S. E. Crans. Pasting Schemes for the monoidal biclosed structure on !-cat. PhD thesis, 1995. Home/Search Document Details and Download
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Cubical Sets are Generalized Transition Systems - Goubault (2001) (1 citation) (Correct)
....precubical sets from the underlying precubical sets, and f n 1 ffi ffl j = ffl j ffi fn (for all n 2 IN, 0 i n) The corresponding category of cubical sets, Upsilon , is isomorphic to the category of presheaves op Set over a small category . This latter can be described in a nice way, see [5]. Therefore, similarly to the case of the category of precubical sets, the category of cubical sets is an elementary topos, which is complete and cocomplete. We do not talk about cubical sets with connections and compositions here [3] but they have a great interest for our purposes, see for ....
S. E. Crans. Pasting Schemes for the monoidal biclosed structure on !-cat. PhD thesis, 1995.
Homotopy Invariants of Higher Dimensional Categories and.. - Gaucher (1999) (12 citations) (Correct)
....j ff i Gamma1 for i j 6 n and ff 2 f Gamma; g 5. ff i ffl i = Id The corresponding category of cubical sets, with an obvious definition of its morphisms, is isomorphic to the category of presheaves Sets op over a small category . This latter can be described in a nice way as follows [Cra95]. The objects of are the sets n = f1; ng where n is a natural number greater or equal than 1 and an arrow f from n to m is a function f from m to n [ f Gamma; g such that f (k) 6 f (k 0 ) 2 n implies k 6 k 0 and f (k) f (k 0 ) 2 n implies k = k 0 . 4 fi v 0 ....
....1 6 i 6 n and for ff = Sigma defined by ( ff i ) l) l if l i, ff i ) l) ff for l = i and ( ff i ) l) l Gamma 1 for l i. Then any morphism of is a composition of ffl i and of ff i . And Q is the unique functor which maps ffl i to ffl i and ff i to ff i [Cra95]. This way, the notion of category can be understood as a generalization of the notion of cubical set. Every cubical set can be seen as an category. The converse is false. We will see in Section 8 why this categoric setting is very well adapted for the development of an analogue of algebraic ....
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Sjoerd E. Crans. Pasting schemes for the monoidal biclosed structure on !cat. Utrecht University, April 1995.
From Concurrency to Algebraic Topology - Gaucher (2000) (3 citations) (Correct)
....construct the free category generated by a cubical set. A cubical set K is indeed a set valued presheaf over some small category whose objects are natural numbers and whose morphisms encode the axioms of cubical sets (exactly in the same way that the small category does for simplicial sets) [7] [16] It is a general fact that a cubical set K is in a canonical way the direct limit of its cubes : K = R n2 K n : n) where the integral sign is the coend construction [19] and K n : n) means the sum of cardinal of K n copies of ( n) It then suces to paste the categories ....
Crans, S., Pasting schemes for the monoidal biclosed structure on !cat (1995), Utrecht University.
Combinatorics Of Branchings In Higher Dimensional Automata - Gaucher (1999) (5 citations) (Correct)
....X and Y are two elements of I n such that t p (X) s p (Y ) for some p, then X[Y 2 I n and X [ Y = X p Y . Moreover, all elements X of I n satisfy the equality X = R(X) The elements of I n correspond to the loop free well formed sub pasting schemes of the pasting scheme n [Joh89] [Cra95] or to the molecules of an complex in the sense of [Ste98] The condition X n Y exists if and only if X Y = t n X = s n Y of [Ste98] is not necessary here because the situation of [Ste98] Figure 2 cannot appear in a composable pasting scheme. The map which sends every category C to N ....
S.E. Crans. Pasting schemes for the monoidal biclosed structure on !cat. Utrecht University, April 1995.
Homotopy Invariants of Multiple Categories and Concurrency in.. - Gaucher (1999) (Correct)
.... exists y such that z = b y (x) and the set E i j of couples (x; z) in (I n ) i Theta (I n ) j such that there exists y such that z = e y (x) Then I n is a loop free well formed pasting scheme (we do not know who proved this fact for the first time : the reader can find some details in (Crans, 1995) and an analoguous result for simplices in (Street, 1987) By abuse of notation, we will denote in the same way the pasting scheme I n and the generated category. The operations s i and t i are those above defined and the composition is the union of sets. The category generated by a 2 cube ....
Crans, S. E. (1995). Pasting schemes for the monoidal biclosed structure on !cat. Utrecht University.
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