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An object I is injective if Hom(−, I) preserves monomorphisms, equivalently, if every morphism A → I extends to a morphism B → I for any monomorphism A → B. Injectives are dual to projectives and generalize divisible groups. Every object embeds into an injective envelope, enabling injective resolutions essential to homology and cohomology theory.
From your study of additive categories and abelian groups, you know what it means for a morphism to be a monomorphism (injective on elements, or more generally left-cancellable). An object I is injective if, whenever you have a monomorphism i: A ↪ B and a morphism f: A → I, you can always find an extension f̃: B → I making the triangle commute: f̃ ∘ i = f. Informally: any map from a subobject A into I can be extended to the whole ambient object B. Injective objects are "extensible targets" — they never block extensions.
The canonical example over ℤ is the group ℚ of rational numbers. Given any subgroup A of an abelian group B and a homomorphism f: A → ℚ, you can always extend f to all of B. The key property enabling this is divisibility: for any x ∈ ℚ and non-zero integer n, there exists y ∈ ℚ with ny = x. When you try to extend f to a new element b ∈ B \ A, you need a consistent value for f̃(b). If nb ∈ A for some n (which happens in quotient situations), divisibility ensures you can divide f(nb) by n inside ℚ to define f̃(b) without contradiction. Injective modules over a ring generalize this: over a principal ideal domain, injective modules are exactly the divisible ones.
Injective envelopes capture the idea of the smallest injective object containing a given object. Every object M in a suitable abelian category embeds into an injective envelope I(M): an injective object in which M sits essentially — meaning every nonzero subobject of I(M) meets M non-trivially. The injective envelope is characterized by being both injective and essential over M, and it is unique up to isomorphism. For abelian groups, the injective envelope of ℤ/nℤ is ℤ[1/p₁, ..., 1/pₖ]/ℤ where the pᵢ divide n — the minimal divisible extension.
The payoff is injective resolutions: for any object M, choose an embedding M ↪ I₀ into its injective envelope, then embed the cokernel into another injective I₁, and continue: 0 → M → I₀ → I₁ → I₂ → ⋯. This is an injective resolution — an exact sequence of injective objects. Injective resolutions are the raw material for derived functors: apply Hom(N, −) to the deleted resolution (drop M from the front), take cohomology, and you get the Ext groups Ext^n(N, M). The failure of Hom to be exact on the right — it only preserves exactness at injective objects — is precisely what Ext measures. Without injective objects and their resolutions, cohomological invariants of modules and sheaves would have no computational foundation.