Rough Objects in Monoidal Closed Categories

Abstract

This chapter will build upon previous achievements on monadic rough objects over the category Set, and show how rough object approximation and algebraic manipulation in general can be enriched by extending constructions to work similarly over monoidal closed categories embracing both algebraic as well as order structures. The chapter will also show how the rough information model in this monoidal closed category extension connects with other information models being relational in their basic original structures. Additionally, the chapter will discuss the potential of real world applications.

Appendix: Category Theory Notations and Constructions

1.1 Basic Concepts and Notations

In a category C with objects A and B, morphisms f from A to B are typically denoted by or . The (A-)identity morphism is denoted and morphism composition uses . The set of C-morphisms from A to B is written as $\operatorname {\mathrm {Hom}}_{{\mathtt {C}}}(A,B)$ or $\operatorname {\mathrm {Hom}}(A,B)$.

The category of sets, Set, is the most typical example of a category, and consists of sets as objects and functions (in ZFC) as morphisms together with the ordinary composition and identity. Other categories may be defined, for example, using Set as a basis: a structure, defined by the given metalanguage, is added on Set-objects, and then morphisms are defined as Set-morphisms preserving these structures. A typical example is to add uncertainty, modelled by a quantale $\ensuremath {\mathfrak {Q}}$, on Set-objects: The objects of the Goguen category ${\mathtt {Set}}(\ensuremath {\mathfrak {Q}})$ are pairs (X, α), where X is an object of Set and is a function (in ZFC). The morphisms are Set-morphisms satisfying . The composition of morphisms is defined as composition of Set-morphisms. Originally, Goguen considered a completely distributive lattice as the underlying lattice in [25] and further properties for Goguen categories can be found in [34].

A (covariant) functor between categories is a mapping that assigns each C-object A to a D-object F(A) and each C-morphism to a D-morphism , such that and $\ensuremath {\mathsf {F}}( \operatorname {\mathrm {id}}_A)= \operatorname {\mathrm {id}}_{\ensuremath {\mathsf {F}}(A)}$. Composition of functors is denoted and the identity functor is written . The (covariant) powerset functor is the typical example of a functor, and is defined by P A being the powerset of A, i.e., the set of subsets of A, and P f(X), for X ⊆ A, being the image of X under f, i.e., P f(X) = {f(x)∣x ∈ X}. A contravariant functor maps to each C-morphism a D-morphism , and for the contravariant powerset functor we have $\overline {\ensuremath {\mathsf {P}}}A=\ensuremath {\mathsf {P}}A$ and $\overline {\ensuremath {\mathsf {P}}}f(Y) =\lbrace x\in X\mid \exists y\in Y: f(x)=y\rbrace$.

A natural transformation between functors assigns to each C-object A a D-morphism such that , for any . The identity natural transformation is defined by $( \operatorname {\mathrm {id}}_{\ensuremath {\mathsf {F}}})_A= \operatorname {\mathrm {id}}_{\ensuremath {\mathsf {F}} A}$. If all τ _A are isomorphisms, τ is called a natural isomorphism, or natural equivalence. For functors F and natural transformations τ we often write F τ and τ F to mean (F τ)_A = F τ _A and (τ F)_A = τ _{F A}, respectively. It is easy to see that given by η _X(x) = {x}, and given by $\mu _X({\mathscr {B}})=\bigcup {\mathscr {B}}(=\bigcup _{B\in \mathscr {B}}B)$ are natural transformations. The (vertical) composition of natural transformations is defined by , for all D-objects A.

Whereas morphisms are typically seen as ‘mappings’ between objects in a category, functors are ‘mappings’ between categories, i.e., morphisms in (quasi-) categories of categories, and natural transformations are ‘mappings’ between functors, i.e., morphisms in functor categories. These notions clearly lead to views on hierarchies of sets, classes and conglomerates, where foundational issues enter the scene, and our approach roughly follows Grothendieck’s [5] view of set-theoretic foundations for category theory.

