Philosophizing Forgetful Functors: This Functor Forgets only Properties: Namely, the Property of Being Abelian + This Functor Forgets Both Structure (the generating set) and Properties (the property of being a free group).

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forgetful functor is a functor which is defined by ‘forgetting’ something. For example, the forgetful functor from Grp to Set forgets the group structure of a group, remembering only the underlying set.

In common parlance, the term ‘forgetful functor’ has no precise definition, being simply used whenever a functor is obviously defined by forgetting something. Many forgetful functors of this sort have left or right adjoints (and many are actually monadic or comonadic), leading to the paradigmatic adjunction “free ⊣ forgetful.”

On the other hand, from the perspective of stuff, structure, propertyevery functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties). From this perspective, the forgetful functor from GrpGrp to SetSet forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the free group functor) is also forgetful: if you identify SetSet with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.

There are many cases in which we want to say that one kind of mathematical object has more structure than another kind of mathematical object. For instance, a topological space has more structure than a set. A Lie group has more structure than a smooth manifold. A ring has more structure than a group. And so on. In each of these cases, there is a sense in which the first sort of object – say, a topological space – results by taking an instance of the second sort – say, a set – and adding something more – in this case, a topology. In other cases, we want to say that two different kinds of mathematical objects have the same amount of structure. For instance, given a Boolean algebra, one can construct a special kind of topological space, known as a Stone space, from which one can uniquely reconstruct the original Boolean algebra; and vice-versa.

These sorts of relationships between mathematical objects are naturally captured in the language of category theory, via the notion of a forgetful functor. For instance, there is a functor F : Top → Set from the category Top, whose objects are topological spaces and whose arrows are continuous maps, to the category Set, whose objects are sets and whose arrows are functions. This functor takes every topological space to its underlying set, and it takes every continuous function to its underlying function. We say this functor is forgetful because, intuitively speaking, it forgets something: namely the choice of topology on a given set.

The idea of a forgetful functor is made precise by a classification of functors due to Baez et al. (2004). This requires some machinery. A functor F : C → D is said to be full if for every pair of objects A, B of C, the map F : hom(A, B) → hom(F (A), F (B)) induced by F is surjective, where hom(A, B) is the collection of arrows from A to B. Likewise, F is faithful if this induced map is injective for every such pair of objects. Finally, a functor is essentially surjective if for every object X of D, there exists some object A of C such that F(A) is isomorphic to X.

If a functor is full, faithful, and essentially surjective, we will say that it forgets nothing. A functor F : C → D is full, faithful, and essentially surjective if and only if it is essentially invertible, i.e., there exists a functor G : D → C such that G ◦ F : C → C is naturally isomorphic to 1C, the identity functor on C, and F ◦ G : D → D is naturally isomorphic to 1D. (Note, then, that G is also essentially invertible, and thus G also forgets nothing.) This means that for each object A of C, there is an isomorphism ηA : G ◦ F (A) → A such that for any arrow f : A → B in C, ηB ◦ G ◦ F(f) = f ◦ ηA, and similarly for every object of D. When two categories are related by a functor that forgets nothing, we say the categories are equivalent and that the pair F, G realizes an equivalence of categories.

Conversely, any functor that fails to be full, faithful, and essentially surjective forgets something. But functors can forget in different ways. A functor F : C → D forgets structure if it is not full; properties if it is not essentially surjective; and stuff if it is not faithful. Of course, “structure”, “property”, and “stuff” are technical terms in this context. But they are intended to capture our intuitive ideas about what it means for one kind of object to have more structure (resp., properties, stuff) than another. We can see this by considering some examples.

For instance, the functor F : Top → Set described above is faithful and essentially surjective, but not full, because not every function is continuous. So this functor forgets only structure – which is just the verdict we expected. Likewise, there is a functor G : AbGrp → Grp from the category AbGrp whose objects are Abelian groups and whose arrows are group homomorphisms to the category Grp whose objects are (arbitrary) groups and whose arrows are group homomorphisms. This functor acts as the identity on the objects and arrows of AbGrp. It is full and faithful, but not essentially surjective because not every group is Abelian. So this functor forgets only properties: namely, the property of being Abelian. Finally, consider the unique functor H : Set → 1, where 1 is the category with one object and one arrow. This functor is full and essentially surjective, but it is not faithful, so it forgets only stuff – namely all of the elements of the sets, since we may think of 1 as the category whose only object is the empty set, which has exactly one automorphism.

