunctions from triangle isomorphisms

A very elementary result in category theory allow us to promote equivalences of categories to adjoint equivalences by changing one of the two natural isomorphisms. Generalizing this, consider the ...
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1answer
54 views

$\text{colim}$ defines a functor, which is a left adjoint to the diagonal functor

I'd like to prove an analogous result for colimits. I suppose the conclusion of the proposition for the limits should be that $\text{colim}:[\mathbf I,\mathscr A]\to\mathscr A$ defines a functor, and ...
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0answers
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Adjoint functors preserve limits/colimits

Here's another theorem from Leinster's book (p. 159) where I got stuck: Just as in my previous question, I don't see how this sequence of isomorphisms establishes the claimed result. To prove the ...
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How to understand the effect of adjoint functors?

I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a ...
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A verification of a detail in the proof of Leinster's Theorem 2.2.5

In the proof of the theorem that states that there is a bijection between adjunctions between functors and pairs of functors satisfying the triangle identities (Theorem 2.2.5), one part of the proof ...
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1answer
35 views

Proving the triangle identities diagrammatically

Leinster gives a proof of the triangle identities (p. 52): However, I'm trying not to use (2.3) directly (because there's no way to memorize those identities), instead, I'm trying to draw naturality ...
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1answer
23 views

Showing that $\epsilon F$ is a natural transformation

I'm trying to prove that that the naturality squares commute for the natural transformation $\epsilon F$ defined by $(\epsilon F)_A=\epsilon_{F(A)}:FGF(A)\to F(A)$ where $\epsilon$ is the counit of ...
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1answer
40 views

Proving that the counit of adjunction is a natural transformation

Let $F:\mathcal A\to\mathcal B, G:\mathcal B\to \mathcal A$ be functors such that $F$ is left adjoint to $G$. I'm trying to prove that the counit of adjunction $\epsilon: FG\to 1_\mathscr{B}$ is a ...
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Restriction-extension identities using six functors formalism

Recently fellow user Thorgott pointed out to me that flat restrictions of flat modules remain flat. That is, let $f : A \to B$ be a flat morphism of rings and $M$ be a flat $B$-module. Then ...
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1answer
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A question about adjoint functors:

I am trying to replicate the solution to this question. But, in the process I got stuck. Here is my question: Suppose $F: C\rightleftarrows D :G$ is an adjoin pair with unit $\eta :1_C\Rightarrow GF$ ...
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Adjointenes of unknown functors

I would like to understand what are here on the page $266$ the types of items in the adjoint euqation $$\mu^{Lim}_{\cal K}\vdash Lim\ \eta_{\cal K}^{Lim}.$$ They should be functors by the definition ...
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2answers
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Left adjoint to inclusion functor of torsion-free abelian groups in abelian groups

Let $\text{Ab}$ be the category of abelian groups, $\text{TFAb}$ the full subcategory of torsion-free abelian groups, and $F \colon \text{TFAb} \to \text{Ab}$ the inclusion functor. I'm trying to show ...
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0answers
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Non-evil definition of a Kleisli object in a weak 2-category

Let $t : a \to a$ be a monad in a weak 2-category. According to nLab, the 1-dimensional universal property of a Kleisli object $(f_t : a \to a_t, \lambda : f_t t \to f_t)$ is that for any right $t$-...
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Sufficient conditions for the counit to be an isomorphism

Let $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{A}$ be a pair of adjoint functors on some categories. The following is well-known: The right adjoint functor $G$ is fully ...
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1answer
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$F: \mathcal{C} \to \mathcal{D}$ equivalence of categories + $\mathcal{C}$ complete implies $\mathcal{D}$ complete.

Let $F: \mathcal{C} \to \mathcal{D}$ be an equivalence of categories where $\mathcal{C}$ is a complete category, i.e. any functor $F: I \to \mathcal{C}$ with $I$ small admits a limit. I'm trying to ...
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1answer
49 views

Right adjoints are isomorphic

Proposition Let $F:\mathcal{C}\to\mathcal{D}$ and $G,G':\mathcal{D}\to\mathcal{C}$ be functors. If $F\dashv G$ and $F\dashv G'$ then $G\simeq G'$. proof. Since $F\dashv G$, there exists a natural ...
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Monoidal categories: strictness from coherence

A version of Mac Lane's Coherence Theorem states that every formal diagram (i.e., a diagram that involves only the associativity isomorphism, the unit isomorphisms, their inverses, identity morphisms, ...
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0answers
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Pointwise adjointness theorem

I'm confused in the way Barr and Wells proof the uniqueness of adjoints. Once they prove that using Yoneda lemma the two candidates $F,F'$ to be adjoint to some functor $U$ satisfy $F A \cong F' A$ ...
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1answer
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Kleisli adjunction in a (weak) 2-category

In the 2-category of 1-categories, each monad $T$ on a category $\mathcal C$ determines a Kleisli category $\mathcal{C}_T$ and the so-called Kleisli adjunction between categories $\mathcal C$ and $\...
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2answers
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Composition of adjunctions in a (weak) 2-category

Let $(f_1, g_1, \varepsilon_1, \eta_1)$ and $(f_2, g_2, \varepsilon_2, \eta_2)$ be adjunctions in a (weak) 2-category. Then there is an adjunction $(f_2 \circ f1, g1 \circ g2, \varepsilon, \eta)$. I ...
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Does the embedding of the Kleisli category of a monad into its Eilenberg-Moore category have a right adjoint?

Let $(T,m,e)$ be a monad on a category $\mathcal{A}$. There is a full and faithful functor $J_T$ from the Kleisli category $\operatorname{Kl}(T)$ of $T$ to its Eilenberg-Moore category $\operatorname{...
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Product functor and diagonal functor

Let $C$ be a category and consider the product category $C \times C$. There is a diagonal functor associating to each object $X$ of $C$ the pair $(X,X)$ as an object of $C \times C$. On the other ...
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1answer
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Dualizing 2-categorical results in the context of locally small categories

Monad, comonad and adjunction are 2-categorical notions. Results about them can be dualized as shown in this answer. In the second part the answer, the dualization is successfully applied to the ...
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1answer
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Possible adjoint to Yoneda embedding and Repeated Yoneda embedding?

While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague): Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\...
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Geometric morphisms and presheaves

I would like concrete descriptions of the adjoint functors that arise in a geometric morphism induced by a functor between the base categories of categories of presheaves. Suppose we have functor $F$...
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1answer
104 views

Mac Lane & Moerdijk's Exercise II.7.

This is Exercise II.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 question, it is new to MSE. The Details: From p. ...
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1answer
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Existence of adjoint functors between topological categories

We have natural functors: $ Mfd\hookrightarrow Top ~~~~$ from the category of smooth manifolds to that of topological spaces, $ LieGrp\hookrightarrow TopGrp ~~~~$ from the category of Lie ...
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0answers
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Layman Intuition of Adjunctions (Category Theory)

I've been trying to get to grips with adjunctions lately, but as with most of my experience with Cat Theory, I'm struggling to bring it back to a solid understanding. What follows is my best guess at ...
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1answer
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Limits via universal arrows and functor categories

I would like to understand is some detail the connection between the 2 snippets taken from McLane's book CWM. Namely, I do not follow the connection between the functor $S$ and categories and arrows $...
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3answers
200 views

How to interpret this definition of adjoint functors?

Firstly consider the four definitions in the question: How to define rigorously [...]. Also consider the following definition: Definition: Let $C,D$ be two categories and $F,G:[C]\to [D]$ be two ...

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