In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, monoidal functors are
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s between
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ''coherence maps''—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors
* The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible.
* The coherence maps of strong monoidal functors are invertible.
* The coherence maps of strict monoidal functors are identity maps.
Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.
Definition
Let
and
be monoidal categories. A lax monoidal functor from
to
(which may also just be called a monoidal functor) consists of a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
together with a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
:
from functors
to
and a morphism
:
,
called the coherence maps or structure morphisms, which are such that for every three objects
,
and
of
the diagrams
:,
: and
commute in the category
. Above, the various natural transformations denoted using
are parts of the monoidal structure on
and
.
Variants
* The dual of a monoidal functor is a comonoidal functor; it is a monoidal functor whose coherence maps are reversed. Comonoidal functors may also be called opmonoidal, colax monoidal, or oplax monoidal functors.
* A strong monoidal functor is a monoidal functor whose coherence maps
are invertible.
* A strict monoidal functor is a monoidal functor whose coherence maps are identities.
* A braided monoidal functor is a monoidal functor between
braided monoidal categories In mathematics, a ''commutativity constraint'' \gamma on a monoidal category ''\mathcal'' is a choice of isomorphism \gamma_ : A\otimes B \rightarrow B\otimes A for each pair of objects ''A'' and ''B'' which form a "natural family." In particu ...
(with braidings denoted
) such that the following diagram commutes for every pair of objects ''A'', ''B'' in
:
:
* A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are
symmetric monoidal categories.
Examples
* The underlying functor
from the category of abelian groups to the category of sets. In this case, the map
sends (a, b) to
; the map
sends
to 1.
* If
is a (commutative) ring, then the free functor
extends to a strongly monoidal functor
(and also
if
is commutative).
* If
is a homomorphism of commutative rings, then the restriction functor
is monoidal and the induction functor
is strongly monoidal.
* An important example of a symmetric monoidal functor is the mathematical model of
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
, which has been recently developed. Let
be the category of
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
s of ''n-1,n''-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension ''n'' is a symmetric monoidal functor
* The
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
functor is monoidal as
via the map