# Tagged Questions

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### Does the inclusion of presheaves into families of sets have a left adjoint?

Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this ...
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### Are there non-trivial infinite chains of adjoint functors?

There are self-adjoint functors $A \dashv A$. There are also functors $A$ that are both left- and right-adjoint to another functor $B$. $$A \dashv B \dashv A$$ There are also finite cyclic chains of ...
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### Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...
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### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...
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### Adjoint Functors as Initial Objects of Some Category

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
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In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states: Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback ...
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In this nlab page there is defined the notion of a lax 2-adjoint. My question is to what extent would an adjoint of this type be unique?
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Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
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### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly. Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...
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We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...
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### Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...
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### Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair $$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$ where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
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### Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
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### Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch? There is ...
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### Does a scheme have a “separification”?

Background: (1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
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### Categories where morphisms are pairs of adjoint functors

Is there a particular name for those categories (of categories) where morphisms "come as adjoint pairs", as in the case of toposes and model categories? Is there any attempt to study them as general ...
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### Is there a monad on Set whose algebras are Tychonoff spaces?

Compact Hausdorff spaces are algebras of the ultrafilter monad on Set. Is the category of Tychonoff spaces also monadic over Set?
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### Is there an analog of adjoint functor theorem for adjunctions of two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection  \mathscr C(L(A,...
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### Do generalizations of adjoint functors, such as adjunctions in 2-categories, and multivariable adjunctions, have formulations in terms of something like universal morphisms?

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In answering this MO question, the issue was raised of characterizing when a given endofunctor $R:C\to C$ has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, i.e. which admit a ...
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My first question here. Accordingly to M. Barr "Coequalizers and free triples" by a free triple (or free monad) generated by an endofunctor $R: X\rightarrow{X}$ we mean a triple $T=(T,\eta,\nu)$ and ...