Identity FunctorEdit

The identity functor is a foundational concept in category theory, appearing whenever one wants to talk about what it means for a process to do nothing. Given a category category C, the identity functor Id_C: C → C leaves every object and every morphism exactly where it is, acting as a perfect baseline against which all other endofunctors can be compared. In particular, Id_C maps each object X to itself and each morphism f: X → Y to itself, while respecting identities and composition: Id_C(id_X) = id_X and Id_C(g ∘ f) = Id_C(g) ∘ Id_C(f).

Definition - For a category category C, the identity functor Id_C is defined by: - On objects: Id_C(X) = X for all objects X in C. - On morphisms: Id_C(f: X → Y) = f for all morphisms f: X → Y. - It preserves identities and composition in the sense that Id_C(id_X) = id_X and Id_C(g ∘ f) = Id_C(g) ∘ Id_C(f) for all composable morphisms f and g. Cross-links: object (category theory), morphism (category theory), identity morphism, composition (category theory)

Basic properties - The identity functor is a strict functor and serves as the unit for the operation of composing endofunctors. If F: C → D is any functor, then F ∘ Id_C = F and Id_D ∘ F = F. This makes Id_C the neutral element for functor composition on either side. - In the 2-category Cat (the category of all categorys), Id_C is the identity 1-cell at C. It participates in all higher-dimensional structure as the canonical filler that does not alter the data of objects and arrows. - Id_C preserves every limit and every colimit that exists in C. Since the diagram and its maps are not changed by Id_C, the universal properties defining limits or colimits remain intact under the identity. - Id_C preserves isomorphisms: if f is an isomorphism in C, then Id_C(f) = f is an isomorphism in C. Consequently, the identity functor reflects isomorphisms in a straightforward way. - There is a natural perspective in which Id_C is self-adjoint: Id_C is both left and right adjoint to itself, with unit and counit given by the identity natural transformations. This reflects the symmetrical role Id_C plays with respect to morphisms and their inverses. Cross-links: functor, End(C) (as endofunctors), Cat, limit, colimit, isomorphism (category theory), adjunction

Examples - In the category Set, Id_Set simply maps each set to itself and each function to itself. It acts as the trivial, non-changing endofunctor on sets. - In the category of groups Grp, Id_Grp leaves every group and every group homomorphism unchanged. - In the category of topological spaces Top, Id_Top leaves spaces and continuous maps unchanged. - In a poset viewed as a category, Id_C acts as the identity on the underlying objects and order relations, reinforcing that the identity is compatible with the order-theoretic structure. - For a category with a single object whose endomorphisms form a monoid, Id_C picks out the identity endomorphism on that object, mirroring the general principle that the identity is the neutral element in the internal monoid of endomorphisms. Cross-links: Set, Grp, Top, Poset

Interaction with limits, colimits, and structure - Because the identity functor does not alter the data of diagrams, computing limits or colimits after applying Id_C yields the same result as before. In particular, for any diagram D in C, Id_C(lim D) ≅ lim(Id_C ∘ D) and Id_C(colim D) ≅ colim(Id_C ∘ D). - In monoidal or structured settings, Id_C plays the role of the unit object with respect to the composition of endofunctors. If one treats endofunctors on C as objects in a monoidal category under composition, Id_C is the unit, analogous to 1 in a multiplicative monoid. - The identity functor interacts trivially with natural transformations: if α: F ⇒ G is a natural transformation between functors F, G: C → D, then α ∘ Id_C = α and Id_D ∘ α = α in the obvious sense, reflecting the non-disturbing nature of Id_C with respect to other functors. Cross-links: limit (category theory), colimit (category theory), monoidal category, natural transformation

Self-adjunction and higher perspectives - Id_C is self-adjoint, as noted above: Id_C ⊣ Id_C with unit and counit the identity transformations. This encapsulates the idea that identity is its own best approximation to an inverse in the adjunction sense. - In higher-category theory and homotopy-type-theoretic contexts, one may study variants of identity on objects or on arrows, or identity up to equivalence. The core role of Id_C as a baseline endofunctor remains central, even as the surrounding notions become more flexible. - The identity functor is also the unit in the 2-category of endofunctors on C under functor composition. This places Id_C at the heart of how endofunctors organize themselves algebraically. Cross-links: adjunction, Cat, endofunctor, monoidal category

See also - category - functor - natural transformation - endofunctor - monoidal category - adjunction - Yoneda lemma