M. Jibladze. A presentation of the initial lift algebra. Journal of Pure and Applied Algebra, 116:185-198, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The operation extends to a functor : C C, where, on f : X Y , the morphism action is de ned by (Lf) e) ff(x) j x 2 eg: Further, the endofuctor carries a monad structure. The unit is singleton f g : and the multiplication is union LLX LX. As in [18], the endofunctor has a nal coalgebra, F LF (necessarily an isomorphism) de ned by: F = fc : j 8n : N: c(n 1) c(n)g (c) f(n 7 c(n 1) j c(0)g: 7 Because F is small and the functor preserves subset inclusions, there exists a smallest subalgebra, I, of ....

M. Jibladze. A presentation of the initial lift algebra. Journal of Pure and Applied Algebra, 116:185-198, 1997.

....1) if n 0 Hence, F L(F ) is a terminal L coalgebra. It is a useful comment that in fact, F is a retract of Sigma , by the map 7 n:N: kn (k) L has also an initial algebra (L(I) I) I can be constructed as the least L subalgebra of (L(F ) F ) Mamuka Jibladze ([11]) has given a beautiful formula for I: I = f 2 F j : 8n:N: n) OE) OE) OEg Note that oe : L(F ) F restricts to oe : L(I) I which is the L algebra structure on I. There are several ways of proving that I is in fact the initial L algebra ( 11] 27] The proof below is new and ....

....of (L(F ) F ) Mamuka Jibladze ( 11] has given a beautiful formula for I: I = f 2 F j : 8n:N: n) OE) OE) OEg Note that oe : L(F ) F restricts to oe : L(I) I which is the L algebra structure on I. There are several ways of proving that I is in fact the initial L algebra ([11], 27] The proof below is new and highlights the role of an induction principle that Jibladze s formula plays. 6 Theorem 1.4 L(I) I is the initial L algebra. Proof. Let (L(X) X) be any L algebra. First we prove that there is at most one algebra map from I to X. Any such h : I X ....

M. Jibladze. A Presentation of the Initial Lift Algebra. Journal of Pure and Applied Algebra, 116, 1997.

....The L operation extends to a functor L : C C, where, on f : X Y , the morphism action Lf : LX LY is defined by (Lf) e) f(x) e . Further, the endofuctor L carries a monad structure. The unit is singleton LX , and the multiplication is union : LLX LX . As in [14], the endofunctor L has a final coalgebra, # : F LF (necessarily an isomorphism) defined by: F = c : # #n : N. c(n 1) c(n) #(c) n ## c(n 1) c(0) Because F is small, there exists a smallest subalgebra, # : LI I, of # 1 , defined internally in C as the ....

M. Jibladze. A presentation of the initial lift algebra. Journal of Pure and Applied Algebra, 116:185--198, 1997.

....g and f : B 0 Gamma A there exists a unique classifying map f : B Gamma LA such that the following diagram is a pullback B 0 f A B f LA jA where f is given explicitly as f (b) h p(b) u:def(p(b) f(hb; ui) i. 5. 3 Initial L Algebra and Terminal L Coalgebra Following [21, 7] one may show the existence of initial and terminal L algebras in Set. Theorem 5.1 In Set there exists an initial L algebra OE : L Gamma and a terminal L coalgebra AE : Gamma L where is a subobject of S(N) and is the least sub L algebra of AE Gamma1 . As OE and AE are known to ....

M. Jibladze. A presentation of the initial lift-algebra. JPAA, 116(2--3):199--220, 1997.

....by Proposition 3.5 (using # 1 as the small mono) By the construction of I, the unique algebra homomorphism # : I # F (from # to # 1 ) is mono. In the special case of an elementary topos, the above constructions of F and I were found independently in 1995 by the present author and M. Jibladze [12]. One can view I as the object obtained from 0 by freely iterating the L functor. When Axioms 1 and 2 hold, I plays the role of a generic # chain in C, and I # # F exhibits F as its chain completion. This intuition plays a fundamental role in developing a basic notion of chain ....

M. Jibladze. A presentation of the initial lift algebra. Journal of Pure and Applied Algebra, 116:185--198, 1997.

....the monad of computations. We refer the reader to the literature cited above and to [24] for more on the subject of SDT. Here we recall some of the basic definitions and the facts we need in the paper. The underlying functor L of a lifting monad always has an initial algebra and a final coalgebra [11, 13]; call them : LI I and : F LF respectively. By a lemma noticed by Lambek, and are isomorphisms. Let c: I F be the canonical map determined by inverting one of the two and recalling universality of the other; i.e. the unique map making the following diagram commute LI = Lc LF ....

