text stringlengths 128 2.05k |
|---|
Finally a word concerning terminology: by the phrase [MATH] preserves colimits (of some type) in the monoidal category [MATH] we mean that, for each [MATH] in [MATH] , the functors [MATH] and [MATH] preserve these colimits. |
Coequalizers in [MATH] Somewhat surprisingly, coequalizers in [MATH] can be constructed the same way as in the special case of the category of unital rings, that is, as in [MATH] where [MATH] denotes the monoidal category of abelian groups. We therefore recall this simple construction as follows: If |
[EQUATION] is a coequalizer diagram in [MATH] , then [MATH] , where [MATH] is the two-sided ideal [MATH] in [MATH] In other words, [MATH] |
Denoting by [MATH] the multiplication of [MATH] and, for any [MATH] -morphism [MATH] , by [MATH] the [MATH] -morphism [EQUATION] |
one obtains [MATH] such that [EQUATION] is a coequalizer diagram in [MATH] Following we will show that this essentially can be generalized to an arbitrary monoidal category [MATH] Starting with a [MATH] -monoid |
[MATH] and a [MATH] -morphism [MATH] we denote, using the same notation as above, by [MATH] the [MATH] -morphism [EQUATION] Facts The following facts are easy to verify: |
1. For every monoid morphism [MATH] the following hold (up to identification of [MATH] and [MATH] ): [EQUATION] The following is then are immediate consequences. |
(a) If [MATH] is another [MATH] -morphism then [EQUATION] where the implication [MATH] even holds, if [MATH] only is a [MATH] -morphism. |
(b) For every monoid morphism [MATH] one has [EQUATION] 1 lemma Let [MATH] be a monoidal category with coequalizers preserved by [MATH] and let [MATH] be a monoid in [MATH] If [MATH] are [MATH] -morphisms and |
[EQUATION] is a coequalizer diagram in [MATH] , then [MATH] carries a unique monoid structure such that [MATH] is a monoid morphism. |
Proof. Observe first that the following diagrams commute. [EQUATION] Since by assumption [MATH] is a coequalizer of [MATH] and [MATH] , one obtains a unique [MATH] -morphism [MATH] satisfying |
[EQUATION] By commutativity of Diagrams ( ) and Equation ( ) the following diagram commutates. [EQUATION] Since [MATH] is a (regular) epimorphism and [MATH] is the coequalizer of [MATH] and [MATH] by assumption, there exists a unique [MATH] -morphism [MATH] satisfying [MATH] |
Using the facts that [MATH] is a monoid and [MATH] is a regular epimorphism (hence [MATH] an epimorphism for each [MATH] ) one shows easily that [MATH] is a monoid and and [MATH] a monoid morphism. |
2 Corollary If [MATH] and [MATH] are [MATH] -morphisms such that, for every monoid morphism [MATH] [EQUATION] Then the coequalizers of [MATH] and [MATH] coincide. |
Proof. Let [MATH] and [MATH] be coequalizers of [MATH] and [MATH] , respectively. Since [MATH] and [MATH] are monoid morphisms by Lemma , one obtains by the equivalences ( ) and ( ) the equivalences |
[EQUATION] which implies the claim. 3 Remark Equivalence ( ) holds in particular, if [MATH] and [MATH] are the homomorphic extensions [MATH] of [MATH] and [MATH] |
4 Theorem Let [MATH] be a monoidal category with coequalizers preserved by [MATH] . Then the category [MATH] has coequalizers and the forgetful functor [MATH] preserves regular epimorphisms. |
In more detail: If [MATH] are monoid morphisms and [EQUATION] is a coequalizer diagram in [MATH] then, with the monoid structure of Lemma on [MATH] , then |
[EQUATION] is a coequalizer diagram in [MATH] Proof. Let [MATH] be monoid morphisms and [MATH] the coequalizer of [MATH] and [MATH] in [MATH] . By Lemma |
[MATH] carries a unique monoid structure such that [MATH] is a monoid morphism. If [MATH] is a monoid morphism such that [MATH] , then [MATH] by the equivalence ( ), such that there exists a unique [MATH] -morphism [MATH] with [MATH] . It remains to prove that [MATH] is a monoid morphism. But this is clear since [MATH]... |
5 Remark Given two parallel pairs of morphisms in a category with coequalizers [EQUATION] one obtains a multiple coequalizer of the morphisms [MATH] as follows: Form the coequalizer [MATH] of [MATH] and then the coequalizer [MATH] of [MATH] ; then [MATH] is the required multiple coequalizer. |
