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Category Theory via C# (8) Advanced LINQ to Monads
Monad is a powerful structure, with the LINQ support in C# language, monad enables chaining operations to build fluent workflow, which can be pure. With these features, monad can be used to manage I/O, state changes, exception handling, shared environment, logging/tracing, and continuation, etc., in the functional paradigm.

Category Theory via C# (7) Monad and LINQ to Monads
As fore mentioned endofunctor category can be monoidal (the entire category. Actually, an endofunctor In the endofunctor category can be monoidal too. This kind of endofunctor is called monad. Monad is another important algebraic structure in category theory and LINQ. Formally, monad is an endofunctor equipped with 2 natural transformations:

Category Theory via C# (6) Monoidal Functor and Applicative Functor
Given monoidal categories (C, ⊗, I_{C}) and (D, ⊛, I_{D}), a strong lax monoidal functor is a functor F: C → D equipped with:

Category Theory via C# (5) Bifunctor
A functor is the mapping from 1 object to another object, with a “Select” ability to map 1 morphism to another morphism. A bifunctor (binary functor), as the name implies, is the mapping from 2 objects and from 2 morphisms. Giving category C, D and E, bifunctor F from category C, D to E is a structurepreserving morphism from C, D to E, denoted F: C × D → E:

Category Theory via C# (4) Natural Transformation
If F: C → D and G: C → D are both functors from categories C to category D, the mapping from F to G is called natural transformation and denoted α: F ⇒ G. α: F ⇒ G is actually family of morphisms from F to G, For each object X in category C, there is a specific morphism α_{X}: F(X) → G(X) in category D, called the component of α at X. For each morphism m: X → Y in category C and 2 functors F: C → D, G: C → D, there is a naturality square in D:

Category Theory via C# (3) Functor and LINQ to Functors
In category theory, functor is a mapping from category to category. Giving category C and D, functor F from category C to D is a structurepreserving morphism from C to D, denoted F: C → D:

Category Theory via C# (2) Monoid
Monoid is an important algebraic structure in category theory. A monoid M is a set M equipped with a binary operation ⊙ and a special element I, denoted 3tuple (M, ⊙, I), where

Category Theory via C# (1) Fundamentals
Category theory is a theoretical framework to describe abstract structures and relations in mathematics, first introduced by Samuel Eilenberg and Saunders Mac Lane in 1940s. It examines mathematical concepts and properties in an abstract way, by formalizing them as collections of items and their relations. Category theory is abstract, and called "general abstract nonsense" by Norman Steenrod; It is also general, therefore widely applied in many areas in mathematics, physics, and computer science, etc. For programming, category theory is the algebraic theory of types and functions, and also the rationale and foundation of LINQ and any functional programming. This chapter discusses category theory and its important concepts, including category, morphism, natural transform, monoid, functor, and monad, etc. These general abstract concepts will be demonstrated with intuitive diagrams and specific C# and LINQ examples. These knowledge also helps building a deep understanding of functional programming in C# or other languages, since any language with types and functions is a categorytheoretic structure.