.... as a retract of Sigma N (the object of admissible subsets of the natural numbers object N) as follows fp 2 Sigma N j 8n 2 N: p n 3 p n 1 g Sigma N oo oo , see [11] The initial L algebra LI = I is the smallest L subalgebra of the (inverse of the) final L coalgebra, see [13] for an explicit description. In our situation, it is obtained from the colimit of the diagram 0 1 L Sigma Delta Delta Delta Sigma n L n 1 Delta Delta Delta (3) which coincides in H and b L) because L: H H preserves non empty connected ....

M. Jibladze. A presentation of the initial lift-algebra. J. Pure Appl. Alg., 116:185--198, 1997.

....there exist a terminal L coalgebra AE: L and an initial L algebra OE: L . The latter can be constructed as the least sub L algebra of AE Gamma1 . Thus, the unique L algebra morphism : ae is monic. Proof. We first construct the terminal L coalgebra AE: L following (Jibladze, 1997). We define as the subset of S N consisting of all f 2 S N with def(f(n 1) def(f(n) for all n 2 N. We define AE: L as AE(f) hf(0) z:def(f(0) n:N: f(n 1)i for every f 2 . Suppose ff: A LA is some L coalgebra. An L coalgebra morphism from ff to AE is given by a map ....

....we have AE Gamma1 (hu; oi) 0) u AE Gamma1 (hu; oi) n 1) u p:def(u) o(u) n) for all u 2 S, o 2 def(u) and n 2 N. We will now construct an initial L algebra as the least sub L algebra of the L algebra AE Gamma1 : L . Again, this construction was originally given in (Jibladze, 1997), however, our version of the proof makes use of our previous observation that L algebras can be considered as algebras in the sense of universal algebra. Let be the least subset P of closed under all operations AE Gamma1 u , i.e. AE Gamma1 u (f) 2 P for all u 2 S and f 2 P def(u) ....

Jibladze, M. (1997). A presentation of the initial lift-algebra. J. Pure Appl. Algebra, 116(1-3), 185--198.

....if n 0 Hence, F L(F ) is a terminal L coalgebra. It is a useful comment that in fact, F is a retract of Sigma N , by the map 7 n:N: V kn (k) L has also an initial algebra (L(I) oe I) I can be constructed as the least L subalgebra of (L(F ) oe F ) Mamuka Jibladze ([11]) has given a beautiful formula for I: I = f 2 F j 8OE: Omega : 8n:N: n) OE) OE) OEg Note that oe : L(F ) F restricts to oe : L(I) I which is the L algebra structure on I. There are several ways of proving that I is in fact the initial L algebra ( 11] 27] The proof below is ....

....oe F ) Mamuka Jibladze ( 11] has given a beautiful formula for I: I = f 2 F j 8OE: Omega : 8n:N: n) OE) OE) OEg Note that oe : L(F ) F restricts to oe : L(I) I which is the L algebra structure on I. There are several ways of proving that I is in fact the initial L algebra ([11], 27] The proof below is new and highlights the role of an induction principle that Jibladze s formula plays. Theorem 1.4 L(I) oe I is the initial L algebra. Proof. Let (L(X) g X) be any L algebra. First we prove that there is at most one algebra map from I to X. Any such h : I X ....

M. Jibladze. A Presentation of the Initial Lift Algebra. Journal of Pure and Applied Algebra, 116, 1997.

....oe : LI I and a final coalgebra : F LF. Moreover, the object F is a retract of Sigma N , and the unique algebra homomorphism from oe to Gamma1 is a mono : I F. These results were proved in 1995 by the author and Mamuka Jibladze independently. For a published account see [11]. The initial algebra of L as a functor, interacts nicely with the monad structure on L. We call a morphism ff : LX X a monad algebra if it is an EilenbergMoore algebra for the monad (L; j; 16] The morphism 0 = oe ffi ffi Loe Gamma1 is a monad algebra LI I. We write up : I ....

M. Jibladze. A presentation of the initial lift algebra. Journal of Pure and Applied Algebra, 116:185--198, 1997.

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