Multicategory
In
Definition
A (non-symmetric) multicategory consists of
- a collection (often a proper class) of objects;
- for every finite sequenceof objects () and every object Y, a set of morphisms from to Y; and
- for every object X, a special identity morphism (with n = 1) from X to X.
Additionally, there are composition operations: Given a sequence of sequences of objects, a sequence of objects, and an object Z: if
- for each , fj is a morphism from to Yj; and
- g is a morphism from to Z:
then there is a composite morphism from to Z. This must satisfy certain axioms:
- If m = 1, Z = Y0, and g is the identity morphism for Y0, then g(f0) = f0;
- if for each , nj = 1, , and fj is the identity morphism for Yj, then ; and
- an associativitycondition: if for each and , is a morphism from to , then are identical morphisms from to Z.
Comcategories
A comcategory (co-multi-category) is a
where O% is the set of all finite ordered sequences of elements of O. The dual image of a multiarrow f may be summarized
A comcategory C also has a multiproduct with the usual character of a composition operation. C is said to be associative if there holds a multiproduct axiom in relation to this operator.
Any multicategory, symmetric or non-symmetric, together with a total-ordering of the object set, can be made into an equivalent comcategory.
A multiorder is a comcategory satisfying the following conditions.
- There is at most one multiarrow with given head and ground.
- Each object x has a unit multiarrow.
- A multiarrow is a unit if its ground has one entry.
Multiorders are a generalization of partial orders (posets), and were first introduced (in passing) by Tom Leinster.[1]
Examples
There is a multicategory whose objects are (small)
There is a multicategory whose objects are
More generally, given any monoidal category C, there is a multicategory whose objects are objects of C, where a morphism from the C-objects X1, X2, ..., and Xn to the C-object Y is a C-morphism from the monoidal product of X1, X2, ..., and Xn to Y.
An operad is a multicategory with one unique object; except in degenerate cases, such a multicategory does not come from a monoidal category.
Examples of multiorders include pointed multisets (sequence A262671 in the OEIS), integer partitions (sequence A063834 in the OEIS), and combinatory separations (sequence A269134 in the OEIS). The triangles (or compositions) of any multiorder are morphisms of a (not necessarily associative) category of contractions and a comcategory of decompositions. The contraction category for the multiorder of multimin partitions (sequence A255397 in the OEIS) is the simplest known category of multisets.[2]
Applications
Multicategories are often incorrectly considered to belong to higher category theory, as their original application was the observation that the operators and identities satisfied by higher categories are the objects and multiarrows of a multicategory. The study of n-categories was in turn motivated by applications in algebraic topology and attempts to describe the homotopy theory of higher dimensional manifolds. However it has mostly grown out of this motivation and is now also considered to be part of pure mathematics.[1]
The correspondence between contractions and decompositions of triangles in a multiorder allows one to construct an associative algebra called its incidence algebra. Any element that is nonzero on all unit arrows has a compositional inverse, and the Möbius function of a multiorder is defined as the compositional inverse of the zeta function (constant-one) in its incidence algebra.
History
Multicategories were first introduced under that name by
References
- ^ Bibcode:2004hohc.book.....L., Example 2.1.7, page 37
- ^ Wiseman, Gus. "Comcategories and Multiorders". Google Docs. Retrieved 9 May 2016.
- ISSN 0075-8434.
- Garner, Richard (2008). "Polycategories via pseudo-distributive laws". S2CID 17057235.