Koslowski, J. Monads and interpolads in bicategories. Theory Appl. Categ. 3, 8 (1997), 182212.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... complicated than would be necessary for the purely closed linear setting this is discussed in (Cockett Seely 1997a) The reader should be advised that our diagrams are to be read from the top down, in contrast to the string diagrams proposed by Joyal and Street and employed by the second author in (Koslowski 1997; Koslowski 1998) This paper is dedicated to Prof. Joachim Lambek, as part of the celebration of his 75th birthday. Jim Lambek has been a central gure in categorical proof theory from its beginning as an identi able eld of study. His 1958 paper (Lambek 1958) on the syntactic calculus, while ....

....# # # # ## L B(X;Y ) B(Y;Z) B(Z;W ) 1 ## 1 ## B(X;Y ) B(Y;W ) ## B(X;Z) B(Z;W ) B(X;W ) # # # # ## R These must satisfy several coherence conditions: The usual diagrams to make a bicategorical composition. These may be found in many references on bicategories; see (Koslowski 1997) for example. The usual diagrams to make a bicategorical composition. These are the analogous diagrams to those above. Cockett, Koslowski, Seely 10 The diagrams (as given in (Cockett Seely 1992) expressing the linear distributivity of the tensor over the cotensor. These may be found in ....

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J. Koslowski (1997) Monads and interpolads in bicategories. Theory and Applications of Categories 3 182-212.

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Koslowski, J. Monads and interpolads in bicategories. Theory Appl. Categ. 3, 8 (1997), 182212.

....their transformations , other possibilities should not be discounted. In particular, how do the poly functors and poly modules of [6] t into the picture In the quest for categories without identities , also called taxonomies , a 24 weakening of the notion of monad was proposed in [11]: an interpolad ha; i on A is an endo 1 cell A a A equipped with a multiplication aa a that is a coequalizer of a and a (and hence in particular associative) After replacing the monads S and T by interpolads, a super cartesian ST will still give rise to a composition of S T spans, ....

Koslowski, J., Monads and interpolads in bicategories, Theory Appl. Categ. 3 (1997), pp. 182-212.

....and their transformations , other possibilities should not be discounted. In particular, how do the poly functors and poly modules of [6] t into the picture In the quest for categories without identities , also called taxonomies , a weakening of the notion of monad was proposed in [9]: an interpolad ha; i on A is an endo 1 cell A a A equipped with a multiplication aa a that is a coequalizer of a and a (and hence in particular associative) After replacing S and T by interpolads, a strongly cartesian span ST will still give rise to a composition of S T spans, but ....

Koslowski, J., Monads and interpolads in bicategories, Theory Appl. Categ. 3 (1997), pp. 182-212.

.... T L 3 T ks R T T T T ks T 3 T T This is indeed the case, but crucially depends on the fth module equation. The essential associativity of both compositions follows in analogy to the case of modules in an ordinary bicategory, cf. e.g. [Koslowski 1997]. Thus we obtain two bicategory structures, Mod(B) and Mod(B) respectively. To de ne linear distributions M (M 0 M 00 ) 3 (M M ) M 00 for linear modules T M S M 0 R M 00 Q 19 we consider the following diagram: M S (M 0 M 00 ) ....

J. Koslowski (1997) Monads and interpolads in bicategories. Theory and Applications of Categories 3 182-212.

....are more endo Chu cells on x than those constrained by symmetry, and there are also Chu cells between di erent objects of V . A generalization of Barr s approach [4] connecting the bicategorical cyclic Chu construction with the construction of the bicategory of monads (or even interpolads, cf. [12]) and modules can be found in [13] Perhaps more surprising is the fact that the notion of Chu cell makes sense between arbitrary 1 cells of B . The (vertical) composition of such Chu cells can be viewd as a rather elegant process that eliminates all technical obscurity often associated with the ....

Koslowski, J. Monads and interpolads in bicategories. Theory Appl. Categ. 3, 8 (1997), 182-212.

....Definition 4.06. 4 The identity crisis The natural question, as to whether identities exist for the tree composition defined in Section 3, turns out to have a negative answer. Another example of a naturally defined composition operation that is associative but lacks identities is discussed in [7]. However, the problem of the missing identities can be traced to two related features of our set up. The corresponding changes, besides yielding identities, also lead to substantial simplifications. 4.00 Proposition. If A 2 (O O)tr is an identity for tree composition, then O = Proof: Let ....

Koslowski, J. Monads and interpolads in bicategories. preprint, Aug. 1996.

....denoted by A . C . 4 The identity crisis The natural question, as to whether identities exist for the tree composition defined in Section 3, turns out to have a negative answer. Another example of a naturally defined composition operation that is associative but lacks identities is discussed in [7]. However, the problem of the missing identities can be traced to a single feature of our set up. The corresponding change, besides yielding identities, also leads to substantial simplifications. 4.00 Proposition. If B 2 T (P P ) is an identity for tree composition, then P = Proof: Let P ....

Koslowski, J. Monads and interpolads in bicategories. preprint, Aug. 1996.

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