2
votes
1answer
40 views

Adjoint functors for the power set monad

There is the power set functor, $T$, which gives raise to a monad: For a set $X$, we set $TX:=\mathcal P(X)$ and for $f:X\to Y$, we set $T(f):=S\mapsto f(S)$, where $f(S)$ denotes the direct image. ...
5
votes
1answer
96 views

Right-adjoint to the inverse image functor

Let $X$ be a set. We can turn $\mathcal P(X)$ (the power set of $X$) into a category by taking inclusion maps as morphisms. Now consider a function $f : X \to Y$, which induces the functor $f^{-1} : ...
1
vote
1answer
49 views

Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
0
votes
0answers
66 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
3
votes
1answer
59 views

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as ...
2
votes
1answer
58 views

Proof that left adjoints preserve direct limits

I am reading Rotman's book on Homological algbra and have a slightly different proof of the statement in the title of this question. Am writing my attempt below. Could someone please advise me if I am ...
6
votes
1answer
127 views

Adjoint Functor Theorem

The Freyd's Adjoint Theorem states that given a complete locally small category $\mathcal{C}$, a continuous functor $G: \mathcal{C} \to \mathcal{D}$ has a left adjoint if and only if it satisfies a ...
2
votes
1answer
22 views

Proving that the transformation obtained from an adjoint pair is natural

I am reading Homological Algebra by J.J. Rotman and am unable to do this problem. Given an adjoint pair $(F,G)$ where $F : \mathcal{C} \to \mathcal{D} $ and $G : \mathcal{D} \to \mathcal{C} $ are two ...
4
votes
0answers
51 views

Closure operators and complete lattices.

A closure operator on a set $A$ is a function $C: \mathcal{P}(A) \to \mathcal{P}(A)$ satisfying following axioms: $X ⊆ Y \implies C(X) ⊆ C(Y)$ $X ⊆ C(X)$ It may also satisfy some additional ...
3
votes
2answers
98 views

Right adjoint to forgetful functor $\mathbf{Cat} \to \mathbf{Graph}$

There is a forgetful functor $U:\mathbf{Cat} \to \mathbf{Graph}$, which assigns a (small) category to its underlying (small) graph. Also, it has a left adjoint $F:\mathbf{Graph} \to \mathbf{Cat}$, ...
4
votes
2answers
67 views

Duality 2-functor on adjunctions

This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...
0
votes
1answer
57 views

Condition for a reflective subcategory of a cartesian closed category to be an exponential ideal

Here's the question I think I'm asking, with background below if necessary: Question: The reflector $L$ left adjoint to the inclusion of a reflective subcategory $\mathcal L\to\mathcal E$ is ...
1
vote
1answer
53 views

Proof that sheafification induces isomorphism on stalks using adjoints

Let $\mathcal{F}$ be a presheaf on some topological space $X$. It is not hard to prove directly that the map $\mathcal{F}\rightarrow \mathcal{F}^{sh}$ induces an isomorphism of stalks (Here ...
7
votes
2answers
95 views

Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
2
votes
1answer
25 views

Composition of Functors which have adjoints has also an adjoint [duplicate]

The exercise is the following: Suppose $F$ has right adjoint $G$ and $H$ has right adjoint $J$ ($F:\mathbb{A}\rightarrow\mathbb{B}$ and $H:\mathbb{B}\rightarrow\mathbb{C}$ with $\mathbb{A,B,C}$ ...
4
votes
2answers
56 views

Adjoint to a functor $\textbf{PoSets}\rightarrow\textbf{PreOrd}$

i have a question about adjoints in category Theory. Let $\textbf{Posets}$ the category of Posets (thus Sets with binary relation $\leq$ which is reflexiv, transitiv and antisymmetric) and let ...
3
votes
2answers
95 views

Equivalence of the definition of Adjoint Functors via Universal Morphisms and Unit-Counit

