nForum - Search Results Feed (Tag: adjoint) 2021-12-07T16:35:50-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Frobenius reciprocity https://nforum.ncatlab.org/discussion/2396/ 2011-01-12T16:46:06-05:00 2020-08-28T14:40:32-04:00 zskoda https://nforum.ncatlab.org/account/10/ Stub Frobenius reciprocity. ]]> Elaboration (references) that adjoint functors can be used for (symbolic) optimization https://nforum.ncatlab.org/discussion/9843/ 2019-04-24T03:01:41-04:00 2019-04-24T04:12:44-04:00 tomr https://nforum.ncatlab.org/account/909/ Wiki has interesting chapter https://en.wikipedia.org/wiki/Adjoint_functors#Solutions_to_optimization_problems that adjoint functors can be used for optimization, I guess more in the sense of finding ...

Wiki has interesting chapter https://en.wikipedia.org/wiki/Adjoint_functors#Solutions_to_optimization_problems that adjoint functors can be used for optimization, I guess more in the sense of finding optimal objects, structures. Is this original idea whose first exposition is in the wiki article or maybe there are available some references and elaborations of this idea? It would be good to know them? References will suffice, I can study them further.

Also, I guess, such optimization can use for solving the “optimal, paradox free deontic logic” as sketched in my previous question https://nforum.ncatlab.org/discussion/9838/category-of-institutions

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Q-category https://nforum.ncatlab.org/discussion/2650/ 2011-04-25T12:56:24-04:00 2017-09-14T08:32:34-04:00 zskoda https://nforum.ncatlab.org/account/10/ There is a small error in the current proof that the category of endofunctors on a Q-category is a Q-category. I am going to correct it as soon as I find my way through the notation (I used it ...

There is a small error in the current proof that the category of endofunctors on a Q-category is a Q-category. I am going to correct it as soon as I find my way through the notation (I used it different on the paper). It now reads

The $(C^R \dashv C^L)$-unit is the dual $C^\eta$ of the original counit $\eta$

$C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{L R}$

and the counit is the dual of the original unit

$C^\epsilon : C^R\circ C^L = C^{R L}\to Id_{C^{\bar{A}}} \,.$

The wrong thing is that $C^L\circ C^R = C^{RL}$, not $C^{LR}$ and that is why the unit and counit got interchanged; they should not get interchanged, but $C^L$ and $C^R$ should. I am going to sort this out. Thus $C^\eta$ where $\eta$ is unit goes $C^\eta : Id_{C^A}\to C^{RL}$.

Edit: the correct version is now below.

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HETEROMORPHISMS https://nforum.ncatlab.org/discussion/3568/ 2012-02-15T17:30:13-05:00 2017-06-24T09:47:02-04:00 jim_stasheff https://nforum.ncatlab.org/account/12/ Has anyone commented already of Ellerman's Adjoint functors and heteromorphisms?or other papers about them? Has anyone commented already of Ellerman's Adjoint functors and heteromorphisms?
or other papers about them? ]]>
coreflective subcategory - examples https://nforum.ncatlab.org/discussion/7606/ 2017-02-18T20:03:35-05:00 2017-02-18T22:14:03-05:00 Bartek https://nforum.ncatlab.org/account/1354/ Included Lie integration of finite-dimensional real Lie algebras as an example of a coreflective subcategory. The coreflector is Lie differentiation. Included Lie integration of finite-dimensional real Lie algebras as an example of a coreflective subcategory. The coreflector is Lie differentiation. ]]> relative adjoint https://nforum.ncatlab.org/discussion/3515/ 2012-01-31T12:49:55-05:00 2016-09-18T03:31:48-04:00 eparejatobes https://nforum.ncatlab.org/account/373/ Hi everyone! I’ve created relative adjoint functor, and linked to it from the local definition of adjoint functor (a partially defined adjoint yields an adjoint relative to the inclusion of a full ...

