Categories, Products, Coproducts and Functors
Handbook guide to categories, products, coproducts and functors with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Categories
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Products and Coproducts
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Functors
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definitions and Comments
A category C consists of objects A, B, C, . . . and morphisms f : A →B (where A and B are objects). If f : A →B and g : B →C are morphisms, one has a notion of composition, in other words, there is a morphism gf = g ◦f : A →C, such that the following axioms are satisfied. (i) Associativity: If f : A →B, g : B →C, h : C →D, then (hg)f=h(gf); (ii) Identity: For each object A there is a morphism 1A : A →A such that for each morphism f : A →B, one has f1A = 1Bf = f. A remark for those familiar with set theory: For each pair (A, B) of objects, the collection of morphisms f : A →B is required to be a set rather than a proper class.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definitions and Comments
A morphism f is called monic if it is left cancellable, epic if it is right cancellable. In all the categories listed in (10.1.1), a morphism f is monic ifff is injective as a mapping of sets. If f is surjective, then it is epic, but the converse can fail. See Problems 2 and 7-10 for some of the details.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definition
Let A be an object in a category. If for every object B, there is a unique morphism from A to B, then A is called an initial object. If for every object B there is a unique morphism from B to A, then A is called a terminal object. A zero object is both initial and terminal.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Duality If C is a category, the opposite or dual category Cop has the same objects
Duality If C is a category, the opposite or dual category Cop has the same objects as C. The morphisms are those of C with arrows reversed; thus f : A →B is a morphism of Cop ifff : B →A is a morphism of C. If the composition gf is permissible in C, then fg is permissible in Cop. To see how the duality principle works, consider first prove that if A and B are initial objects of C, then A and B are isomorphic.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Kernels and Cokernels If f : A →B is an R-module homomorphism, then its
Kernels and Cokernels If f : A →B is an R-module homomorphism, then its kernel is, as we know, {x ∈A : f(x) = 0}. The cokernel of f is defined as the quotient group B/im(f). Thus f is injective iffits kernel is 0, and f is surjective iffits cokernel is 0. We will generalize these notions to an arbitrary category that contains zero objects.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Definition
A product of objects Ai in a category C is an object A, along with morphisms pi : A →Ai, with the following universal mapping property. Given any object S of C and morphisms fi : S →Ai, there is a unique morphism f : S →A such that pif = fi for all i. In a definition via a universal mapping property, we use a condition involving morphisms, along with a uniqueness statement, to specify an object and morphisms associated with that object. One has already seen this idea in connection with kernels and cokernels in the previous section, and in the construction of the tensor product in Section 8.7.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
If (A, pi, i ∈I) and (B, qi, i ∈I) are products of the objects Ai, then A and B are isomorphic.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Definition
A coproduct of objects Mj in a category C is an object M, along with morphisms ij : Mj →M, with the following universal mapping property. Given any object N of C and morphisms fj : Mj →N, there is a unique morphism f : M →N such that fij = fj for all j Exactly as in (10.2.2), any two coproducts of a given collection objects are isomorphic. The discussion of diagram (2) shows that in the category of R-modules, the coproduct is the direct sum, which is isomorphic to the direct product if there are only finitely many factors. In the category of sets, the coproduct is the disjoint union.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
The Functor HomR(M,
The Functor HomR(M, ) We are going to construct a mapping from the category of R-modules to the category of abelian groups. Since a category consists of both objects and morphisms, our map will have two parts: (i) Associate with each R-module N the abelian group HomR(M, N). (ii) Associate with each R-module homomorphism h : N →P a homomorphism h∗from the abelian group HomR(M, N) to the abelian group HomR(M, P). The following diagram suggests how h∗should be defined. f h M → N → P Take h∗(f) = hf.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
The Functors M ⊗R
The Functors M ⊗R and ⊗R N To avoid technical complications, we consider tensor products of modules over a commutative ring R. First we discuss the tensor functor T = M ⊗R The relevant diagram is given below. g f N → P → Q If N is an R-module, we take T(N) = M⊗RN. If g : N →P is an R-module homomorphism, we set T(g) = 1M ⊗g : M ⊗R N →M ⊗R P, where 1M is the identity mapping on M [A useful starting fact is that (1M ⊗g)(x ⊗y) = x ⊗g(y).] Then T(fg) = 1M ⊗fg = (1M ⊗f)(1M ⊗g) = T(f)T(g) and T(1N) = 1T (N) so T is a functor from R-mod to R-mod. The functor S = ⊗R N is defined in a symmetrical way.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Natural Transformations
Natural Transformations Again we will introduce this idea with an explicit example. The diagram below summarizes the data. tA F(A) → G(A) Ff ↓ ↓Gf F(B) → G(B) tB (1) We start with abelian groups A and B and a homomorphism f : A →B. We apply the forgetful functor, also called the underlying functor U. This is a fancy way of saying that we forget the algebraic structure and regard A and B simply as sets, and f as a mapping between sets.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Quick-reference relationships
Problem-solving workflow
Fix the coefficient ring and variance
State whether modules are left/right modules and whether a functor is covariant or contravariant.
Write the maps, not just the objects
Kernels, images, exactness and universal properties depend on the actual homomorphisms.
Use the appropriate universal property
Direct sums, products, tensor products, projectives, injectives and limits are best handled by their mapping property.
Check exactness at each position
Verify image equals kernel rather than relying on the appearance of a diagram.
Choose a resolution only when needed
Derived constructions should be tied to projective or injective resolutions and independence from the chosen resolution.
Test naturality and compatibility
For induced maps, ensure compositions and commutative squares behave as required.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test kernel, ideal, polynomial, basis, matrix, field, module, homomorphism. These checks are used here as verification themes rather than copied as answer text.
The supplied worked solutions for this section repeatedly test order, kernel, ideal, homomorphism, injective, prime, factor, functor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
- Confusing direct sum with direct product for infinite families.
- Reading exactness from a diagram without checking image equals kernel.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 10.1 | Categories | 198–200 |
| 10.2 | Products and Coproducts | 201–202 |
| 10.3 | Functors | 203–205 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.