Shahada ya kwanza (mwanzo)Category TheoryImepangwa
Universal properties
Describing objects by their relationships rather than their elements: universal properties of products, quotients, free objects and fields of fractions, and the fact that an object with a universal property is unique up to unique isomorphism.
math.cat.universal-properties
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 16 (hadi chini kabisa)
Utajifunza
- State the universal properties of products in Set, Grp and Top, of quotient sets and quotient groups, and of free monoids and free groups.
- Prove that any two objects with the same universal property are isomorphic by a unique isomorphism compatible with the structure maps.
- Recognise when a familiar construction, such as the tensor product or the field of fractions, is best defined by a universal property.
Fasili
- A universal arrow from an object X of C to a functor U : D → C is an object F of D with a morphism η : X → U(F) such that every morphism X → U(E) equals U(g) ∘ η for a unique g : F → E.
- A free monoid on a set S is a monoid F(S) with a map S → F(S) through which every map from S to a monoid factors uniquely via a monoid homomorphism.
Nadharia
- Objects defined by a universal property are unique up to a unique isomorphism compatible with the structure maps.
- The free monoid on S is the monoid of finite words in the elements of S under concatenation.
- The quotient map X → X/~ is universal among maps out of X that are constant on equivalence classes.
Mawazo potofu ya kawaida
- Thinking a universal property constructs an object; it characterises one, and existence must be proved separately. For example, the fields 𝔽₂ and 𝔽₃ have no product in the category of fields.
- Forgetting the uniqueness clause, which is what makes the object unique up to isomorphism.
Inaelekea kwa
Ugunduzi wa hiari
- Limits and colimits Products, equalisers, pullbacks and their duals are all limits or colimits: one universal property covering many constructions.Njia ya Maajabu →
- Kan extensions Limits and colimits are themselves special Kan extensions, along the functor to the category with one object and one morphism.Njia ya Maajabu →
- ← Relations, equivalence relations and partitions
- ← Composition
- ← Constructing ℤ, ℚ and ℝ
- ← Metric spaces and point-set topology
Usomaji zaidi
- Tom Leinster, Basic Category Theory (2014)
- Emily Riehl, Category Theory in Context (2016)