In particular, every category with coequalizers has such multiple coequalizers and any such is a composite of ordinary ones. Applications |
Monadicity Applying the result above we first provide two similar monadicity criteria for the forgetful functor [MATH] , provided that this has a left adjoint. |
6 Proposition Let [MATH] be a monoidal category with regular factorizations and denumerable coproducts and assume that these as well as regular epimorphisms are preserved by [MATH] . Then the forgetful functor [MATH] is regularly monadic. |
Proof. By the standard construction of free monoids [MATH] has a left adjoint. Monadicity follows by the Beck-Paré-Theorem (the argument given in |
for the case of semigroups applies by replacing [MATH] by [MATH] ; it only requires that [MATH] preserves regular epimorphisms ). |
The respective monad [MATH] acts on a morphism [MATH] by [MATH] , hence, maps regular epimorphisms to regular epimorphisms by assumption. |
7 Proposition Let [MATH] be a monoidal category with coequalizers and assume that these are preserved by [MATH] and that [MATH] has a left adjoint. Then the functor [MATH] is monadic (and preserves regular epimorphisms). |
Proof. Monadicity follows as above. [MATH] preserves regular epimorphisms by Theorem 8 Remarks Note the differences between these results: |
1. Proposition requires the free monoid functor to be given in the standard way, while Proposition works for an arbitrary left adjoint of [MATH] (see e.g. |
for important examples). 2. Proposition requires [MATH] to have regular factorizations, while Proposition only requires the existence of coequalizers in [MATH] |
3. Proposition requires [MATH] to preserve coequalizers, while Proposition only requires preservation of regular epimorphisms. Moreover, the assumption on coequalizers in Proposition can be restricted to reflexive coequalizers if [MATH] has binary coproducts which are preserved by [MATH] . In fact, such a category has ... |
General Colimits in [MATH] Not much seems to be known about the existence of colimits in [MATH] . The only results we are aware of are |
If [MATH] is cocomplete and [MATH] preserves colimits, then the category [MATH] has all pushouts of the form [EQUATION] where [MATH] is a morphism in [MATH] |
and [MATH] is the free monoid functor (see ). If [MATH] is locally [MATH] -presentable and [MATH] preserves [MATH] -directed colimits, then [MATH] is locally presentable and, hence, cocomplete in particular (see |
). Using the monadicity criteria above we obtain the following results and so generalize the result of substantially. 9 Proposition |
In the situation of Proposition [MATH] has coequalizers and all other colimits which exist in [MATH] . In particular, if [MATH] is cocomplete then so is [MATH] |
Proof. Since every category with regular factorizations has coequalizers (see , 20.33] ) the result follows immediately. 10 Proposition |
In the situation of Proposition the category [MATH] has coequalizers. Moreover, if [MATH] is cocomplete then so is [MATH] Proof. |
The first result follows from Theorem and implies the second by , 4.3.4] Lifting adjunctions Let [MATH] be a monoidal functor. It is well known that [MATH] induces a functor [MATH] such that the diagram |
[EQUATION] commutes, where the vertical arrow denote the respective forgetful functors (denoted by [MATH] if necessary). It is of quite some interest (see e.g. |
) to know under which conditions the functor [MATH] has a left adjoint if [MATH] has one. A standard approach to this problem would be to apply Dubuc’s Adjoint Triangle Theorem, which would require both forgetful functors to have left adjoints and [MATH] to have coequalizers of reflexive pairs. Tambara |
claimed without a proof that it suffices to assume that [MATH] is cocomplete and that [MATH] preserves all colimits. A proof of this claim is contained in |
The following is a generalization of this result in that we do not assume the free monoids over [MATH] to be given by MacLane’s standard construction. |
11 Theorem Let [MATH] be a monoidal functor with left adjoint [MATH] . Assume that [MATH] has coequalizers which are preserved by [MATH] and that the forgetful functor [MATH] has left adjoint [MATH] Then the functor [MATH] has a left adjoint. |
