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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...