S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equivalence relation de ned on each hom set. The cocompletion is then the equiv category modulo equivalence of morphisms. A di erent construction of the free cocompletion is well known in mathematics and uses the category Set C op of presheaves over C. It is described in, for instance, [Mac71, MM92]. A presheaf in itself is somewhat cumbersome as a formal colimit, more so than a diagram, but the description of a presheaf is greatly simpli ed by the use of generators and relations. The trick is taken from universal algebra, and exploits the fact that there is a many sorted algebraic theory ....

....makes Set C op a free cocompletion of C. Theorem 2.0.2 Set C op is cocomplete (i.e. has all small colimits) Any functor F from C to a cocomplete category D factors, uniquely up to natural isomorphism, as y ; G where G : Set C op D preserves colimits. Proof. See, for instance, [MM92]. This tells us that each presheaf can be understood as a formal colimit of a diagram in C, and for our present purposes it is as well to understand how this works. Every presheaf P can be expressed as a colimit of representable presheaves as follows. Associated with P is its category of ....

MacLane, S. and Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory, SpringerVerlag, 1992.

....duality is described in detail in section 2.7 (except that we deal with # instead of # and the corresponding Theory and Applications of Categories, Vol. 8, No. 7 119 part of T instead of T ) This is the duality Joyal describes in section 1. 1 of [J] Also, the same duality appears in [SGL], p. 455. Our treatment of this duality serves a purely expository purpose. We display the ingredients of the schizophrenic object in detail to make the generalization to the # dimensional case easier to follow. In the remaining parts of the paper we show that the higher dimensional analogs of ....

....non empty linear orders and monotone maps and D 1 is equivalent to the category of finite linear orders with (necessarily di#erent) endpoints and monotone functions preserving endpoints. Thus, it is well know that they are dually equivalent. An easy explanation of this fact can be found in [SGL]. Below, we will exhibit a more involved but, as we believe, more instructive explanation of this fact, that can be generalized to the higher dimensional categories, of which it is a special case. Before we prove the duality theorem we need to develop some notation. Both elements on level 1 in a ....

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S. MacLane, I.Moerdijk, Sheaves in Geometry and Logic: a first introduction to topos theory, Springer-Verlag, New York, (1992). Theory and Applications of Categories, Vol. 8, No. 7 243

....those presheaves corresponding to event structures, are more difficult to explain, though some are representable via models such as general Petri nets. 10 Proofs for presheaf models can be found in (Joyal, Nielsen, and Winskel 1994) A good introduction to presheaves can be found in Chapter 1 of (MacLane and Moerdijk 1992). Models for Concurrency 51 The embeddings of theorem 45 extend the Yoneda embedding of P # b P, regarding a path object P as the presheaf P( P ) M( P ) because, in these cases, the subcategory P ## M is full. Now, if we regard presheaves as the model M # and the image of P under the ....

MacLane, S. and I. Moerdijk (1992). Sheaves in geometry and logic: a first introduction to topos theory. Springer.

....The inclusions L ae F P S(L) ae S(F) ae S(P) induce equivalences sh(L; can) oe sh(F; can) H sh(S(L) can) 6 oe sh(S(F) can) 6 oe sh(S(P) can) 6 PROOF. The vertical equivalences are obvious. For the others, we apply the Comparison Lemma, see [9], using the following characterization. Lemma 4 The collection of non empty families R of monos in S(L) with common codomain D such that (i) for every x; y 2 D there exists f 2 R such that fx; yg im(f) and (ii) for every unbounded countable chain hx k i k in D there exists f 2 R such that fx ....

....topology. Theorem 11 The inclusions L L 2 F 2 S(L) ae S(L 2 ) ae S(F 2 ) induce H sh(L; can) faithful oe sh(L 2 ; can) H 2 sh(S(L 2 ) can) 6 oe sh(S(F 2 ) can) 6 10 PROOF. It is similar to that of Theorem 3, again using the Comparison Lemma of [9]. Also in S(L 2 ) epimorphisms are retractions, and the inverse image of a mono intersecting the image of a map is computed on the underlying set. The canonical topology can be characterized as follows. Lemma 12 The collection of non empty families R of monos in S(L 2 ) with common codomain D ....

S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, 1992.

....is what is needed) the diagrams are the right objects but the diagram morphisms are too concrete the equivalence relation must be factored out. There is an alternative construction of cocompletions using presheaves instead of diagrams, which is described by, for instance, MacLane and Moerdijk [MM92]. If C is a small category, then the Yoneda embedding y is a full and faithful functor from C to the cocomplete category Set C op of presheaves. Every presheaf P can be expressed as a colimit of representable presheaves (i.e. those of the form y(X) as follows. Associated with P is its category ....

MacLane, S. and Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.

....on A of norm 1, then the restriction F jM of F to M is also a linear functional of norm 1. The Hahn Banach theorem says that the restriction function F 7 Gamma F jM is surjective. For T 1 spaces, surjectivity on points follows from formal injectivity (see for instance MacLane and Moerdijk [10]) Classically, the T 1 property on the space of functionals follows easily from the fact that R is Hausdorff. The localic Hahn Banach theorem is then formulated as formal injectivity: THEOREM 6.1 Let M be a linear subspace of the linear space A, w an element and U a subset of the base, both ....

S. MacLane, L. Moerdijk. "Sheaves in Geometry and Logic : A First Introduction to Topos Theory", Springer-Verlag, New York, 1992.

....such as that of real numbers, it is very difficult to get ones hands on any but a small fraction of their members. The categorical approach is that sets are defined by the relations between them, namely the functions, and this view has been strengthened by the success of topos theory. The book [14] is a good introduction to topos theory for those with a foundation in category theory. For an article relating the history of topos theory to notions of the foundations of sets, see [15] The author emphasises that the notion of topos was defined by Grothendieck as a replacement for the notion ....

....of membership is not the primary aspect. In the case of graphs, the elements have to be the vertices, but these capture only a small part of the structure. For more information on this approach in graph theory, see [12] while for the general body of theory, see the book by Mac Lane and Moerdijk [14]. The exponential law in DG has a number of consequences. One is that there is a composition morphism DIGRPH(B;C) Theta DIGRPH(A;B) DIGRPH(A;C) which is associative and with identity. Hence END(A) DIGRPH(A;A) has the structure of both a monoid and a graph. In the category of sets, monoids ....

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S.MacLane and I.Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, Springer-Verlag, 1992.

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S. MacLane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, 1992.

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McLane, S. and I. Moerdijk, "Sheaves in Geometry and Logic: A First Introduction to Topos Theory," Springer, 1994.

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S. McLane and I. Moerdijk, "Sheaves in Geometry and Logic: A First Introduction to Topos Theory," Springer, 1994.

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