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### Adjunctions capturing duality between ideals and saturated monoids in a commutative ring?

Let $R$ be a commutative ring. A saturated monoid in $R$ is a multiplicative submonoid $S\subset R$ which is closed under divisors, i.e $xy\in S\implies x\in S$. This is the converse of the analogous ...
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### Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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### Does the functor sending a DGA to its zeroth component admit a right adjoint?

Let $A$ be a ring and write $\underline{A}^\bullet$ for the associated trivial DGA. We have a functor $$\mathrm{ev}_0\colon\mathbf{dgAlg}_{\underline{A}^\bullet}\longrightarrow\mathbf{Alg}_A$$ sending ...
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### Left and right Kan extensions

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between small categories. We define the functor \begin{align*} f:\hat{\mathcal{D}}&\longrightarrow\hat{\mathcal{C}} \\ G&\longmapsto G\circ F^{\...
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Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
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### Adjunctions between Groupoids and Hilbert spaces

I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
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### Question on Eilenberg-Watts theorem

I'm not sure if this is a research level question, but: Let $F:Rep_A \to Rep_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has ...
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### Explicit expression of the unstraightening functor

Hard as I tried, I couldn't find a proof of Remark 2.2.2.11 in Higher Topos Theory, or prove it myself. It seems to need an explicit formulation for the unstraightening functor, so my question is: is ...
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### Adjoints of scalar extension and scalar coextension

Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts): $h^*$: Scalar extension by ...
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### “Equivalence” is to “group” as “adjoint” is to …?

The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces ...
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### Adjoints for the functor ${\bf Top}\to {\bf Conv}$

Let $X$ be a set and let $\Phi(X)$ denote the collection of filters on $X$. For $x\in X$ we denote by $P_x$ the filter $P_x=\{A\subseteq X:x\in A\}$. A convergence space is a pair $(X,\to)$, where $X$ ...
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### Translating parabolic induction as $\Lambda G^F/U^F\otimes_{\Lambda L^F}-$ to $\hom_{\Lambda L^F}(\Lambda U^F/G^F,-)$?

Suppose $P=L\ltimes U$ is an $F$-stable parabolic subgroup of a finite group of Lie type $G$, with $F$-stable Levi complement $L$. Here $F$ is a Frobenius endomorphism, and $G^F$ is the subgroup of ...
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Forgive me if this question is not well thought out. I don't know how else to ask it. The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
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I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\... 1answer 159 views ### About small-orthogonality classes of a locally presentable category Let$\mathcal{A} \subset \mathcal{K}$be two locally presentable categories.$\mathcal{A}$reflective and closed under filtered colimits. Then$\mathcal{A}$is a small-orthogonality class. Let$...
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I heard a professor say that $\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same. What does it mean, intuitively, that a category is ...
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### Why quasi-inverse functors are adjoint pairs?

Let $C$,$D$ be two category, and $F:C \rightarrow D$, $G:D \rightarrow C$ are two functors such that $FG \simeq Id_{D}$ and $GF \simeq Id_{C}$, Show that $(F,G)$ is an adjoint pair. To show this, we ...
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Has the monad induced by the free commutative monoid functor already been studied anywhere? Does it have any particular properties (other than not being cartesian)? I would prefer a reference on ...
The $n$Lab writes (prop. 2.2 in https://ncatlab.org/nlab/show/category+of+monoids) : Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the ...