A monad (or triple, or algebraic theory) over a category C is written as F = (F, η, μ), where is a (covariant) functor, and and are natural transformations for which and hold. A Kleisli category C _F for a monad F over a category C is defined as follows: Objects in C _F are the same as in C, and the morphisms are defined as , that is morphisms $f\colon X \rightharpoondown Y$ in C _F are simply morphisms in C, with being the identity morphism on X. Composition of morphisms is defined as

The category Rel with sets as objects and binary relations as morphisms, is isomorphic with the Kleisli category of the powerset monad over Set. This invites to viewing Kleisli morphisms as a general notion for relations in the sense of intuitively being “substitutions”.

Powerset monads and their many-valued extensions are in close connection to fuzzification and are good candidates to represent situations with incomplete or imprecise information. The many-valued covariant powerset functor L for a completely distributive lattice $\ensuremath {\mathfrak {L}}=(L, \vee , \wedge )$ is obtained by L X = L ^X, i.e. the set of functions (or $\ensuremath {\mathfrak {L}}$-sets) , and following [25], for a morphism in Set, by defining L f(α)(y) =∨_f(x)=y α(x). Further, if we define by

$$\displaystyle \begin{aligned} \eta_X(x)(x^{\prime})=\begin{cases} \top & \text{if } x=x^{\prime} \\ \bot & \text{otherwise} \end{cases} \end{aligned}$$

and by

$$\displaystyle \begin{aligned} \mu_X({\mathscr{M}})(x)= \bigvee_{\alpha\in \ensuremath{\mathsf{L}}X} A(x)\wedge {\mathscr{M}}(\alpha) \end{aligned}$$

then L = (L, η, μ) is a monad.

1.2 Sorted Categories

In the one-sorted (and crisp) case for signatures we typically work in Set, but in the many-sorted (and crisp) case we need the “sorted category of sets” for the many-sorted term functor. We start this section by a more general view by considering “a sorted category of objects”.

Let S be an index set (in ZFC), the indices are called sorts (or types), and we do not assume any order on S. For a category C, we write C _S for the product category ∏_S C. The objects of C _S are tuples (X _s)_s∈S such that $X_{\mathtt {s}} \in\operatorname {\mathrm {Ob}}({\mathtt {C}})$ for all s ∈ S. We will also use X _S as a shorthand notation for these tuples. The morphisms between objects (X _s)_s∈S and (Y _s)_s∈S are tuples (f _s)_s∈S such that $f_{\mathtt {s}} \in\operatorname {\mathrm {Hom}}_{{\mathtt {C}}}(X_{\mathtt {s}}, Y_{{\mathtt {s}}})$ for all s ∈ S, and similarly we will use f _S as a shorthand notation. The composition of morphisms is defined sortwise (componentwise), i.e., .

Functors are lifted to functors F = (F _s)_s∈S from C _S to D _S. so that e.g. the regular powerset functor P _S = (P)_s∈S and the regular many-valued powerset functor L _S = (L)_s∈S, both are lifted to functors on Set _S.

Products and coproducts, $\prod$ and $\coprod$, are handled sortwise. We also have a “subobject relation”, thus, (X _s)_s∈S ⊆ (Y _s)_s∈S if and only if X _s ⊆ Y _s for all s ∈ S. It is clear that all limits and colimits exist in Set _S, because operations on Set _S-objects are defined sortwise for sets. Further, the product ∏_{i ∈ I} F _i and coproduct ∐_{i ∈ I} F _i of covariant functors F _i over Set _S are defined as

$$\displaystyle \begin{aligned} (\prod_{i\in I}\ensuremath{\mathsf{F}}_i) (X_{{\mathtt{s}}})_{{{\mathtt{s}}}\in S} = \prod_{i\in I} \ensuremath{\mathsf{F}}_i (X_{{\mathtt{s}}})_{{{\mathtt{s}}}\in S} \end{aligned}$$

and

$$\displaystyle \begin{aligned} (\coprod_{i\in I}\ensuremath{\mathsf{F}}_i) (X_{{\mathtt{s}}})_{{{\mathtt{s}}}\in S} = \coprod_{i\in I} \ensuremath{\mathsf{F}}_i (X_{{\mathtt{s}}})_{{{\mathtt{s}}}\in S} \end{aligned}$$

with morphisms being handled accordingly.

