Alphabeta MathLimits and Colimits
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Groups and Presentations
- Group Homomorphisms and the Isomorphism Theorems
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
Category Theory
Real Analysis
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Constant diagrams, cones, cocones, and their morphisms
Definition
Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties
Theorem
Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps
Proposition
A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category
Proposition
Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects
Definition
Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations
Definition
Equalizers and coequalizers as limits and colimits of a parallel pair
Definition
Pullbacks and pushouts as limits and colimits of cospans and spans
Definition
Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers
Lemma
The legs of a limiting cone are jointly monic, and the legs of a colimiting cocone are jointly epic
Corollary
Every equalizer is a monomorphism, and every coequalizer is an epimorphism
Lemma
A pullback of a monomorphism is a monomorphism, and a pushout of an epimorphism is an epimorphism
Lemma
Horizontally pasted commutative squares commute
Theorem
Pullback and pushout pasting, with cancellation of the square adjacent to the outer edge
Definition
Assuming Choice, cardinality of a small category and κ-small diagrams
Definition
Finite, small, and large limits and colimits; complete and cocomplete categories
Lemma
A cone over an identity diagram is weakly initial, and the identity diagram has a limit exactly when the category has an initial object
Theorem
Set has all small limits, realized as compatible tuples in a set-indexed product
Theorem
Set has all small colimits, realized as a quotient of a set-indexed disjoint union
Theorem
Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category
Theorem
Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category
Corollary
A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers
Theorem
Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals
Definition
Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors
Lemma
A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits
Proposition
Fully faithful functors reflect limits and colimits
Proposition
Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense
Proposition
A functor that creates limits of a given shape lifts their existence and preserves the created limits, and dually for colimits
Theorem
Chosen limits and colimits of a fixed small shape assemble into limit and colimit functors
Remark
A limit for each diagram need not provide a chosen limit functor without a simultaneous choice of representatives
Theorem
Every covariantly representable functor to Set preserves all existing small limits
Corollary
Hom(X,−) is continuous, while Hom(−,X) sends every existing small colimit to a limit of sets
Theorem
For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise
Corollary
If A is small, then [A,C] is complete or cocomplete whenever C is respectively complete or cocomplete
Corollary
For a small category, the Yoneda functor preserves and reflects all existing small limits
Theorem
Iterated small limits commute: either order is canonically isomorphic to the limit over the product category
Definition
Filtered categories and filtered colimits
Lemma
Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage
Theorem
Filtered colimits commute with finite limits in Set
Definition
Final and initial functors via nonempty connected comma categories
Theorem
Assuming Choice, precomposition with a final functor does not change colimits, and precomposition with an initial functor does not change limits
Theorem
Grp is complete and cocomplete
Theorem
Top is complete and cocomplete, and its underlying-set functor preserves all small limits and colimits
Theorem
For every ring R, the category R-Mod is complete and cocomplete
Proposition
A poset category is complete exactly when every small family has an infimum, and cocomplete exactly when every small family has a supremum
Theorem
Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder
Corollary
Assuming Choice, every small complete category and every small cocomplete category is a preorder
Sources
- E. Riehl, Category Theory in Context, Definitions 3.1.1 and 3.1.5
- E. Riehl, Category Theory in Context, Definitions 3.1.6 and 3.1.11
- E. Riehl, Category Theory in Context, Proposition 3.1.7
- E. Riehl, Category Theory in Context, Remark 3.1.8
- E. Riehl, Category Theory in Context, Example 3.1.14
- E. Riehl, Category Theory in Context, Definitions 3.1.9 and 3.1.13
- E. Riehl, Category Theory in Context, Definition 3.1.15
- E. Riehl, Category Theory in Context, Definition 3.1.23
- E. Riehl, Category Theory in Context, Example 3.1.26
- E. Riehl, Category Theory in Context, Exercise 3.1.iv
- E. Riehl, Category Theory in Context, Exercise 3.1.v
- S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Ch. 1
- E. Riehl, Category Theory in Context, Exercise 3.1.vi
- E. Riehl, Category Theory in Context, Definition 3.7.2
- E. Riehl, Category Theory in Context, Definitions 3.2.1 and 3.2.3
- E. Riehl, Category Theory in Context, Lemma 3.7.1
- E. Riehl, Category Theory in Context, Theorem 3.2.4
- E. Riehl, Category Theory in Context, Proposition 3.6.1
- E. Riehl, Category Theory in Context, Theorem 3.5.11 and Remark 3.2.13
- E. Riehl, Category Theory in Context, Theorem 3.5.11
- The Stacks Project, Categories, Lemmas 4.18.2 to 4.18.4
- E. Riehl, Category Theory in Context, Theorem 3.5.17
- E. Riehl, Category Theory in Context, Definition 3.4.1 and Remark 3.4.2
- E. Riehl, Category Theory in Context, Definition 3.4.1
- E. Riehl, Category Theory in Context, Lemma 3.4.3
- E. Riehl, Category Theory in Context, Lemma 3.4.5 and Definition 3.4.7
- E. Riehl, Category Theory in Context, Definition 3.4.7 and Proposition 3.4.9
- E. Riehl, Category Theory in Context, Lemma 3.4.6
- E. Riehl, Category Theory in Context, Remark following Lemma 3.4.6
- E. Riehl, Category Theory in Context, Theorem 3.5.5
- E. Riehl, Category Theory in Context, Theorems 3.5.5 and 3.5.6
- E. Riehl, Category Theory in Context, Proposition 3.3.1
- E. Riehl, Category Theory in Context, Corollary 3.3.3
- E. Riehl, Category Theory in Context, Theorem 3.8.1
- E. Riehl, Category Theory in Context, Definition 3.8.7
- The Stacks Project, Categories, Lemma 4.19.2
- E. Riehl, Category Theory in Context, Theorem 3.8.9
- T. Leinster, Basic Category Theory, Definition 6.3.1
- T. Leinster, Basic Category Theory, Theorem 6.3.4
- The Stacks Project, Categories, Example 4.19.5
- E. Riehl, Category Theory in Context, Section 3.6
- E. Riehl, Category Theory in Context, Proposition 3.6.2
- E. Riehl, Category Theory in Context, Example 3.1.24
- E. Riehl, Category Theory in Context, Proposition 3.7.3
- E. Riehl, Category Theory in Context, Corollary after Proposition 3.7.3