category of topological spaces

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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 ...
, the category of topological spaces, often denoted Top, is the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
whose
object Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
s are
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s and whose
morphism 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 mod ...
s are
continuous map In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s using the techniques of
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, categ ...
is known as categorical topology. N.B. Some authors use the name Top for the categories with
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
s, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces.

# As a concrete category

Like many categories, the category Top is a
concrete category 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 mod ...
, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
:''U'' : Top → Set to the
category of sets In the mathematical field of 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 foundatio ...
which assigns to each topological space the underlying set and to each continuous map the underlying function. The forgetful functor ''U'' has both a
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
:''D'' : Set → Top which equips a given set with the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is ...
, and a
right adjoint 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 m ...
:''I'' : Set → Top which equips a given set with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
. Both of these functors are, in fact, right inverses to ''U'' (meaning that ''UD'' and ''UI'' are equal to the identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Top is also ''fiber-complete'' meaning that the category of all topologies on a given set ''X'' (called the ''
fiber Fiber or fibre (from la, fibra, links=no) is a #Natural fibers, natural or Fiber#Artificial fibers, artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The stronge ...
'' of ''U'' above ''X'') forms a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
when ordered by inclusion. The
greatest element 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 ...
in this fiber is the discrete topology on ''X'', while the
least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, ...
is the indiscrete topology. Top is the model of what is called a topological category. These categories are characterized by the fact that every structured source $\left(X \to UA_i\right)_I$ has a unique initial lift $\left( A \to A_i\right)_I$. In Top the initial lift is obtained by placing the
initial topology In general topology In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including ...
on the source. Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).

# Limits and colimits

The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top. In fact, the forgetful functor ''U'' : Top → Set uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in Top are given by placing topologies on the corresponding (co)limits in Set. Specifically, if ''F'' is a
diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
in Top and (''L'', ''φ'' : ''L'' → ''F'') is a limit of ''UF'' in Set, the corresponding limit of ''F'' in Top is obtained by placing the
initial topology In general topology In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including ...
on (''L'', ''φ'' : ''L'' → ''F''). Dually, colimits in Top are obtained by placing the final topology on the corresponding colimits in Set. Unlike many ''algebraic'' categories, the forgetful functor ''U'' : Top → Set does not create or reflect limits since there will typically be non-universal
cones A cone is a three-dimensional space, three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the Apex (geometry), apex or vertex (geometry), vertex. A cone is ...
in Top covering universal cones in Set. Examples of limits and colimits in Top include: *The
empty set 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 ...
(considered as a topological space) is the
initial object In category theory, a branch of mathematics, an initial object of a category (mathematics), category is an object in such that for every object in , there exists precisely one morphism . The dual (category theory), dual notion is that of a t ...
of Top; any
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics) In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which t ...
topological space is a
terminal object 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 top ...
. There are thus no
zero object 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 top ...
s in Top. *The
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
in Top is given by the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
on the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
. The
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of Set (mathematics), sets and disjoint union (topology), of topological spaces, the free product of Group (mathematics), group ...
is given by the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of th ...
of topological spaces. *The equalizer of a pair of morphisms is given by placing the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relat ...
on the set-theoretic equalizer. Dually, the
coequalizer 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 to ...
is given by placing the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
on the set-theoretic coequalizer. *
Direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be Group (mathematics), groups, Ring (mathematics), rings, Vector spac ...
s and
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related mathematical object, objects, the precise gluing process being specified by morphisms between the objects. Thu ...
s are the set-theoretic limits with the final topology and
initial topology In general topology In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including ...
respectively. * Adjunction spaces are an example of pushouts in Top.

# Other properties

*The
monomorphism In the context of abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, fiel ...
s in Top are the
injective 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 ...
continuous maps, the
epimorphism 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 top ...
s are the
surjective 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 m ...
continuous maps, and the
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
s are the
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
s. *The extremal monomorphisms are (up to isomorphism) the subspace embeddings. In fact, in Top all extremal monomorphisms happen to satisfy the stronger property of being regular. *The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular. *The split monomorphisms are (essentially) the inclusions of retracts into their ambient space. *The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts. *There are no
zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Definitions Suppose C is a Category (mathematics), category, and ''f'' : ''X'' → ''Y'' ...
s in Top, and in particular the category is not preadditive. *Top is not cartesian closed (and therefore also not a
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category (mathematics), category that behaves like the category of Sheaf (mathematics), sheaves of Set (mathematics), sets on a topological space (or more generally: on a Site (math ...
) since it does not have
exponential object In mathematics, specifically in category theory, an exponential object or map object is the category theory, categorical generalization of a function space in set theory. Category (mathematics), Categories with all Product (category theory), fini ...
s for all spaces. When this feature is desired, one often restricts to the full subcategory of
compactly generated Hausdorff space In topology, a compactly generated space is a topological space whose topology is coherent topology, coherent with the family of all compact space, compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies th ...
s CGHaus or the category of compactly generated weak Hausdorff spaces. However, Top is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of
convergence space 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 mod ...
s.

# Relationships to other categories

*The category of pointed topological spaces Top is a
coslice category In mathematics, specifically 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 foundation ...
over Top. * The
homotopy category In mathematics, the homotopy category is a category (mathematics), category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) cat ...
hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of Top. One can likewise form the pointed homotopy category hTop. *Top contains the important category Haus of Hausdorff spaces as a
full subcategory 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 m ...
. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with
dense Density (volumetric mass density or specific mass) is the substance's mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of ma ...
images An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
in their
codomain In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
s, so that epimorphisms need not be
surjective 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 m ...
. *Top contains the full subcategory CGHaus of
compactly generated Hausdorff space In topology, a compactly generated space is a topological space whose topology is coherent topology, coherent with the family of all compact space, compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies th ...
s, which has the important property of being a
Cartesian closed category In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...
while still containing all of the typical spaces of interest. This makes CGHaus a particularly ''convenient category of topological spaces'' that is often used in place of Top. * The forgetful functor to Set has both a left and a right adjoint, as described above in the concrete category section. * There is a functor to the category of locales Loc sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of
sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every irreducible closed subset h ...
s and spatial locales. *The homotopy hypothesis relates Top with ∞Grpd, the category of ∞-groupoids. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo weak homotopy equivalence.

* * * * *

# References

* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990)
''Abstract and Concrete Categories''
(4.2MB PDF). Originally publ. John Wiley & Sons. . (now free on-line edition). * * * * Herrlich, Horst:
Topologische Reflexionen und Coreflexionen
'. Springer Lecture Notes in Mathematics 78 (1968). * Herrlich, Horst: ''Categorical topology 1971–1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383. * Herrlich, Horst & Strecker, George E.
Categorical Topology – its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971
In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255–341. {{refend
Topological spaces 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 mod ...
General topology