The category ${\mathtt {Set}}(\ensuremath {\mathfrak {Q}})_S$ is called the many-sorted Goguen category. Objects in this category are tuples of pairs ((X _s, α _s))_s∈S as objects, where for each s ∈ S, is a function (in ZFC). So, fixing s ∈ S we consider pairs (X _s, α _s) as objects in ${\mathtt {Set}}(\ensuremath {\mathfrak {Q}})$. Now, the ${\mathtt {Set}}(\ensuremath {\mathfrak {Q}})$-morphisms form morphisms .

1.3 Term Constructions

Here we recall the term functor construction and for clarity we present it in the one sorted situation, using the construction presented in [18]. The many sorted extension is found in [19].

Let $\varOmega =\bigcup _{n=0}^\infty \varOmega _n$ be an operator domain, where Ω _n contains n-ary operators. The term functor T _Ω: Set →Set is given as ${ \ensuremath {\mathsf {T}}}_\varOmega (X)=\bigcup _{k=0}^\infty { T}^k_\varOmega (X)$, where

$$\displaystyle \begin{aligned} \begin{array}{rcl} \ensuremath{\mathsf{T}}^0_\varOmega (X) &\displaystyle = &\displaystyle X, \\ \ensuremath{\mathsf{T}}^{k+1}_\varOmega (X) &\displaystyle = &\displaystyle \{ (n, \omega, (m_i)_{i\le n}) \mid \omega \in\varOmega_n, n\in N, m_i\in \ensuremath{\mathsf{T}}^k_\varOmega (X) \}. \end{array} \end{aligned} $$

In this context it is more convenient to write terms as (n, ω, (x _i)_i≤n) instead of the more common ω(x ₁, …, x _n). It is clear that (T _Ω X, (σ _ω)_{ω ∈ Ω}) is an Ω-algebra, if σ _ω((m _i)_i≤n) = (n, ω, (m _i)_i≤n) for ω ∈ Ω _n and m _i ∈T _Ω X. Morphisms $X\stackrel {f}{\rightarrow } Y$ in S et are extended in the usual way to the corresponding Ω-homomorphisms $(\ensuremath {\mathsf {T}}_{\varOmega }X, (\sigma _{\omega })_{ \omega \in \varOmega }) \stackrel {\ensuremath {\mathsf {T}}_{\varOmega }f}{\longrightarrow } (\ensuremath {\mathsf {T}}_{\varOmega }Y, (\tau _{\omega })_{\omega \in \varOmega })$, where T _Ω f is given as the Ω-extension of $X\stackrel {f}{\rightarrow } Y \hookrightarrow T_\varOmega Y$ associated to (T _Ω Y, (τ _nω)_{(n,ω) ∈ Ω}). To obtain the term monad, define $\eta _X^{\ensuremath {\mathsf {T}}_{\varOmega }}(x)=x$, and let $\mu _X^{\ensuremath {\mathsf {T}}_{\varOmega }}=id^{\star }_{T_{\varOmega }X}$ be the Ω-extension of $id_{T_{\varOmega }X}$ with respect to (T _Ω X, (σ _nω)_{(n,ω) ∈ Ω}). This gives us the (one-sorted) term monad $\ensuremath {\mathbf {T}}_{\varOmega }= (\ensuremath {\mathsf {T}}_{\varOmega },\eta ^{T_{\varOmega }},\mu ^{T_{\varOmega }})$.