Hi everyone!

I’ve created relative adjoint functor, and linked to it from the local definition of adjoint functor (a partially defined adjoint yields an adjoint relative to the inclusion of a full subcategory).

$L {\,\,}_J\!\dashv R$ and $L \dashv_J R$ is as far as I know nonstandard notation, but I think it’s ok, even if the left subscript feels a bit kludgy. I will add more stuff in the next few days.

PS: Thanks a lot to all the nLab contributors; in the past few years I’ve learn a lot through here :) I now have the time and a little bit of confidence to contribute, so any pointers, tips, formatting, style suggestions, whatever will be greatly appreciated

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Adjoints in a commutative square of functors https://nforum.ncatlab.org/discussion/5170/ 2013-08-14T23:06:01-04:00 2013-08-15T02:12:18-04:00 ceciliaflori https://nforum.ncatlab.org/account/914/ Let be a commutative square of categories and functors. Assume that L 1L_1 and L 2L_2 have right adjoints R 1R_1 and R 2R_2, respectively. Under which conditions do we have FR 1&cong;R ...

Let

be a commutative square of categories and functors. Assume that $L_1$ and $L_2$ have right adjoints $R_1$ and $R_2$, respectively. Under which conditions do we have $FR_1\cong R_2G$?

The thing is that we have a concrete situation in which this does seem to be the case, but we would like to have an easy-to-check criterion which implies it.

In our case, all four categories are actually functor categories and the right adjoints correspond to taking Kan extensions.

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Frobenius reciprocity for 2-representations of finite groups https://nforum.ncatlab.org/discussion/4432/ 2012-10-24T21:35:46-04:00 2012-11-22T14:11:47-05:00 domenico_fiorenza https://nforum.ncatlab.org/account/37/ At Frobenius reciprocity I can see Frobenius reciprocity can be formulated in very general terms and that the classical adjoint pair in the theory of finite dimensional representations of finite ...

At Frobenius reciprocity I can see Frobenius reciprocity can be formulated in very general terms and that the classical adjoint pair in the theory of finite dimensional representations of finite groups $Hom_H(W,Res(V))=Hom_G(Ind(W),V)$ with $H$ a subgroup of $G$ is just a very particular case of the general theory. However, this familiar case has (to me) the advantage that I know how to explicitly compute $Ind(W)$ for a given representation $W$ of $H$ and how to prove the adjointness between $Res$ and $Ind$ by hand. So I’m wondering about what can be said about linear 2-representations of finite groups (with, I guess, some finiteness assumptions I’m not able to specify at the moment). For instance, if we take as a model for $2Vect$ the 2-category algebras/bimodules/bimodule morphisms then I have a clear idea of what the explcit data of a representation of a finite group $G$ with values in $2Vect$ are, and can easily write a restriction functor $Res$ from 2-representations of $G$ with values in $2Vect$ to 2-representations of a subgroup $H$ with values in $2Vect$. Does ths have an adjoint $Ind$? How is this explicitely described? (I should be able to work this out by myself by thinking to it enough, but I would like not to loose time on this if it is already well known and a pointer to the literature will solve this)

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differential monad https://nforum.ncatlab.org/discussion/2696/ 2011-05-05T17:46:19-04:00 2011-05-06T14:32:22-04:00 zskoda https://nforum.ncatlab.org/account/10/ I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective ...

I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

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New microstubs S-category, separable coring and finally some substantial material at separable functor at last. The monograph by Caenapeel, Militaru and Zhu listed at separable functor studies Frobenius functors and separable functors in parallel; there are relations in a number of interesting situations. Frobenius functors are those where left and right adjoint are the same (hence in particular we have adjoint n-tuple for every $n$). Separable is a notion which is about certain spliting condition. This spliting is of the kind as spliting in Galois theory, I mean the Grothendieck’s version of classical Galois theory involves separable algebras at one side of Galois equivalence.