# Tagged Questions

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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. ...
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### 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 ...
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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 ...
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### 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 ...
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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?
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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 ...
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### 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: $$\eta : I \to GF, \ \epsilon \colon FG \to I, \ \bar{\eta} : I \to HG, \ \bar{\epsilon} \colon GH \to I.$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f$ ...
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### 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 ...
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### 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 ...