In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a comma category (a special case being a slice category) is a construction 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, ca ...
. It provides another way of looking at
morphisms: instead of simply relating objects of a
category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by
F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
s and
colimits. The name comes from the notation originally used by Lawvere, which involved the
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13).
Definition
The most general comma category construction involves two
functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general.
General form
Suppose that
,
, and
are categories, and
and
(for source and target) are
functors:
We can form the comma category
as follows:
*The objects are all triples
with
an object in
,
an object in
, and
a morphism in
.
*The morphisms from
to
are all pairs
where
and
are morphisms in
and
respectively, such that the following diagram
commutes:
Morphisms are composed by taking
to be
, whenever the latter expression is defined. The identity morphism on an object
is
.
Slice category
The first special case occurs when
, the functor
is the
identity functor, and
(the category with one object
and one morphism). Then
for some object
in
.
In this case, the comma category is written
, and is often called the ''slice category'' over
or the category of ''objects over
''. The objects
can be simplified to pairs
, where
. Sometimes,
is denoted by
. A morphism
from
to
in the slice category can then be simplified to an arrow
making the following diagram commute:
Coslice category
The
dual concept to a slice category is a coslice category. Here,
,
has domain
and
is an identity functor.
In this case, the comma category is often written
, where
is the object of
selected by
. It is called the ''coslice category'' with respect to
, or the category of ''objects under
''. The objects are pairs
with
. Given
and
, a morphism in the coslice category is a map
making the following diagram commute:
Arrow category
and
are
identity functors on
(so
).
In this case, the comma category is the arrow category
. Its objects are the morphisms of
, and its morphisms are commuting squares in
.
Other variations
In the case of the slice or coslice category, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of
adjoint functors. For example, if
is the
forgetful functor mapping an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
to its
underlying set
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
, and
is some fixed
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(regarded as a functor from 1), then the comma category
has objects that are maps from
to a set underlying a group. This relates to the left adjoint of
, which is the functor that maps a set to the
free abelian group having that set as its basis. In particular, the
initial object of
is the canonical injection
, where
is the free group generated by
.
An object of
is called a ''morphism from
to
'' or a ''
-structured arrow with domain
''.
An object of
is called a ''morphism from
to
'' or a ''
-costructured arrow with codomain
''.
Another special case occurs when both
and
are functors with domain
. If
and
, then the comma category
, written
, is the
discrete category whose objects are morphisms from
to
.
An
inserter category In category theory, a branch of mathematics, the inserter category is a variation of the comma category where the two functors are required to have the same domain category.
Definition
If ''C'' and ''D'' are two categories and ''F'' and ''G'' are ...
is a (non-full) subcategory of the comma category where
and
are required. The comma category can also be seen as the inserter of
and
, where
and
are the two projection functors out of the
product category .
Properties
For each comma category there are forgetful functors from it.
* Domain functor,
, which maps:
** objects:
;
** morphisms:
;
* Codomain functor,
, which maps:
** objects:
;
** morphisms:
.
* Arrow functor,
, which maps:
** objects:
;
** morphisms:
;
Examples of use
Some notable categories
Several interesting categories have a natural definition in terms of comma categories.
* The category of
pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.
Maps between pointed sets (X, x_0) and (Y, y_0) – called based ...
s is a comma category,
with
being (a functor selecting) any
singleton set, and
(the identity functor of) the
category of sets. Each object of this category is a set, together with a function selecting some element of the set: the "basepoint". Morphisms are functions on sets which map basepoints to basepoints. In a similar fashion one can form the category of
pointed spaces
.
*The category of associative algebras over a ring
is the coslice category
, since any ring homomorphism
induces an associative
-algebra structure on
, and vice versa. Morphisms are then maps
that make the diagram commute.
* The category of
graphs is
, with
the functor taking a set
to
. The objects
then consist of two sets and a function;
is an indexing set,
is a set of nodes, and
chooses pairs of elements of
for each input from
. That is,
picks out certain edges from the set
of possible edges. A morphism in this category is made up of two functions, one on the indexing set and one on the node set. They must "agree" according to the general definition above, meaning that
must satisfy
. In other words, the edge corresponding to a certain element of the indexing set, when translated, must be the same as the edge for the translated index.
* Many "augmentation" or "labelling" operations can be expressed in terms of comma categories. Let
be the functor taking each graph to the set of its edges, and let
be (a functor selecting) some particular set: then
is the category of graphs whose edges are labelled by elements of
. This form of comma category is often called ''objects
-over
'' - closely related to the "objects over
" discussed above. Here, each object takes the form
, where
is a graph and
a function from the edges of
to
. The nodes of the graph could be labelled in essentially the same way.
* A category is said to be ''locally cartesian closed'' if every slice of it is
cartesian closed
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
(see above for the notion of ''slice''). Locally cartesian closed categories are the
classifying categories of
dependent type theories.
Limits and universal morphisms
Limits and
colimits
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
in comma categories may be "inherited". If
and
are
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
,
is a
continuous functor, and
is another functor (not necessarily continuous), then the comma category
produced is complete,
and the projection functors
and
are continuous. Similarly, if
and
are cocomplete, and
is
cocontinuous, then
is cocomplete, and the projection functors are cocontinuous.
For example, note that in the above construction of the category of graphs as a comma category, the category of sets is complete and cocomplete, and the identity functor is continuous and cocontinuous. Thus, the category of graphs is complete and cocomplete.
The notion of a
universal morphism
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently f ...
to a particular colimit, or from a limit, can be expressed in terms of a comma category. Essentially, we create a category whose objects are cones, and where the limiting cone is a
terminal object; then, each universal morphism for the limit is just the morphism to the terminal object. This works in the dual case, with a category of cocones having an initial object. For example, let
be a category with
the functor taking each object
to
and each arrow
to
. A universal morphism from
to
consists, by definition, of an object
and morphism
with the universal property that for any morphism
there is a unique morphism
with
. In other words, it is an object in the comma category
having a morphism to any other object in that category; it is initial. This serves to define the
coproduct in
, when it exists.
Adjunctions
Lawvere showed that the functors
and
are
adjoint if and only if the comma categories
and
, with
and
the identity functors on
and
respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of
. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.
Natural transformations
If the domains of
are equal, then the diagram which defines morphisms in
with
is identical to the diagram which defines a
natural transformation . The difference between the two notions is that a natural transformation is a particular collection of morphisms of type of the form
, while objects of the comma category contains ''all'' morphisms of type of such form. A functor to the comma category selects that particular collection of morphisms. This is described succinctly by an observation by S.A. Huq
that a natural transformation
, with
, corresponds to a functor
which maps each object
to
and maps each morphism
to
. This is a
bijective correspondence between natural transformations
and functors
which are
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of both forgetful functors from
.
References
*
*Lawvere, W (1963). "Functorial semantics of algebraic theories" and "Some algebraic problems in the context of functorial semantics of algebraic theories". http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf
External links
* J. Adamek, H. Herrlich, G. Stecker
Abstract and Concrete Categories-The Joy of CatsWildCatsis a category theory package for
Mathematica. Manipulation and visualization of objects,
morphisms, categories,
functors,
natural transformations,
universal properties.
Interactive Web pagewhich generates examples of categorical constructions in the category of finite sets.
{{DEFAULTSORT:Comma Category
Categories in category theory