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An Introduction to Category Theory and Categorical Logic Wolfgang Jeltsch Category theory An Introduction to Category Theory basics Products, and Categorical Logic coproducts, and exponentials Categorical logic Functors and Wolfgang Jeltsch natural transformations Monoidal TT¨ categories and U K¨ uberneetika Instituut monoidal functors Monads and Teooriaseminar comonads April 19 and 26, 2012 References

An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and comonads Monads and comonads References References

An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal Monoidal categories and monoidal functors categories and monoidal functors Monads and comonads Monads and comonads References References

An Introduction to From set theory to universal algebra Category Theory and Categorical Logic ◮ classical set theory (for example, Zermelo–Fraenkel): Wolfgang Jeltsch ◮ sets Category theory ◮ functions from sets to sets basics ◮ composition of functions yields function Products, coproducts, and ◮ identity functions exist exponentials ◮ adding structure and preserving it: Categorical logic Functors and ◮ vector spaces natural ◮ linear maps from vector spaces to vector spaces transformations ◮ composition of linear maps yields linear map Monoidal categories and ◮ identity functions are linear maps monoidal functors ◮ generalization of this idea in universal algebra: Monads and comonads ◮ certain algebras with the same signature References ◮ homomorphisms from such algebras to other such algebras ◮ composition of homomorphisms yields homomorphism ◮ identity functions are homomorphisms

An Introduction to Beyond universal algebra Category Theory and Categorical Logic Wolfgang Jeltsch ◮ topology based on the Kuratowski axioms: ◮ topological space is a set X and a closure operator Category theory basics Products, cl : P ( X ) → P ( X ) coproducts, and exponentials that fulfills certain axioms Categorical logic ◮ continuous function from ( X , cl) to ( X ′ , cl ′ ) Functors and is a function f : X → X ′ with natural transformations Monoidal f (cl( A )) ⊆ cl ′ ( f ( A )) categories and monoidal functors ◮ does not fit into the universal algebra framework: Monads and comonads ◮ closure operator operates on sets References instead of single elements ◮ continuity axiom uses ⊆ instead of = ◮ will fit into the categorical framework

An Introduction to No elements anymore Category Theory and Categorical Logic Wolfgang Jeltsch Category theory basics ◮ revision control system darcs: Products, coproducts, and ◮ repository states exponentials ◮ patches that turn repository states into repository states Categorical logic ◮ composition of patches yields patch Functors and natural ◮ empty patches exist transformations ◮ repository states do not have elements Monoidal categories and ◮ will fit into the categorical framework nevertheless monoidal functors Monads and ◮ more about a categorical approach to darcs in [Swierstra] comonads References

An Introduction to Categories Category Theory and Categorical ◮ components of a category: Logic ◮ a class of objects Wolfgang Jeltsch ◮ class of morphisms, each having a unique domain Category theory and a unique codomain, which are objects basics ◮ composition of morphisms: Products, coproducts, and f : A → B g : B → C exponentials gf : A → C Categorical logic ◮ identity morphisms: Functors and natural transformations id A : A → A Monoidal ◮ axioms that have to hold: categories and monoidal functors ◮ composition is associative Monads and ◮ id is left and right unit comonads ◮ classes of objects and morphism are not necessarily sets: References allows categories of sets, vector spaces, etc. ◮ composition is partial: codomain and domain must match ◮ above constructions lead to categories Set , Vec , etc.

An Introduction to Duality Category Theory and Categorical Logic Wolfgang Jeltsch ◮ axioms still hold after doing the following: Category theory ◮ swapping domain and codomain of each morphism basics ◮ changing the argument order of composition Products, coproducts, and ◮ opposite category C op for every category C : exponentials ◮ objects of C op are the ones of C Categorical logic ◮ morphisms f : A → B of C op are the morphism Functors and natural f : B → A of C transformations ◮ compositions gf in C op are the compositions fg in C Monoidal ◮ identities in C op are the same as in C categories and monoidal functors ◮ consequences: Monads and comonads ◮ for every categorical notion N , there is a dual notion N op References such that something is an N op in C if it is an N in C op ◮ for every theorem, there is a dual theorem that refers to the dual notions

An Introduction to Products of categories Category Theory and Categorical Logic Wolfgang Jeltsch ◮ product category C × D for any two categories C and D : ◮ objects Category theory basics ( A , B ) Products, coproducts, and where A is an object of C , and B is an object of D exponentials ◮ morphisms Categorical logic ( f , g ) : ( A , B ) → ( A ′ , B ′ ) Functors and natural where f : A → A ′ and g : B → B ′ transformations ◮ compositions and identities defined componentwise: Monoidal categories and monoidal functors ( f ′ , g ′ )( f , g ) = ( f ′ f , g ′ g ) Monads and comonads id ( A , B ) = (id A , id B ) References ◮ neutral element is the category 1: ◮ exactly one object ◮ exactly one morphism (the identity of that object)

An Introduction to Categories and elements Category Theory and Categorical Logic Wolfgang Jeltsch Category theory basics Products, coproducts, and ◮ in general, no notion of element of an object exponentials Categorical logic ◮ however, elements can be recovered for specific kinds Functors and of categories natural transformations ◮ furthermore, some concepts that seem to require Monoidal categories and the notion of element actually do not monoidal functors Monads and comonads References

An Introduction to Injectivity Category Theory and Categorical Logic Definition (Injectivity) Wolfgang Jeltsch A function f : A → B is injective if and only if Category theory basics ∀ x 1 , x 2 ∈ A . f ( x 1 ) = f ( x 2 ) ⇒ x 1 = x 2 . Products, coproducts, and exponentials Theorem Categorical logic Functors and A function f : A → B is injective if and only if natural transformations ∀ C . ∀ g 1 , g 2 : C → A . fg 1 = fg 2 ⇒ g 1 = g 2 . Monoidal categories and monoidal functors Monads and ◮ above definition relies on the notion of element comonads ◮ theorem gives us another property for defining injectivity: References ◮ does not mention elements, but only sets and functions (point-free style) ◮ can therefore be generalized to arbitrary categories ◮ leads to the notion of monomorphism

An Introduction to Surjectivity Category Theory and Categorical Logic Definition (Surjectivity) Wolfgang Jeltsch A function f : A → B is surjective if and only if Category theory basics ∀ y ∈ B . ∃ x ∈ A . f ( x ) = y . Products, coproducts, and exponentials Categorical logic Theorem Functors and A function f : A → B is surjective if and only if natural transformations Monoidal ∀ C . ∀ g 1 , g 2 : B → C . g 1 f = g 2 f ⇒ g 1 = g 2 . categories and monoidal functors Monads and comonads ◮ theorem gives us point-free definition References ◮ generalization to arbitrary categories leads to the notion of epimorphism ◮ point-free style makes it clear that monomorphism and epimorphism (injectivity and surjectivity) are duals

An Introduction to Isomorphisms Category Theory and Categorical Logic Wolfgang Jeltsch Category theory basics Products, ◮ generalization of bijections coproducts, and exponentials ◮ morphism f : A → B is an isomorphism Categorical logic if there is an f − 1 : B → A such that Functors and natural ff − 1 = id B transformations f − 1 f = id A Monoidal categories and ◮ objects A and B are isomorphic ( A ∼ monoidal functors = B ) Monads and if there exists an isomorphism f : A → B comonads References

An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal Monoidal categories and monoidal functors categories and monoidal functors Monads and comonads Monads and comonads References References