Proof. We use the following notations. 1. Unit and counit of the adjunction [MATH] are denoted by [MATH] and [MATH] , respectively. For every [MATH] in [MATH] |
[MATH] and [MATH] are the multiplication and unit, respectively, of the free monoid [MATH] 2. [MATH] denotes the multiplication and [MATH] the unit of the monoidal structure of [MATH] [MATH] and [MATH] denote the opmonoidal structure of [MATH] [MATH] and [MATH] |
denote the unit and counit, respectively, of the adjunction [MATH] 3. Unit and counit of the adjunction [MATH] are denoted by [MATH] and [MATH] , respectively. In particular, for any [MATH] -object [MATH] |
[EQUATION] Following the analysis of adjoint triangles in one should try to obtain the left adjoint of [MATH] as follows: 1. Find in [MATH] , for each [MATH] -monoid [MATH] , a suitable morphism |
[EQUATION] 2. Find in [MATH] a morphism [MATH] with [EQUATION] [MATH] so defined has the potential of being [MATH] -universal for [MATH] , hence the family [MATH] to be the unit of the desired adjunction. |
Now [MATH] has to be an epimorphism since the forgetful functor [MATH] is faithful (see ). A natural choice of [MATH] , thus, would be to consider a (multiple) coequalizer of [MATH] -morphisms which in some way reflect the monoid structure of [MATH] and the monoidal structure of [MATH] (equivalently, the opmonoidal str... |
). 1. [MATH] [MATH] 2. [MATH] [MATH] and consider the homomorphic extensions of these maps, that is, the monoid morphisms 3. [MATH] with [MATH] and [MATH] |
4. [MATH] with [MATH] and [MATH] Now let [MATH] be the multiple coequalizer (which exists by assumption — see Remark of the morphisms [MATH] [MATH] [MATH] [MATH] |
in [MATH] According to step 2. we check that the following [MATH] -morphism is a morphism of [MATH] -monoids [MATH] [EQUATION] Compatibility with the multiplications is equivalent to commutativity of the outer frame of the following diagram, which is clear by standard arguments for monoidal functors and the fact that [... |
Preservation of units follows by a similar argument. [EQUATION] It remains to prove that [MATH] is [MATH] -universal for [MATH] . Here we again use elements of the proof given in |
First define, for a [MATH] -monoid [MATH] , a morphism [MATH] in [MATH] . This will in fact be the counit of the desired adjunction. By the easily checked identities |
[EQUATION] one obtains, using the equivalence ( ) and the universal property of [MATH] , a unique monoid morphism [MATH] making the following diagram commute |
[EQUATION] Next one checks, for any morphism [MATH] in [MATH] , the identities [EQUATION] These imply [EQUATION] Hence, [MATH] coequalizes simultaneously the [MATH] -morphisms [MATH] and, thus, by Remark the monoid morphisms [MATH] . Consequently there exist a unique monoid morphism [MATH] making the following diagram ... |
[EQUATION] Then, for every morphism [MATH] in [MATH] , the following diagram obviously commutes (in [MATH] ) and illustrates the required one-to-one correspondence between morphisms [MATH] in [MATH] and morphisms [MATH] in [MATH] |
[EQUATION] 12 Remarks To apply this theorem one typically will use Theorem . In this case, if assuming in addition existence of (countable) coproducts in [MATH] preserved by [MATH] , Theorem 11 specializes to Tambara’s claim. |
# Source: arxiv 1807.00426 # Title: Stationary states of the cubic conformal flow on $\mathbb{S}^3$ # Sections: all # Downloaded: 2026-03-02T08:56:15.015628+00:00 |
Stationary states of the cubic conformal flow on [MATH] Abstract. We consider the resonant system of amplitude equations for the conformally invariant cubic wave equation on the three-sphere. Using the local bifurcation theory, we characterize all stationary states that bifurcate from the first two eigenmodes. Thanks t... |
This research was supported by the Polish National Science Centre grant no. 2017/26/A/ST2/00530. 1. Introduction The conformally invariant cubic wave equation on [MATH] (the unit three-dimensional sphere) is a toy model for studying the dynamics of resonant interactions between nonlinear waves on a compact manifold. Th... |
