Isomorphism of categories

In category theory, an isomorphism of categories is a functor F: \mathcal{C} \to \mathcal{D} between two categories \mathcal{C} and \mathcal{D} that induces a bijection on the collections of objects and a bijection on the hom-sets for every pair of objects, making F fully faithful and bijective on objects, with an inverse functor G: \mathcal{D} \to \mathcal{C} such that the compositions G \circ F = \mathrm{id}_{\mathcal{C}} and F \circ G = \mathrm{id}_{\mathcal{D}} are the respective identity functors.^[1]^[2] This strict invertibility ensures that \mathcal{C} and \mathcal{D} are indistinguishable in structure, preserving compositions, identities, and all categorical relations exactly.^[1] Unlike the weaker notion of an equivalence of categories, which requires only that F be fully faithful and essentially surjective on objects (meaning every object in \mathcal{D} is naturally isomorphic to the image of some object in \mathcal{C}), an isomorphism demands precise bijections without allowance for natural isomorphisms between functors.^[2]^[3] Equivalences capture "sameness up to isomorphism," which is often more useful in practice for comparing categories like the category of sets and the category of complete atomic Boolean algebras, but isomorphisms are rarer and more rigid, typically occurring in contrived or specially constructed settings.^[1]^[2] For instance, the categories of abelian groups (\mathbf{Ab}) and \mathbb{Z}-modules (\mathbb{Z}\mathbf{-Mod}) are isomorphic via the functor that equips each abelian group with its canonical \mathbb{Z}-module structure (with the forgetful functor as inverse, though the structures are identical).^[3] Isomorphisms of categories play a foundational role in formalizing when two categories are identical, facilitating proofs by transporting properties across them and underpinning advanced constructions like the category of categories \mathbf{[CAT](/page/Cat)}, where objects are all categories and morphisms include isomorphisms.^[1]^[2] They preserve and reflect all categorical features, such as limits, colimits, adjunctions, and monoidal structures, ensuring that any diagram or universal property in one category corresponds exactly to its counterpart in the other.^[1] However, their strictness limits their prevalence; most "equivalent" categories in mathematics, such as finite-dimensional vector spaces over a field and matrices, are not strictly isomorphic due to non-bijective choices of objects.^[2] This distinction highlights category theory's emphasis on abstraction, where isomorphisms provide the strongest form of structural identity.^[1]

Definition

Formal Definition

In category theory, a category consists of a collection of objects, a collection of morphisms between those objects, a composition operation for compatible morphisms, and identity morphisms for each object, all satisfying certain axioms such as associativity of composition and the identity laws.^[1] A functor between two categories C and D is a structure-preserving map that assigns objects in C to objects in D and morphisms in C to morphisms in D, while preserving composition and identities.^[1] A functor F: C → D is an isomorphism of categories if there exists a functor G: D → C such that the compositions F ∘ G and G ∘ F are equal to the identity functors on D and C, respectively; that is,

F \circ G = \mathrm{id}_D, \quad G \circ F = \mathrm{id}_C.

^[1] This equality must hold strictly, meaning the functors are identical on the nose, without the intervention of natural isomorphisms, which underscores the rigid invertibility required for such an isomorphism.^[1] The concept of isomorphism of categories originates from the foundational work in category theory developed in the 1940s by Samuel Eilenberg and Saunders Mac Lane, extending the notion of isomorphisms from algebraic structures like groups and rings to the broader framework of categories.^[4]

Role of the Inverse Functor

In the context of category theory, the inverse functor plays a pivotal role in establishing an isomorphism between categories \mathcal{C} and \mathcal{D}. Given a functor F: \mathcal{C} \to \mathcal{D} that is an isomorphism, there exists a functor G: \mathcal{D} \to \mathcal{C} such that the compositions satisfy F \circ G = \mathrm{id}_{\mathcal{D}} and G \circ F = \mathrm{id}_{\mathcal{C}}, where \mathrm{id}_{\mathcal{D}} and \mathrm{id}_{\mathcal{C}} are the respective identity functors.^[1] This construction ensures that G reverses the action of F precisely, mapping objects and morphisms back in a way that restores the original structure without deviation.^[1] The functoriality of the inverse G is essential, as it must preserve the composition of morphisms and the identities in \mathcal{D} exactly as they correspond to those in \mathcal{C}. Specifically, for any morphisms f: d_1 \to d_2 and g: d_2 \to d_3 in \mathcal{D}, G(f \circ g) = G(f) \circ G(g), and for each object d in \mathcal{D}, G(\mathrm{id}_d) = \mathrm{id}_{G(d)}.^[1] This strict preservation distinguishes the inverse from weaker notions, enforcing that G acts as a faithful reversal on the entire categorical framework.^[5] If such an inverse G exists for F, it is unique, as it can be explicitly derived by inverting the bijective actions of F on objects and morphisms.^[5] For instance, on objects, G assigns to each F(c) the unique preimage c, and on morphisms, it applies the inverse bijection induced by F. This uniqueness follows from the requirement that G satisfies the identity composition conditions, leaving no room for alternative constructions.^[5] The strict invertibility via G implies that \mathcal{C} and \mathcal{D} are essentially the same category, differing only by a relabeling of objects and morphisms that preserves all internal structure verbatim.^[1] This equivalence in structure underscores the isomorphism as a rigid notion of sameness, where every categorical feature—such as compositions and identities—is mirrored exactly through the inverse.^[1]