I was reading here about adjoint functors, and I was following the construction of the right adjoint to a left adjoint functor, and I kept getting tripped up over showing that the resulting functor ...
1
vote
1answer
65 views

Adjoint to identity functor must be trivial

If an endofunctor F on some category $\mathfrak{C}$ is (left) adjoint to the identity functor.... then does it necesarilly have to also be the identity functor... Since $\mathfrak{C}(F(X),Y)\cong ...
2
votes
1answer
29 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
1
vote
1answer
78 views

Composition of adjoint functors

Does the composition of adjoint functors again form an adjunction? Say $\langle F_1,G^1\rangle$ is an adjunct pair between two categories A and B and $\langle F_2,G^2\rangle$ is also an adjoint pair ...
2
votes
2answers
91 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
5
votes
1answer
51 views

Isomorphism through adjunction

An adjunction $F \dashv G$ gives a morphism $\phi(f) : A \to G B$ to each morphism $f : F A \to B$. Does $\phi(f)$ have any special property if I know that $f : F A \to B$ is an isomorphism?
3
votes
1answer
76 views

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 ...
6
votes
2answers
75 views

Commutativity of the square diagram coming from an adjoint triple

Suppose we have an adjoint triple $F \dashv G \dashv H$ with the following (co)units: $$\eta : I \to GF, \ \epsilon \colon FG \to I, \ \bar{\eta} : I \to HG, \ \bar{\epsilon} \colon GH \to I.$$ ...
6
votes
2answers
136 views

Equivalence of Categories and of Their Functor Categories

Suppose $A, B, C$ are categories. If $A$ and $B$ are equivalent, is it the case that $C^A$ and $C^B$ are equivalent? Also, is it the case that $A^C$ and $B^C$ are equivalent. I first conjectured that ...
3
votes
1answer
138 views

Right adjoint unique up to isomorphism

i want to prove the following without the Yoneda Lemma (because it is the exercise): Suppose $F\dashv G$ (with unit $\eta$ and counit $\epsilon$) and $F\dashv G'$ (with unit \eta' and conunit ...
3
votes
1answer
58 views

Help with exercise on Reports of the Midwest Category Seminar IV

At the end of the LMN 137, "Reports of the Midwest Category Seminar IV", there is a list of exercises. "5. Considering a left-adjoint as male and a right adjoint as female, give the correct term for ...
2
votes
1answer
64 views

Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $$ {\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring}, $$ where ...
1
vote
1answer
274 views

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 ...
2
votes
1answer
35 views

A case where we have a functor, and we are looking for the right adjoint

Most of the examples of adjoint functors I saw ''in the wild'' have a right adjoint forgetting a part of structure, and left adjoint recovering it in the most efficient/general way. Often, a functor ...
-1
votes
2answers
122 views

Polynomial ring and the free algebra

In the Algebra book of Mac Lane there is an exercise in Chap. IV which tells me to construct a polynomial ring $A[X]$ for any set (not necessarily finite) $X$ ($A$ a ring), and to give correct the ...
3
votes
2answers
233 views

Does the adjugant of additive functors between abelian categories preserve the abelian structure of the hom-set?

I think the following is a counter-example. I noticed it when trying to prove that the sheafification functor induces isomorphism on the stalks (Vakil 2.4M). As in Vakil (2.6.3) the stalk functor is ...
4
votes
1answer
89 views

Derive adjoint unit from counit

As an exercise (2.4.12#5) in Pierce's Basic Category Theory for Computer Scientists, I'm trying to derive the unit natural transformation $\eta : I_{\textbf{C}} \xrightarrow{\cdot} G \circ F$ given ...
9
votes
2answers
215 views

When does a left adjoint between Heyting algebras preserve 1?

This is an exercise from Johnstone's book Stone Spaces: Let $f: A \to B$ and $g: B \to A$ be order-preserving maps between complete Heyting algebras with $f$ left adjoint of $g$. Show that $f$ ...
1
vote
1answer
103 views

Are coproducts right adjoint of diagonal functor on opposite category?