. In terms of complex amplitudes [MATH] , this system takes the form [EQUATION] where [MATH] are the interaction coefficients. The cubic conformal flow ( 1.1 ) is the Hamiltonian system with the symplectic form |
[MATH] and the conserved energy function [EQUATION] The attention of has been focused on understanding the patterns of energy transfer between the modes. In particular, a three-dimensional invariant manifold was found on which the dynamics is Liouville-integrable with exactly periodic energy flows. Of special interest ... |
, however a complete classification of stationary states was deemed as an open problem. The purpose of this paper is to study existence and stability of all stationary states that bifurcate from the first two eigenmodes of the cubic conformal flow. Thanks to the Hamiltonian formulation, we are able to give the variatio... |
The constrained maximizer of energy can be normalized to the form [EQUATION] where [MATH] is a parameter. This solution was labeled as the ground state in our previous work |
where we proved orbital stability of the ground state in spite of its degeneracy with respect to parameter [MATH] Two constrained minimizers of energy are given by the exact solutions: |
[EQUATION] where [MATH] and [MATH] are parameters, whereas [MATH] are expressed by [EQUATION] The cutoff in the interval [MATH] for [MATH] ensures that [MATH] Since the constrained minimizers of energy are nondegenerate with respect to [MATH] their orbital stability follows from the general stability theory |
By using the local bifurcation methods of this paper, we are able to prove that the stationary states ( 1.3 ) and ( 1.4 ) are respectively maximizer and two minimizers of energy constrained by two other conserved quantities in the limit of small [MATH] |
For the ground state ( 1.3 ), we know from our previous work that it remains a global constrained maximizer of energy for any [MATH] For the stationary states ( 1.4 ) and ( 1.5 ), we have checked numerically that they remain local constrained minimizers of energy for any [MATH] and [MATH] , however, we do not know if a... |
In addition, we have proven existence of another constrained minimizer of energy bifurcating from the second eigenmode The new minimizer is nondegenerate with respect to [MATH] , hence again its orbital stability follows from the general stability theory in |
However, we show numerically that this stationary state remains a local constrained minimizer only near the bifurcation point and becomes a saddle point of energy far from the bifurcation point. |
Bifurcation analysis of this paper for the first two eigenmodes suggests existence of other constrained minimizers of energy bifurcating from other eigenmodes. This poses an open problem of characterizing a global constrained minimizer of energy for the cubic conformal flow ( 1.1 ). Another open problem is to understan... |
The cubic conformal flow ( 1.1 ) shares many properties with a cubic resonant system for the Gross-Pitaevskii equation in two dimensions |
. In particular, the stationary states for the lowest Landau level invariant subspace of the cubic resonant system have been thoroughly studied in |
by using the bifurcation theory from a simple eigenvalue In comparison with Section 6.4 in , where certain symmetries were imposed to reduce multiplicity of eigenvalues, we develop normal form theory for bifurcations of all distinct families of stationary states from the double eigenvalue without imposing any a priori ... |
Another case of a completely integrable resonant system with a wealth of stationary states is the cubic Szegő equation . Classification of stationary states and their stability has been performed for the cubic and quadratic Szegő equations in |
and respectively. One more example of a complete classification of all travelling waves of finite energy for a non-integrable case of the energy-critical half-wave map onto [MATH] is given in |
, where the spectrum of linearization at the travelling waves is studied by using Jacobi operators and conformal transformations. The half-wave map was found to be another integrable system with the Lax pair formulation |
It is unclear in the present time if the cubic conformal flow on [MATH] is also an integrable system with the Lax pair formulation. |