Properties

Bijection on Objects and Morphisms

An isomorphism of categories F: \mathcal{C} \to \mathcal{D} is equipped with an inverse functor G: \mathcal{D} \to \mathcal{C} such that F \circ G = \mathrm{Id}_{\mathcal{D}} and G \circ F = \mathrm{Id}_{\mathcal{C}}.^[6] This inverse ensures that F induces a strict bijection between the class of objects of \mathcal{C}, denoted \mathrm{Ob}(\mathcal{C}), and \mathrm{Ob}(\mathcal{D}). For injectivity, suppose F(c_1) = F(c_2) for objects c_1, c_2 \in \mathrm{Ob}(\mathcal{C}); then c_1 = G(F(c_1)) = G(F(c_2)) = c_2. For surjectivity, given any d \in \mathrm{Ob}(\mathcal{D}), there exists a unique c = G(d) \in \mathrm{Ob}(\mathcal{C}) such that F(c) = F(G(d)) = d.^[7]^[6] On morphisms, F induces a bijection F_*: \mathrm{Hom}_{\mathcal{C}}(a, b) \to \mathrm{Hom}_{\mathcal{D}}(F(a), F(b)) for any objects a, b \in \mathrm{Ob}(\mathcal{C}), with G providing the inverse mapping G_*: \mathrm{Hom}_{\mathcal{D}}(F(a), F(b)) \to \mathrm{Hom}_{\mathcal{C}}(a, b). To see injectivity, if F(f) = F(f') for f, f': a \to b, then f = G(F(f)) = G(F(f')) = f'. For surjectivity, given any g: F(a) \to F(b) in \mathcal{D}, define f = G(g): a \to b in \mathcal{C}; then F(f) = F(G(g)) = g. These bijections hold uniformly for all pairs of objects, as the functoriality of F and G preserves the necessary compositions.^[7]^[6] As a consequence, an isomorphism F effectively relabels the objects and morphisms of \mathcal{C} to those of \mathcal{D} while preserving the category's composition table exactly, without any alteration to the relational structure among them.^[7] The inverse functor G ensures that compositions are mapped exactly, maintaining this one-to-one correspondence.^[6]

Preservation of Categorical Structure

An isomorphism of categories F: \mathcal{C} \to \mathcal{D} with strict inverse G: \mathcal{D} \to \mathcal{C} preserves limits exactly: if a diagram in \mathcal{C} admits a limit \lim_D with projections \pi_i: \lim_D \to D_i, then the image diagram F \circ D in \mathcal{D} admits limit F(\lim_D) with projections F(\pi_i): F(\lim_D) \to F(D_i), and the universal property holds strictly via the bijective correspondence on morphisms induced by F and G.^[1] Similarly, colimits are preserved: a colimit \colim_D in \mathcal{C} with inclusions \iota_i: D_i \to \colim_D maps to the colimit F(\colim_D) in \mathcal{D} with F(\iota_i), ensuring exact cocone universality.^[1] This strict preservation follows from the bijective action of F on objects and morphisms, which maps limiting or colimiting cones directly without alteration. Isomorphisms of categories preserve adjunctions strictly. The defining hom-set bijection of an adjunction transfers exactly to the corresponding bijection in the image category via the bijective correspondence on hom-sets induced by F, with units and counits mapped directly, preserving the triangular identities.^[1] Isomorphisms also preserve other categorical structures precisely. Monomorphisms (injective morphisms) in \mathcal{C} map to monomorphisms in \mathcal{D} under F, as the full and faithful nature ensures that the cancellation property defining monicity holds exactly in the image. Epimorphisms (surjective morphisms) are preserved analogously, with the dual cancellation property transferred bijectively. Products and coproducts follow suit: a product A \times B in \mathcal{C} with projections \pi_A, \pi_B becomes the product F(A) \times F(B) in \mathcal{D} with F(\pi_A), F(\pi_B), satisfying the universal pairing property strictly.^[1] In enriched or cartesian closed settings, exponential objects B^A (internal homs) are mapped exactly, preserving the evaluation and currying isomorphisms via the bijective functor action. This exact preservation underscores the rigidity of category isomorphisms compared to equivalences of categories, which only guarantee structures up to natural isomorphism rather than strict identity.^[1] Isomorphisms maintain not only the existence but also the precise "size" (cardinality of objects and morphisms) and diagram shapes, rendering them particularly strict in concrete categories like \mathbf{Set} or \mathbf{[Ab](/page/AB)}, where set-theoretic bijections align directly. While isomorphisms apply uniformly to small and large categories, their formulation in large settings demands caution with set-theoretic foundations, as the bijection on object classes may involve proper classes rather than sets, potentially invoking axioms like the axiom of choice or global choice principles to ensure well-definedness.^[1]

