수반 함자

노트

위키데이터

• ID : Q357858

말뭉치

1. In category theory, a monad can be constructed from two adjoint functors.
2. With a relation on the morphisms of the category of pointed coalgebras we obtain an adjunction between these two functors.
3. Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint.
4. In this paper, we prove general adjoint functor theorems for functors between ∞ ‐categories.
5. As an application of this result, we recover Lurie's adjoint functor theorems for presentable ∞ ‐categories.
6. I think things like the adjoint functor theorem and Brown reprensentability become very reasonable from this point of view.
7. There is also a symmetrical “dual” concept of the receiving universal, and the pair of sending and receiving universals is what is given by a pair of adjoint functors.
8. Often in an adjunction, one of the universals is the one of interest and the other seems to be more a rather trivial bit of conceptual bookkeeping so that the two will make an adjunction.
9. We are trying to give an elementary introduction to the heteromorphic theory of adjoint functors.
10. This standard definition of an adjunction makes no mention of the heteromorphisms.
11. the Adjoint Functor Theorems (AFTs) aren't as useful as you might think when you first meet them.
12. We can also define the counit of the adjunction.
13. An adjunction is a pair of functors that interact in a particularly nice way.
14. As we'll see next time, an adjunction consists of a pair of functors that satisfy a nearly identical equation.
15. A basic result of category theory is that right adjoint functors preserve all limits that exist in their domain, and, dually, left adjoints preserve all colimits.
16. It is easy to check that these would be the unit and counit of an adjunction L ⊣ R L\dashv R .
17. In the “if” direction, this is an application of the general adjoint functor theorem: any accessible functor satisfies the solution set condition.
18. The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors.
19. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.
20. These maps are called the adjunction maps.
21. In mathematics, specifically category theory, adjunction is a relationship that two functors may have.
22. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint.
23. The slogan is "Adjoint functors arise everywhere".
24. The long list of examples in this article indicates that common mathematical constructions are very often adjoint functors.
25. We show that this is equivalent to the abstract definition, in terms of an adjunction in the 2-category Cat, in Prop.
26. according to Def. , with adjunction units η c \eta_c and adjunction counits ϵ d \epsilon_d according to Def. .
27. that the unit of an adjunction and counit of an adjunction plays a special role.
28. Proposition (collection of universal arrows equivalent to adjoint functor) Let R : 𝒟 → 𝒞 R \;\colon\; \mathcal{D} \to \mathcal{C} be a functor.