Continuing my recent investigations into adjunctions, I've come to understand how products are defined in terms of adjunctions. A category $\mathscr{C}$ has products if there is a right adjoint to ...
6
votes
1answer
85 views

name of the unit of adjunction between $-\times C$ and $\cdot^C$

Answers to a earlier question about the categorical interpretation of first-order quantification led me to learn more about adjoints. Now, I understand that a category $\mathscr{C}$ with products has ...
0
votes
1answer
104 views

Localization of modules as adjunction

Usually, the localization of a $R$-module $M$ by a multiplicative subset $S \subseteq R$ with $1 \in S$ is categorically defined as the initial object of the full subcategory $\mathbf C$ of $M \, ...
1
vote
1answer
60 views

Adjoints preserve limits (resp. colimits) Do they preserves completeness (resp. cocompleteness)?

I know that left adjoints preserve colimits and right adjoints preserve limits. So clearly if the limits (resp. colimits) in both categories exist, the adjoints map them to each other. My question ...
11
votes
2answers
378 views

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, ...
5
votes
2answers
261 views

Left Adjoint of a Representable Functor

Let $\mathcal{C}$ be a category with coproducts. Show that if $G:\mathcal{C} \to \mathbf{Set}$ is representable then $G$ has a left adjoint. I can't seem to wrap my head around this nor why ...
5
votes
1answer
62 views

Does completeness of a category in an adjunction imply completeness for the other?

Assume we have an adjunction $(L,R,\varphi):\mathcal{C}\rightarrow\mathcal{D}$ between two categories, and assume also that $\mathcal{D}$ is complete (i.e. closed under limits). Under what assumptions ...
4
votes
3answers
155 views

How to show two functors form an adjunction

Say I have two functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$. How can I show they form an adjunction without writing explicitly the natural transformations ...
5
votes
3answers
769 views

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 operator, then $\mathcal{F}$ is ...
3
votes
0answers
98 views

Functors $f^*$ and $f_*$ on the category of sheafs of modules (Hartshorne)

Let $f: (X, O_X) \rightarrow (Y, O_Y)$ be a scheme morphism, $F$ - module over $O_X$, $G$ - module over $O_Y$. How to prove, that $$ Hom_{O_X}(f^*G, F) = Hom_{O_Y}(G, f_* F). $$ Please give the most ...
2
votes
1answer
103 views

Questions about adjointness of quantifiers in first-order logic

I have a category theory homework problem which asks: "In first-order logic, why does $\forall$ not have a right adjoint?" The typical argument is that: If for some operator $\cdot$ it could be ...
1
vote
0answers
69 views

Adjoint to the Hom functor in Boolean rigs

What I wanna ask is about analogies to the tensor product for commutative boolean rings. What I mean by commutative boolean ring is set with two operations, + and *, and two identities, 0 and 1, as is ...
2
votes
1answer
81 views

What kind of structure are exponentials in their “contravariant argument”

Given a cartesian closed category $\mathbf C$ (or any closed monoidal category) the covariant part of the internal Hom-functor is defined simply in terms of it being the right adjoint to the product. ...
1
vote
0answers
80 views

Can eigenvectors and eigenvalues be seen as limits is the same sense as fixed points?

Can eigenvectors and eigenvalues be seen as limits in the same sense as fixed points? What I mean by the same sense as fixed points is that if $G$ is group we can consider limits to ${\bf Set}^G$, or ...
7
votes
2answers
456 views

Understanding adjoint functors

To understand adjoint functors I tried to look at an example. Can you tell me if the following is correct? Before I give the example I'd like to recap the definition: Given two categories $C,D$ and ...
1
vote
0answers
106 views

Adjoint prefunctors

I have a functor and a prefunctor (not a functor) "in the inverse direction". Can the notion of adjunction be generalized for prefunctors? I remind that a prefunctor is a functor without the ...