Examples

Trivial Isomorphisms

In category theory, the identity functor provides the simplest example of a categorical isomorphism. For any category \mathcal{C}, the identity functor \mathrm{id}_\mathcal{C}: \mathcal{C} \to \mathcal{C} maps each object and morphism to itself and serves as its own inverse, satisfying \mathrm{id}_\mathcal{C} \circ \mathrm{id}_\mathcal{C} = \mathrm{id}_\mathcal{C}.^[1]^[8] This construction aligns with the formal definition of an isomorphism as a pair of functors that are strictly inverse to each other. Another straightforward case arises from relabeling the objects of a category via a bijection while preserving the morphisms. Given a bijection \sigma on the objects of \mathcal{C}, one can define a functor F: \mathcal{C} \to \mathcal{C} by setting F(A) = \sigma(A) for objects A and F(f: A \to B) = f: \sigma(A) \to \sigma(B) for morphisms f, assuming the hom-sets remain unchanged under this renaming. The inverse functor employs \sigma^{-1} similarly, yielding an isomorphism that merely reindexes the objects without altering the categorical structure.^[8]^[1] Every category is isomorphic to itself through any automorphism, which is an isomorphism from \mathcal{C} to \mathcal{C}; the trivial such automorphisms are precisely the identity functors.^[9] These self-isomorphisms highlight the inherent symmetry in categorical presentations but offer no new structural insights.^[1] In discrete categories, where the only morphisms are identity arrows, any two such categories with the same cardinality of objects are isomorphic via any bijection between their object sets. This bijection extends uniquely to a functor on morphisms, as each must map identities to identities, and its inverse provides the required strict inverse.^[8]^[1] Trivial isomorphisms of these forms are ubiquitous in category theory, existing for every category and often serving to standardize or normalize presentations without introducing complexity. Despite their prevalence, they are generally uninteresting for deeper structural analysis, as they do not reveal non-obvious equivalences.^[1]

Non-Trivial Isomorphisms

Non-trivial isomorphisms of categories, distinct from mere relabelings of objects and morphisms, arise infrequently and typically demand a precise matching of complex internal structures, such as morphism compositions that align exactly beyond superficial bijections. These examples underscore the rigidity of categorical isomorphisms, where the functors must not only be bijective on objects and morphisms but also strictly preserve the entire compositional framework without relying on equivalences that allow for natural transformations. In practice, such isomorphisms are challenging to construct outside of finite or artificially simplified settings, as discrepancies in how morphisms compose often prevent strict equality.^[10] A notable example occurs in the category of posets, where objects are partially ordered sets and morphisms are order-preserving maps; a non-trivial isomorphism exists between the category of all posets and the category of Alexandroff T0 topological spaces (with continuous maps). This isomorphism is induced by the specialization preorder on Alexandroff T0-spaces, which turns closure operators into partial orders, and conversely by equipping posets with the Alexandroff topology where open sets are downward-closed; the functors are inverses, yielding a strict categorical isomorphism that reveals posets and Alexandroff T0-spaces as structurally identical despite their topological versus order-theoretic presentations. For finite cases, the categories of finite posets and finite T0-spaces are likewise isomorphic via the same mechanism, providing a concrete, non-relabeling match where the finite restrictions preserve the bijections on objects and morphisms exactly.^[10]^[11] A classic example from algebra is the forgetful functor from the category of abelian groups (\mathbf{Ab}) to the category of \mathbb{Z}-modules (\mathbb{Z}\mathbf{-Mod}), which is an isomorphism. It bijectively maps abelian groups to \mathbb{Z}-modules (since every abelian group is a \mathbb{Z}-module) and group homomorphisms to module homomorphisms, with a strict inverse given by the free abelian group functor that recovers the additive structure exactly.^[1] In the context of groups, consider the one-object category BG associated to a group G, where the single object represents the group and morphisms are the elements of G with composition given by group multiplication. This category is isomorphic to itself via any automorphism of G, but inner automorphisms—conjugations by elements of G—provide non-trivial isomorphisms for non-abelian groups, as they induce bijections on morphisms that rearrange the multiplication table non-trivially yet preserve composition strictly (e.g., for G = S_3, conjugation by a transposition yields a distinct but isomorphic labeling of permutations). Such isomorphisms highlight how group symmetries can permute the morphism set without altering the categorical structure, though they remain rare for categories beyond these monoidal presentations.^[10] Discussions on platforms like MathOverflow illustrate further instances, such as isomorphisms between categories sharing the same objects but with permuted morphisms that preserve fullness and skeletonicity, where equivalences coincide with strict isomorphisms due to the categories being skeletal (e.g., permuting morphisms in a small category with rigid composition yields an isomorphism if the permutation is an automorphism of the morphism poset). These examples, often drawn from synthetic or small categories, reinforce the scarcity of non-trivial isomorphisms, as constructing them typically necessitates identical composition tables across potentially disparate presentations, limiting their occurrence to specialized algebraic or order-theoretic domains.^[10]

