|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'501'711
Articles rated: 2609
19 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Adjunction contexts and regular quasi-monads | | Robert Wisbauer
; | | Date: |
6 Jan 2011 | | Abstract: | Generalising the unit element in a ring, one may consider (central)
idempotents in the ring. Similarly, the unitality condition required for a
monad $(F,mu,eta)$ on any category was released (by G. B"ohm et al.) to
define {em pre-monads} by imposing weaker requirements on $eta$. Doing so,
the adjointness of the free functor from $A$ to the category of unital
$F$-modules $A_F$ and the forgetful functor is lost. In this paper we
establish, for a premonad $(F,mu,eta)$, a weakened form of adjointness
between the free functor from $A$ to the category $
A_F$ of {em regular
quasi-$F$-modules} with the forgetful functor.
For this we consider, for functors $L:A o B$ and $R:B o A$ between any
categories $A$ and $B$, an {em adjunction context} given by maps
$$xymatrix{Mor_B (L(A),B) ar@<0.5ex>[r]^alpha & Mor_A
(A,R(B))ar@<0.5ex>[l]^eta,}$$ natural in $Ain A$ and $Bin B$. We call
this a {em regular adjunction context} if both $alpha$ and $eta$ are
regular, that is
$alpha = alphacirc etacirc alpha$ and $eta = eta
circalphacirceta$.
From this configuration we derive the notion of a {em regular quasi-monad}
and a {em regular quasi-comonad} leading to {em pre-units} and {em
pre-monads} (as considered by G. B"{o}hm, J.N. Alonso ’Alvarez, and others).
The notions allow to study the lifting of functors between categories to the
corresponding categories of regular quasi-modules.
Along the way, the corresponding notions for {em quasi-comonads} are
formulated. The entwinings of regular quasi-monads and quasi-comonads
considered in the final section provide the techniques to handle {em weak
bialgebras} and {em weak Hopf algebras} on arbitrary categories but this
aspect is not exploited in the present paper. | | Source: | arXiv, 1101.1195 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER