Monoidal categories

Category theory



Recall that a category consists of a collection of morphisms each having a single object as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A multicategory is like a category, except that one allows multiple inputs and a single output.

Another term for multicategory is coloured operad.


In components

A multicategory CC consists of

  • A collection of objects, C 0C_0.
  • A collection of multimorphisms, C 1C_1.
  • A source map s:C 1(C 0)*s: C_1 \to (C_0)* to the collection of finite, possibly empty lists of objects (where (C 0)*(C_0)* is the free monoid generated by C 0C_0), and a target map t:C 1C 0t: C_1 \to C_0. We write f:c 1,,c ncf: c_1, \ldots, c_n \to c to indicate the source and target of a multimorphism ff.
  • Identity and composition laws. The identity law is a map 1 :C 0C 11_{-}: C_0 \to C_1 where 1 c:cc1_c: c \to c. The composition law assigns, to each f:c 1,,c ncf: c_1, \ldots, c_n \to c together with an nn-tuple f i:c ic i:i=1,,n\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle, a composite
    f(f 1,,f n):c 1,,c ncf \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c

    where the source is obtained by concatenating lists in the evident way.

These operations are subject to associativity and identity axioms which the reader can probably figure out, but see for example (Leinster, page 35 ff.), for details.

Many people (especially non-category theorists) use the word multicategory or the word colored operad to mean what we would call a symmetric multicategory / symmetric operad. These are multicategories equipped with an action of the symmetric group S nS_n on the multimorphisms c 1,,c ncc_1, \ldots, c_n \to c such that the composition is equivariant with respect to these actions.

In terms of cartesian monads

An efficient abstract method for defining multicategories and related structures is through the formalism of cartesian monads. For ordinary categories, one uses the identity monad on Set; for ordinary multicategories, one uses the free monoid monad ()*:SetSet(-)*: Set \to Set. This is a special case of the yet more general notion of generalized multicategory.

We summarize here how the theory applies to the case of a cartesian monad TT on a category with pullbacks; see generalized multicategory for the fully general context.

  • First, a TT-span from XX to YY is a span pp from TXT X to YY, that is, a diagram
    TXp 1Pp 2YT X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y

    A TT-span is often written as p:XUnknown characterUnknown characterUnknown characterYp: X ⇸ Y.

When TT is the free monoid monad on SetSet, a TT-span from XX to itself is called a multigraph on XX.

  • TT-spans are the 1-cells of a bicategory. A 2-cell between TT-spans e,f:XUnknown characterUnknown characterUnknown characterYe, f: X ⇸ Y is a 2-cell between ordinary spans from TXT X to YY. To horizontally compose TT-spans e:XUnknown characterUnknown characterUnknown characterYe: X ⇸ Y and f:YUnknown characterUnknown characterUnknown characterZf: Y ⇸ Z, take the ordinary span composite of

    (TXmXT 2XTe 1TETe 2TY)(TYf 1Ff 2Z)(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)

    where m:T 2Tm: T^2 \to T is the monad multiplication. The identity TT-span from XX to itself is the span

    TXuXX1 XXT X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X

    where u:ITu: I \to T is the monad unit. The verification of the bicategory axioms uses the cartesianness of TT in concert with the corresponding axioms on the bicategory of spans.

  • A TT-multicategory is defined to be a monad in the bicategory of TT-spans.

When TT is the free monoid monad on sets, then a TT-multicategory is a multicategory as defined above. For more examples and generalizations, see generalized multicategory.


Relation to operads

A nonpermutative (or Stasheff-) operad in Set may be defined as an ordinary multicategory with exactly one object. Likewise, a symmetric operad in any symmetric monoidal category VV is equivalent to a VV-enriched multicategory with one object.

More generally, the notion of multi-colored planar operad is equivalent to that of multicategory, and the notion of multi-colored symmetric operad is equivalent to that of symmetric multicategory.

Fully generally, for each cartesian monad TT, there is a corresponding notion of TT-operad, namely a TT-multicategory whose underlying TT-span has the form 1Unknown characterUnknown characterUnknown character11 ⇸ 1.

For example, in Batanin’s approach to (weak) \infty-categories, a globular operad is a TT-operad, where TT is the free (strict) ω\omega-category monad on the category of globular sets.

Ordinary (permutative/symmetric) operads, and their generalization to symmetric multicategories, can also be treated in the framework of generalized multicategories, but they require a framework more general than that of cartesian monads.

Relation to monoidal categories

There is a faithful functor from monoidal categories to multicategories, given by forming represented multicategories. Conversely, to any multicategory CC there is an associated (strict) monoidal category F(C)F(C), whose objects (respectively, arrows) are lists of objects (respectively, arrows) of CC, and where the tensor product in F(C)F(C) is given by concatenation.

Examples and special cases

See also the examples at operad.


Revised on June 23, 2016 10:35:58 by David Tanzer? (