WSAA 5
Jiří Rosický
2-categorical tools in the theory of concrete categories
In: Zdeněk Frolík (ed.): Abstracta. 5th Winter School on Abstract Analysis.
Czechoslovak Academy of Sciences, Praha, 1977. pp. 95--97.
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95 FIFTH WINTER SCHOOL (1977)
2-CATEGORICAL TOOLS IN THE THEQHI OF CONCRETE CATEGORIES by
Jiff Rosicktf
Let X be a category. Denote by Dj a 2-category objects of which are couples (AfU) where U: A—*X i s a functor, arrows see couples (Ff \f ) : (A,U) —5 (BfV) where F: A —^B i s a functor and y : U --t VF a natural transformation and 2 - c e l l a ©c : ( Ff* ) — ^ (F* f J) are natural transformations a : F—*F* such that %f * V« . f . Let E^
be a sub-2-category of Dx having the same objects as D^ such that an arrow (F, y ) : (AfU)—^ (B,V) in Dx belongs to Ej i f and only i f VF • U and *f * ^u and any 2 - c e l l in D-^ between arrows of Ex belongs to E-^.
The 2-category Ex i s quite usualf D-^ was considered, for i n - stance, by R.Guitart. There are two important choices of X: the one-morphisms category 4 and the category Set of s e t s . Df « Ej »
* Cat i s the 2-category of categories and Dg t or Ege+ contains the 2-category D or E reap, of concrete categories ( i . e . of f a i t h - f u l functors U: A — ^ S e t ) . Our aim i s to get consequences for con- crete categories by making the theory of categories in D^-. All pre- sented r e s u l t s can be found in [4] • The themes a) and b) are partly considered in [5]»
a) Extensions of functors: Let us have two arrows K: M—* A and T: M—*B in a 2-category C. A couple Lf^ consisting of an arrow L: A—^B and a 2 - c e l l Jf: T —^IK i s c a l l e d a l e f t extension of T along K i f for any S: A—»B and o~i T —*SK there i s a unique 2 - c e l l <* : L—* S such that cc K. X .«-** •
Left extensions in Cat are l e f t Kan extensions of functors.
Left extensions in Ex are useful for the study of l i f t i n g s of fun- c t o r s , extensions of f u l l embeddings e t c . Constructions of them ere
96
described in [3] , [4] and [5] • One construction of left extensions in Dx is given in M 3 • There is another construction of left exten- sions in Dx which calculates La for each a 6 A by a suitable uni- versal property. For that reason left extensions obtained by it will be called pointwise.
b) Liftings of monads: Let K: .M—» A be an arrow in a 2-category C.
A left extension of K along K is a comonad in C which is called a density comonad. The construction of a density comonad in D-^ can be parametrized by comoneds in X* This procedure can be adapted for the study of liftings of comonads (or monads by the duality) in the similar manner as we have treated liftings of functors in a ) . c) Inductive generation: An arrow K: U —->A in a 2-category C is called dense if A^ is the pointwise left extension of K along K.
Dense subcategories in 5 ere precisely inductive generating sub- categories in the sense of [3]« This fact establishes the expected relevance between inductive generation and density in Cat.
d) Pointwise extensions: The preceding theme uses pointwise left extensions which can be defined in any 2-category C (see [6]). This definition is based on comma objects. A couple of arrows in C with a common domain (codomain) is called a span (an opspan) in C.
F O
A comma object for an opspan A --£-J.* C t-*- B is a span D_ D,
A *—— F/G » B together with a 2-cell \ : F DQ— » (&A which are universal among these data.
Pointwise left extensions in Dx in the sense of a) are point- wise in the sense of the theory of 2-categories if one uses comma objects in D^ for opspans in E^. Both comma objects in Ex and comma objects in D~ for general opspans give a too strong concept.
Pointwise left extensions in D^ agree with pointwise left Kan
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extensions.
e) Initial completion; The Yoneda embedding A — * Cat(Aop,Set) can be internally characterized by a suitable universal property using comma objects (see [7] ). The corresponding universal property in Dx (but using the same comma objects as in d) - i.e. comma objects
in D~£ for opspans in E ^ - instead of general ones) determines
"a Yoned8 embedding* in D^. The role of the category of functors from Ao p to Set is in Dx for a given (A,U) played by (^tU^, where A^ has objects (F, J* fx) where F: Ao p— - S e t is a functor, x 6 X
and J : F —* X(U-,x) is a natural transformation and Ux assigns x to (Ff J tx ) . If we restrict ourselves to D, then we have to take only (F,$ ,x) such that a is mono (i.e. F is a subfunctor of X(U-,x)) and the corresponding *Yoneda embedding* (A,U)-—* (A,U) is precisely the initial completion in the sense of [2] *
References:
M ] R. Guitart, Remarques sur les machines et les structures, Cahiers Topo. G£o. Diff. XV-2 (1974), 113-145.
[2] H. Herrlich, Initial completions, Kategorienseminar, Hagen 1976, 3-26.
[3] M. HuSek, Construction of special functors and its applications, Comment. Math. Univ. Carolinae 8 (1967), 555-566.
[4] J. Rosidtf, Extensions of functors and their applications, to appear in Cahiers Topo. G6o. Diff.
[5] J. Rosick^, Liftings of functors in topological situations, to appear in Proc. 4th Prague Toposymposium.
[6] R. H. Street, Fibrations and Yoneda's lemma in a 2-category, Category Seminar, Sydney 1972-3, Lect. Not. 420, 104-134.
[7] R. H. Street, Elementary cosmoi, Category Seminar, Sydney 1972-3, Lecture Notes in Math. 420, 134 - 180.