Equivalence of Categories

In category theory, an equivalence of categories offers a more flexible alternative to strict isomorphism, establishing that two categories \mathcal{C} and \mathcal{D} are essentially the same despite potential differences in their concrete presentations. A functor F: \mathcal{C} \to \mathcal{D} is an equivalence if there exists a functor G: \mathcal{D} \to \mathcal{C} and natural isomorphisms \eta: \mathrm{id}_\mathcal{C} \Rightarrow G \circ F (the unit) and \varepsilon: F \circ G \Rightarrow \mathrm{id}_\mathcal{D} (the counit), demonstrating that F and G act as mutual inverses up to isomorphism.^[7] This structure ensures that structures and relationships in \mathcal{C} correspond precisely to those in \mathcal{D}, but without requiring exact matching of objects or morphisms.^[7] A functor F forms an equivalence precisely when it is full, faithful, and essentially surjective on objects; full and faithful guarantee bijective hom-sets, while essential surjectivity ensures every object in \mathcal{D} is isomorphic to the image of some object in \mathcal{C}.^[7] This characterization highlights how equivalences relax the rigidity of strict isomorphisms, which demand literal bijections on objects and morphisms without intermediary isomorphisms. Equivalences tolerate such "witnessing" isomorphisms, accommodating non-strict alignments; for example, the skeleton of a category—which collapses isomorphic objects into unique representatives—is equivalent to the original but rarely isomorphic due to the loss of redundant objects.^[7] Equivalences are far more prevalent in mathematical applications than strict isomorphisms, as most intuitively "isomorphic" categories differ only in representational choices rather than intrinsic structure. For instance, the category of finite-dimensional vector spaces over a field k (with linear maps as morphisms) is equivalent to the category of finite matrices over k (with matrix multiplication), but not isomorphic, because bases are not canonically fixed and isomorphisms depend on basis selections.^[7] Saunders Mac Lane played a pivotal role in promoting equivalences, emphasizing them as the appropriate notion of categorical sameness to sidestep overly rigid set-theoretic constraints and better reflect mathematical equivalence in practice.^[1]

Full and Faithful Functors

A functor F: \mathcal{C} \to \mathcal{D} between categories is faithful if, for every pair of objects A, B in \mathcal{C}, the induced map on hom-sets F: \hom_{\mathcal{C}}(A, B) \to \hom_{\mathcal{D}}(F(A), F(B)) is injective.^[1] This means that distinct morphisms in \mathcal{C} are mapped to distinct morphisms in \mathcal{D}, preserving the distinction between different arrows without collapsing them.^[1] A functor F: \mathcal{C} \to \mathcal{D} is full if the same induced map on hom-sets is surjective for every pair of objects A, B.^[1] In other words, every morphism in \mathcal{D} between the images F(A) and F(B) arises as the image under F of some morphism in \mathcal{C}.^[1] A functor that is both full and faithful thus induces a bijection on hom-sets, establishing a one-to-one correspondence between the morphisms in \mathcal{C} and those in \mathcal{D} between the corresponding images of objects.^[1] While full and faithful functors provide strong control over morphisms, they are insufficient for an isomorphism of categories without additional conditions, such as bijectivity on objects.^[1] For instance, the inclusion of a full subcategory into a larger category is typically full and faithful, as it restricts to bijections on hom-sets between objects in the subcategory, but it fails to be bijective on the entire collection of objects, preventing a strict inverse functor.^[12] An isomorphism of categories requires a full and faithful functor that is also bijective on objects and admits a strict inverse, ensuring the categories are identical up to relabeling.^[1] Without the strict invertibility, full and faithful functors contribute to weaker notions like equivalences when paired with essential surjectivity on objects. A concrete example is the forgetful functor U: \mathbf{Grp} \to \mathbf{Set}, which sends a group to its underlying set and a group homomorphism to the corresponding function on sets. This functor is faithful, as distinct group homomorphisms induce distinct set functions, but it is not full, since not every set function between underlying sets preserves the group structure.^[1] Such examples illustrate how full and faithful properties lay groundwork for comparing categorical structures, even if they do not yield isomorphisms on their own.