In what follows, we will say that one sort of object has more structure (resp. properties, stuff) than another if there is a functor from the first category to the second that forgets structure (resp. properties, stuff). It is important to note, however, that comparisons of this sort must be relativized to a choice of functor. In many cases, there is an obvious functor to choose – i.e., a functor that naturally captures the standard of comparison in question. But there may be other ways of comparing mathematical objects that yield different verdicts.

For instance, there is a natural sense in which groups have more structure than sets, since any group may be thought of as a set of elements with some additional structure. This relationship is captured by a forgetful functor F : Grp → Set that takes groups to their underlying sets and group homomorphisms to their underlying functions. But any set also uniquely determines a group, known as the free group generated by that set; likewise, functions generate group homomorphisms between free groups. This relationship is captured by a different functor, G : Set → Grp, that takes every set to the free group generated by it and every function to the corresponding group homomorphism. This functor forgets both structure (the generating set) and properties (the property of being a free group). So there is a sense in which sets may be construed to have more structure than groups.

Not Just Any Lair of Filth….Investment Environment = Ratio of Ordinary Profits to Total Capital – Long-Term Interest Rate

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In the stock market the price changes are subject to the law of demand and supply, that the price rises when there is excess demand, and the price falls when there is excess supply. It seems natural to assume that the price raises if the number of the buyer exceeds the number of the seller because there may be excess demand, and the price falls if the number of seller exceeds the number of the seller because there may be excess supply. Thus a trader, who expects a certain exchange profit through trading, will predict every other traders’ behaviour, and will choose the same behaviour as the other traders’ behaviour as thoroughly as possible he could. The decision-making of traders will be also influenced by changes of the firm’s fundamental value, which can be derived from analysis of present conditions and future prospects of the firm, and the return on the alternative asset (e.g. bonds). For simplicity’s sake of an empirical analysis, lets use the ratio of ordinary profits to total capital that is a typical measure of investment, as a proxy for changes of the fundamental value, and the long-term interest rate as a proxy for changes of the return on the alternative asset.

An investment environment is defined as

investment environment = ratio of ordinary profits to total capital – long- term interest rate

When the investment environment increases (decreases) a trader may think that now is the time for him to buy (sell) the stock. Formally let us assume that the investment attitude of trader i is determined by minimisation of the following disagreement function ei(x),

ei(x) = -1/2 ∑j=1Naijxixj – bisxi —– (1)

where aij denotes the strength of trader j’s influence on trader i, and bi denotes the strength of the reaction of trader i upon the change of the investment environment s which may be interpreted as an external field, and x denotes the vector of investment attitude x = (x1, x2, ……xN). The optimisation problem that should be solved for every trader to achieve minimisation of their disagreement functions ei(x) at the same time is formalised by

min E(x) = -1/2 ∑i=1Nj=1Naijxixj – ∑i=1Nbisxi —– (2)

Now let us assume that trader’s decision making is subject to a probabilistic rule. The summation over all possible configurations of agents’ investment attitude x = (x1,…..,xN) is computationally explosive with size of the number of trader N. Therefore under the circumstance that a large number of traders participates into trading, a probabilistic setting may be one of best means to analyse the collective behaviour of the many interacting traders. Let us introduce a random variable xk =(xk1,xk2,……,xkN), k=1,2,…..,K. The state of the agents’ investment attitude xk occur with probability P(xk) = Prob(xk) with the requirement 0 < P(xk) < 1 and ∑k=1KP(xk) = 1. Defining the amount of uncertainty before the occurrence of the state xk with probability P(xk) as the logarithmic function: I(xk) = −logP(xk). Under these assumptions the above optimisation problem is formalised by

min <E(x)> = ∑k=1NP(xk) E(xk) —– (3)

subject to H = − ∑k=1NP(xk)logP(xk), ∑k=-NNP(xk) = 1

where E(xk) = 1/2 ∑i=1NEi(xk)

xk is a state, and H is information entropy. P(xk) is the relative frequency the occurrence of the state xk. The well-known solutions of the above optimisation problem is

P(xk) = 1/Z e(-μE(xk)), Z = ∑k=1Ke(-E(xk)) k = 1, 2, …., K —– (4)

where the parameter μ may be interested as a market temperature describing a degree of randomness in the behaviour of traders. The probability distribution P(xk) is called the Boltzmann distribution where P(xk) is the probability that the traders’ investment attitude is in the state k with the function E(xk), and Z is the partition function. We call the optimising behaviour of the traders with interaction among the other traders a relative expectation formation.

