Covariant functor, identity functor, composite functor, and contravariant functor

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection
• Opposite category $\mathcal C^{\mathrm{op}}$

Used by

• A functor need not preserve monomorphisms Counterexample
• A reflexive coequalizer of sets not preserved by Set(ℕ,-) Counterexample
• The functor D(X)=X⊔ X on Set is not covariantly representable Counterexample
• A presheaf on a topological space Definition
• Absolute Kan extension Definition
• Additive functor Definition
• Cohomological functor on a triangulated category Definition
• Comma category, slice category, and coslice category Definition
• Conservative functor Definition
• Constant diagrams, cones, cocones, and their morphisms Definition
• Diagram as a functor from an indexing category Definition
• Dinatural transformation between functors on CᵒᵖtimesC Definition
• Ends and coends with parameters Definition
• Equivalence, quasi-inverse, and adjoint equivalence of categories Definition
• Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors Definition
• Final and initial functors via nonempty connected comma categories Definition
• Lax, strong, and strict monoidal functors Definition
• Left and right Kan extensions Definition
• Local systems and pullback Definition
• Localization of a category at a class of morphisms Definition
• Monad on a category Definition
• Monoidal category Definition
• Mutually left and mutually right adjoint contravariant functors Definition
• Natural transformation and its components Definition
• Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors Definition
• Product category and its projection functors Definition
• Slice categories, composition, and pullback along a morphism Definition
• Stable natural cohomology operation Definition
• Strict 2-category Definition
• The category of elements of a covariant functor or a presheaf Definition
• The category of right-module endofunctors Definition
• The functor of points of an affine scheme Definition
• The solution-set condition for a functor, stated object by object Definition
• The twisted arrow category and its projection to CᵒᵖtimesC Definition
• Universal arrows from an object to a functor and from a functor to an object Definition
• Whiskering and horizontal composition of natural transformations Definition
• Actions of a group G on sets are functors BG toSet Example
• Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
• For a fixed space X, product with X defines an endofunctor of Top Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example

…and 22 more results.

Dependency tree · two levels

4 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)