Isomorphism, groupoid, and connected category

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

• A colimit of a set-valued functor is the set of connected components of its category of elements Corollary
• Objects a and b are isomorphic exactly when C(-,a) and C(-,b) are naturally isomorphic Corollary
• Conservative functor Definition
• Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors Definition
• Filtered categories and filtered colimits Definition
• Final and initial functors via nonempty connected comma categories Definition
• Natural isomorphism Definition
• Skeletal category and skeleton Definition
• Split monomorphism, split epimorphism, retraction, and section Definition
• Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms Definition
• An action groupoid has the acted-on set as objects, orbits as connected components, and stabilizers as automorphism groups Example
• The fundamental groupoid of a topological space Example
• The inclusion of groupoids into categories is left adjoint to the maximal-subgroupoid functor Example
• A morphism is an isomorphism exactly when postcomposition, equivalently precomposition, induces bijections on every hom-collection Lemma
• Every fully faithful functor reflects isomorphisms Proposition
• Every functor preserves isomorphisms Proposition
• The isomorphisms in a category form its maximal subgroupoid Proposition
• Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects Proposition
• A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice Theorem
• A splitting of an idempotent is simultaneously an equalizer and a coequalizer and is unique up to unique isomorphism Theorem
• An ambient object lies in the essential image of a reflective inclusion exactly when its reflection unit is invertible Theorem
• Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps Theorem
• Initial and terminal objects are unique up to a unique isomorphism Theorem
• Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it Theorem
• Representing objects are unique up to a unique isomorphism compatible with their universal elements Theorem
• The counit of a reflection is an isomorphism Theorem
• The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)