Phenomenological Model for Stock Portfolios. Note Quote.

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The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years. The non-Gaussian property of the probability distributions of price changes, in stock markets and foreign exchange markets, has been one of main problems in this field. From the analysis of the high-frequency time series of market indices, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution, while the tail deviate from it and shows another power law asymptotic behavior. In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distributionup to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. The scaling property on the sampling time interval of data is also well described by the crossover of the two distributions. Several stochastic models of the fluctuation dynamics of stock prices are proposed, which reproduce power law behavior of the probability density. The auto-correlation of financial time series is also an important problem for markets. There is no time correlation of price changes in daily scale, while from more detailed data analysis an exponential decay with a characteristic time τ = 4 minutes was found. The fact that there is no auto-correlation in daily scale is not equal to the independence of the time series in the scale. In fact there is auto-correlation of volatility (absolute value of price change) with a power law tail.

Portfolio is a set of stock issues. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are phenomenologically determined by empirical data. They are derived from the covariance of sequences of up and down spins using fluctuation-response theorem. We start with the Hamiltonian expression of our system that contain N stock issues. It is a function of the configuration S consisting of N coded price changes Si (i = 1, 2, …, N ) at equal trading time. The interaction coefficients are also dynamical variables, because the interactions between stocks are thought to change from time to time. We divide a coefficient into two parts, the constant part Jij, which will be phenomenologically determined later, and the dynamical part δJij. The Hamiltonian including the interaction with external fields hi (i = 1,2,…,N) is defined as

H [S, δ, J, h] = ∑<i,j>[δJij2/2Δij – (Jij + δJij)SiSj] – ∑ihiSi —– (1)

The summation is taken over all pairs of stock issues. This form of Hamiltonian is that of annealed spin glass. The fluctuations δJij are assumed to distribute according to Gaussian function. The main part of statistical physics is the evaluation of partition function that is given by the following functional in this case

Z[h] = ∑{si} ∫∏<i,j> dδJij/√(2πΔij) e-H [S, δ, J, h] —– (2)

The integration over the variables δJij is easily performed and gives

Z[h] = A {si} e-Heff[S, h] —– (3)

Here the effective Hamiltonian Heff[S,h] is defined as

Heff[S, h] = – <i,j>JijSiSj – ∑ihiSi —– (4)

and A = e(1/2 ∆ij) is just a normalization factor which is irrelevant to the following step. This form of Hamiltonian with constant Jij is that of quenched spin glass.

The constant interaction coefficients Jij are still undetermined. We use fluctuation-response theorem which relates the susceptibility χij with the covariance Cij between dynamical variables in order to determine those constants, which is given by the equation,

χij = ∂mi/∂hj |h=0 = Cij —– (5)

Thouless-Anderson-Palmer (TAP) equation for quenched spin glass is

mi =tanh(∑jJijmj + hi – ∑jJij2(1 – mj2)mi —– (6)

Equation (5) and the linear approximation of the equation (6) yield the equation

kik − Jik)Ckj = δij —– (7)

Interpreting Cij as the time average of empirical data over a observation time rather than ensemble average, the constant interaction coefficients Jij is phenomenologically determined by the equation (7).

The energy spectra of the system, simply the portfolio energy, is defined as the eigenvalues of the Hamiltonian Heff[S,0]. The probability density of the portfolio energy can be obtained in two ways. We can calculate the probability density from data by the equation

p(E) ΔE = p(E – ΔE/2 ≤ E ≤ E + ΔE/2) —– (8)

This is a fully consistent phenomenological model for stock portfolios, which is expressed by the effective Hamiltonian (4). This model will be also applicable to other financial markets that show collective time evolutions, e.g., foreign exchange market, options markets, inter-market interactions.

Category-less Category Theory. Note Quote.

