Questions tagged [adjoint-functors]
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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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Iterated adjoint functors
Let $F_0 : C \to D$ be a functor. If it exists, let $G_0 : D \to C$ be its left adjoint. If it exists, let $F_1 : C \to D$ be its left adjoint. And so forth. In situations where the infinite ...
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Monoidal categories whose tensor has a left adjoint
Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some ...
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How do I check if a functor has a (left/right) adjoint?
Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, ...
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What is an intuitive view of adjoints? (version 1: category theory)
In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.
I know the definition (several ...
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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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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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Can adjoint linear transformations be naturally realized as adjoint functors?
Last week Yan Zhang asked me the following: is there a way to realize vector spaces as categories so that adjoint functors between pairs of vector spaces become adjoint linear operators in the usual ...
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Surmounting set-theoretical difficulties in algebraic geometry
The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
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If a right adjoint to the product functor exists, must it be the diagonal?
Let $C$ be a category with binary products. The product functor $\times : C^2 \to C$ is right adjoint to the diagonal $\Delta: C \to C^2$. If $C$ has biproducts, then $\times$ is also left adjoint to $...
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What's an example of an "adjunction up to adjunction"?
(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...
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Characterization of functors whose right adjoint is monadic?
Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...
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Adjoints to forcing
Forcing over a partial order $P$ can be viewed in a category theoretic sense as constructing the presheaf topos ${\bf Set}^{P^{op}}$ over the partial order (viewed as a category) then passing through ...
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Which spectra have a universal connective quotient?
Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...
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When is the Eilenberg-Moore category of a relative monad between two topoi a topos?
In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.
Now how does this ...
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A specific property of bi-adjunction
Let $$I: C \rightleftarrows D: F$$ be biadjoint [1] functors between categories $C, D$. That is, $I$ is the left and also the right adjoint of $F$ (thus vice versa). Put in notations, it's
$$ \cdots \...
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Adjunctions: Algebras of the induced monad VS. Coalgebras of the induced comonad.
Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent?
The example i have ...
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Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
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Different levels of isomorphism/equivalence/adjunction between bicategories
What are all the different levels of 'isomorphism/equivalence/adjunction' we can have between bicategories? Do any of them 'collapse' to one-another?
When working with $1$-categories, we have four ...
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Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...
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When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the ...
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Which spectra have a homotopy-universal connective quotient?
Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...
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How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
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Another adjoint pair: Definable sets and set-builder formulas
I see adjointness between the two concepts of "being a definable set" and "being a set-builder formula":
A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\...
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Uniqueness of comparison functors
Given an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ with unit $\eta$ and counit $\epsilon$, we naturally have a monad $(G\circ F,\eta,G\epsilon_F)$ on $\mathcal{C}$ and a comparison ...
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Reference request for (co-)free constructions
Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces ...
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Adjoints of exact functors between semisimple abelian categories
Motivated by the answer to this question, I will ask the following question: Let $\mathcal{A}$ and $\mathcal{B}$ be small semisimple abelian categories. Let $U:\mathcal{A} \to \mathcal{B}$ be a ...
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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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