Organization of the paper. Symmetry, conserved quantities, and some particular stationary states for the cubic conformal flow ( 1.1 are reviewed in Section based on the previous works |
Local bifurcation results including the normal form computations for the lowest eigenmode are contained in Section 3. Variational characterization of the bifurcating families from the lowest eigenmode including the proof of extremal properties for the stationary states ( 1.3 ) and ( 1.4 is given in Section 4. Similar b... |
are reported in Section 7. Notations. We denote the set of nonnegative integers by [MATH] and the set of positive integers by [MATH] A sequence [MATH] is denoted for short by [MATH] The space of square-summable sequences on [MATH] is denoted by [MATH] It is equipped with the inner product [MATH] and the induced norm |
[MATH] . The weighted space [MATH] denotes the space of squared integrable sequences with the weight [MATH] . We write [MATH] to state [MATH] for some universal (i.e., independent of other parameters) constant [MATH] Terms of the Taylor series in [MATH] |
of the order [MATH] are denoted by [MATH] 2. Preliminaries Here we recall from some relevant properties of the cubic conformal flow ( 1.1 ) and its stationary states. The cubic conformal flow ( 1.1 ) enjoys the following three one-parameter groups of symmetries: |
[EQUATION] where [MATH] [MATH] , and [MATH] are real parameters. By the Noether theorem, the latter two symmetries give rise to two conserved quantities: |
[EQUATION] It is proven in Theorem 1.2 of that [MATH] and the equality is achieved if and only if [MATH] for some [MATH] with [MATH] |
As is shown in Appendix A of (see also for generalizations), there exists another conserved quantity of the conformal flow ( 1.1 ) in the form |
[EQUATION] This quantity is related to another one-parameter group of symmetries: [EQUATION] where [MATH] is arbitrary and [MATH] is a difference operator given by |
[EQUATION] Note that the difference operator [MATH] is obtained from [MATH] acting on any function [MATH] on phase space, where the Poisson bracket is defined by |
[EQUATION] thanks to the symplectic structure [MATH] of the conformal flow ( 1.1 ). The factor [MATH] in the definition of [MATH] is chosen for convenience. There exists another one-parameter group of symmetries obtained from [MATH] which is not going to be used in this paper. |
Stationary states of the cubic conformal flow ( 1.1 ) are obtained by the separation of variables [EQUATION] where the complex amplitudes [MATH] are time-independent, while parameters [MATH] and [MATH] are real. Substituting ( 2.9 ) into ( 1.1 ), we get a nonlinear system of algebraic equations for the amplitudes: |
[EQUATION] Some particular solutions [MATH] of the stationary system ( 2.10 ) are reviewed below. 2.1. Single-mode states The simplest solutions of ( 2.10 ) are the single-mode states given by |
[EQUATION] with [MATH] , where [MATH] is an amplitude and [MATH] is fixed. Thanks to the transformations ( 2.1 ) and ( 2.2 ), we can set [MATH] so that [MATH] |
2.2. Invariant subspace of stationary states As is shown in , system ( 2.10 ) can be reduced to three nonlinear equations with the substitution |
[EQUATION] where [MATH] are complex parameters satisfying the nonlinear algebraic system [EQUATION] and we have introduced [MATH] so that [MATH] It is clear from the decay of the sequence ( 2.12 ) as [MATH] that [MATH] must be restricted to the unit disk: [MATH] |
The analysis of solutions to system ( 2.13 )–( 2.15 simplifies if we assume that the parameters [MATH] [MATH] [MATH] are real-valued, which can be done without loss of generality. To see this, note that [MATH] can be made real-valued by the transformation ( 2.3 ). If [MATH] then equation ( 2.15 ) implies [MATH] and hen... |
There exist exactly four families of solutions to the system ( 2.13 )–( 2.15 ). 2.2.1. Ground state Equation ( 2.14 ) is satisfied if [MATH] . This implies that [MATH] |
from equation ( 2.13 ) and [MATH] from equation ( 2.15 with [MATH] and [MATH] . Parameterizing [MATH] and bringing all together yield the geometric sequence |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.