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Definition:

Let us axiomatically define a theory which we shall call an objectless or object free category theory. In this theory, the only primitive concepts (besides the usual logical concepts and the equality concept) are:

(I) α is a morphism,
(II) the composition αβ is defined and is equal to γ, The following axioms are assumed:

1. Associativity of compositions: Let α, β, γ be morphisms. If the compositions βα and γβ exist, then

• the compositions γ(βα) and (γβ)α exist and are equal;
• if γ(βα) exists, then γβ exists, and if (γβ)α exists then βα exists.

2. Existence of identities: For every morphism α there exist morphisms ι and ι′, called identities, such that

• βι = β whenever βι is defined (and analogously for ι′),

• ιγ = γ whenever ιγ is defined (and analogously for ι′).

• αι and ι′α are defined.

Lemma:

Identities ι and ι′ of axiom (2) are uniquely determined by the morphism α.

Proof:

Let us prove the uniqueness for ι (for ι′ the proof goes analogously). Let ι1 and ι2 be identities, and αι1 and αι2 exist. Then αι1 = α and (αι12 = αι2. From axiom (1) it follows that ι1ι2 is determined. But ι1ι2 exists if an only if ι1 = ι2. Indeed, let us assume that ι1ι2 exist then ι1 = ι1ι2 = ι2. And vice versa, assume that ι1 = ι2. Then from axiom (2) it follows that there exists an identity ι such that ιι1 exists, and hence is equal to ι (because ι1 is an identity). This, in turn, means that (ιι12 exists, because (ιι12 = ιι2 = ιι1 = ι. Therefore, ι1ι2 exists by Axiom 1.

Let us denote by d(α) and c(α) identities that are uniquely determined by a morphism α, i.e. such that the compositions αd(α) and c(α)α exist (letters d and c come from “domain” and “codomain”, respectively).

Lemma 2.2 The composition βα exists if and only if c(α) = d(β), and consequently,

d(βα) = d(α) and c(βα) = c(β).

Proof. Let c(α) = d(β) = ι, then βι and ια exist. From axiom (1) it follows that there exists the composition (βι)α = βα. Let us now assume that βα exists, and let us put ι = c(α). Then ια exists which implies that βα = β(ια) = (βι)α. Since βι exists then d(β) = ι.

Definition:

If for any two identities ι1 and ι2 the class ⟨ι12⟩ = {α : d(α) = ι1, c(α) = ι2},

is a set then objectless category theory is called small.

Definition:

Let us choose a class C of morphisms of the objectless category theory (i.e. C is a model of the objectless category theory), and let C0 denote the class of all identities of C. If ι123 ∈ C0, we define the composition

mC0ι123 : ⟨ι1, ι2⟩ × ⟨ι2, ι3⟩ → ⟨ι1, ι3

by mC0 (α, β) = βα. Class C is called objectless category.

Proposition:

The objectless category definition is equivalent to the standard definition of category.

Proof:

To prove the theorem it is enough to reformulate the standard category definition in the following way. A category C consists of

(I) a collection C0 of objects,
(II) for each A,B ∈ C0,
a collection ⟨A,B⟩ C0 of morphisms from A to B,

(III) for each A,B,C ∈ C0, if α ∈ ⟨A,B⟩ C0 and β ∈ ⟨B,C⟩ C0, the composition

mC0 : ⟨A,B⟩ C0 × ⟨B,C⟩ C0 → ⟨A,C⟩ C0

is defined by mC0A,B,C (α, β). The following axioms are assumed

1. Associativity: If α ∈ ⟨A,B⟩C0, β ∈ ⟨B,C⟩C0 , γ ∈ ⟨C,D⟩C0 then γ(βα) = (γβ)α.

2. Identities: For every B ∈ C0 there exists a morphism ιB ∈ ⟨B,B⟩C0 such that

A∈C0α∈⟨A,B⟩C0 ιBα = α, ∀C∈C0β∈⟨B,C⟩C0 βιB = β.

To see the equivalence of the two definitions it is enough to suitably replace in the above definition objects by their corresponding identities.

This theorem creates three possibilities to look at the category theory: (1) the standard way, in terms of objects and morphisms, (2) the objectless way, in terms of morphisms only, (3) the hybrid way in which we take into account the existence of objects but, if necessary or useful, we regard them as identity morphisms.