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K. Dosen

Publications and source records attributed to K. Dosen.

34 records · Page 2Linked to original sources
arXiv2007

Simplicial Endomorphisms

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in Temperley-Lieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and self-contained manner.

math.GT↗
arXiv2007

Bicartesian Coherence

Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality. (A proof of coherence for categories with binary products and sums simpler than that presented in this paper may be found in Section 9.4 of {\it Proof-Theoretical Coherence}, revised version of September 2007, http://www.mi.sanu.ac.yu/~kosta/coh.pdf.)

math.CT↗
arXiv2007

Relevant Categories and Partial Functions

A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination 'relevant' comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets, is a category that is relevant, but not cartesian closed.

math.CT↗
arXiv2007

Associativity as Commutativity

It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for symmetric or braided strictly monoidal categories, where associativity arrows are identities. Mac Lane's pentagonal coherence condition for associativity is decomposed into conditions concerning commutativity, among which we have a condition analogous to naturality and a degenerate case of Mac Lane's hexagonal condition for commutativity. This decomposition is analogous to the derivation of the Yang-Baxter equation from Mac Lane's hexagon and the naturality of commutativity. The pentagon is reduced to an inductive definition of a kind of commutativity.

math.CT↗
arXiv2007

Medial Commutativity

It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type ${(A\otimes B)\otimes(C\otimes D)\to(A\otimes C)\otimes(B\otimes D)}$, called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane's pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor $\otimes$. This preservation boils down to an isomorphic representation of the Yang-Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane's pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor $\otimes$. In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups $S_{n \choose i}$ for $0\leq i\leq n$.

math.CT↗
arXiv2006

Coherent Bicartesian and Sesquicartesian Categories

Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every bicartesian closed category, and, in particular, the category Set, is such a category.) This coherence amounts to the existence of a faithful functor from categories of this sort freely generated by sets of objects to the category of relations on finite ordinals, and it yields a very easy decision procedure for equality of arrows. Restricted coherence holds also for bicartesian categories where, in addition to this equality for projections, we have that the first and the second injection to the sum of the terminal object with itself are the same. The printed version of this paper (in: R. Kahle et al. eds, Proof Theory in Computer Science, Lecture Notes in Computer Science, vol. 2183, Springer, Berlin, 2001, pp. 78-92) and versions previously posted here purported to prove unrestricted coherence for the bicartesian categories mentioned above. Lemma 5.1 of these versions, on which the proof of coherence for sesquicartesian categories relied too, is however not correct. The present version of the paper differs from the previous ones also in terminology.

math.CT↗
arXiv2005

Symmetric Self-Adjunctions: A Justification of Brauer's Representation of Brauer's Algebras

A classic result of representation theory is Brauer's construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this paper is to provide a motivation for this result through the categorial notion of symmetric self-adjunction.

math.RT↗
arXiv2005

Coherence for Star-Autonomous Categories

This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. The proof of this theorem is based on a categorial cut-elimination result, which is presented in some detail.

math.CT↗
arXiv2005

Coherence of Proof-Net Categories

The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence theorem is also proved in the presence of arrows corresponding to the mix principle of linear logic. The notion of proof-net category catches the unit free fragment of the notion of star-autonomous category, a special kind of symmetric monoidal closed category.

math.CT↗
arXiv2005

Negation and Involutive Adjunction

This note analyzes in terms of categorial proof theory some standard assumptions about negation in the absence of any other connective. It is shown that the assumptions for an involutive negation, like classical negation, make a kind of adjoint situation, which is named involutive adjunction. The notion of involutive adjunction amounts in a precise sense to adjunction where an endofunctor is adjoint to itself.

math.LO↗
arXiv2005

Coherence and Confluence

Proofs of coherence in category theory, starting from Mac Lane's original proof of coherence for monoidal categories, are sometimes based on confluence techniques analogous to what one finds in the lambda calculus, or in term-rewriting systems in general. This applies to coherence results that assert that a category is a preorder, i.e. that ``all diagrams commute''. This note is about this analogy, paying particular attention to cases where the category for which coherence is proved is not a groupoid.

math.CT↗
arXiv2004

Identity of Proofs Based on Normalization and Generality

In general proof theory there are two approaches to the question of identity criteria for proofs. The first approach, which stems from Prawitz, Kreisel and Lambek, and is based on normalization of proofs, gives good results in intuitionistic, but not in classical logic. The second approach, which stems from Lambek, Mac Lane and Kelly, and is inspired by the generality of proofs, seems to be more promissing in classical logic.

math.LO↗
arXiv2003

A Note on the Set-Theoretic Representation of Arbitrary Lattices

Every lattice is isomorphic to a lattice whose elements are sets of sets, and whose operations are intersection and an operation extending the union of two sets of sets A and B by the set of all sets in which the intersection of an element of A and of an element of B is included. This representation spells out precisely Birkhoff's and Frinks's representation of arbitrary lattices, which is related to Stone's set-theoretic representation of distributive lattices.

math.LO↗
arXiv2001

Abstraction and Application in Adjunction

The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional abstraction and application to an argument in the postulates of the lambda calculus. Such an inversion principle arises also in two adjoint situations involving a cartesian closed category and its polynomial extension. Composing these two adjunctions, which stem from the deduction theorem of logic, produces the adjunction connecting product and exponentiation, i.e. conjunction and implication.

math.CT↗
arXiv2001

Kauffman Monoids

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.

math.GT↗