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Z. Petric

Publications and source records attributed to Z. Petric.

33 records · Page 2Linked to original sources
arXiv2008

Self-Adjunctions and Matrices

It is shown that the multiplicative monoids of Temperley-Lieb algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of the Brauer representation of the Brauer centralizer algebra.

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↗
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↗
arXiv2000

G-dinaturality

An extension of the notion of dinatural transformation is introduced in order to give a criterion for preservation of dinaturality under composition. An example of an application is given by proving that all bicartesian closed canonical transformations are dinatural. An alternative sequent system for intuitionistic propositional logic is introduced as a device, and a cut elimination procedure is established for this system.

math.CT↗
arXiv2000

Coherence in Substructural Categories

It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with ``graphs'' (g-natural transformations), and corresponding morphism theorems are given as consequences. Using these results, some basic relations between the free categories of these classes are obtained.

math.CT↗