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(2014-11-24)  
Describing objects externally.

Many interesting mathematical patterns are based onrelationships between objects rather than whatever concrete meaning isfound inside  those objects.

Category theory  focuses on this external  viewpoint,illustrated by the description ofsets solely in terms of thefunctions between them  (such isthe prototypical example of a category, called  Set,  presentedbelow).

Category theory is an enlightening way to describe mathematical structures. Over its first 70 years of existence, it has proved very useful forformulating the general concepts worth studying. A more controversial  and problematic aspectis the effort to turn category theory  into an axiomatic alternative to set theory  as the logical foundation of all mathematics...

Definition of a category :

A category  consists of twoclasses respectively dubbed objects  and arrows (or morphisms )  obeying the four postulates below. (Traditionally, the names of objects are in UPPERcase; arrow names are in lowercase.)

• Any arrow f  goes from a source  object Ato a target  object B. (If  f  is called a morphism, A is its domain  and B is its codomain.)

A		f		B

• A well-defined composition  operator exists among arrows: For every arrow f  from A to B andevery arrow g  from B to C, their composition g  f (pronounce "g  after f ")  is an arrow  from A to C.

A		f		B
					g
		gf
				C

• The composition of arrows isassociative:  h  (g  f )  =  (h  g)  f 

A	f		B
						hg
	gf				g
			C				D
				h

• For any object B, there's an identity arrow  1_B  from B to B  such that:
  • 1_B  f  = f  for every arrow f  of target B.
  • g  1_B  = g  for every arrow g  of source B.
  Such an identity is necessarily unique  (the proof is an easy exercise).

In a category C,  the class of all morphism from object  X  to object  Y is best denoted  C(X,Y) .  Other possible notations include:

hom_C (X.Y)      hom (X.Y)      Mor_C (X.Y)      Mor (X.Y)

Such a thing is sometimes called an hom-set, although it's not always a set  (it could be a proper class). When all of those are indeed sets  the category is said tobe locally small.

Diagrams and commuting  diagrams :

A triangle formed by three objects (vertices) and three arrows (edges)is said to commute when the arrow going from the double source to the double targetis actually thecomposition of the other two. Our last postulate about the existence of an identity arrow for every object  B could thus be expressed by stating that the following two triangles commute :

A	f		B		B
				1B	1B			g
	f
			B		B				C
							g

More generally, a diagram is said to commute  whenthe compositions of displayed arrows along two distinct pathssharing the same origin and the same destination are always equal. The simplest examples are just terser versions of the previous triangular diagrams:

A	f		B		B		g		C
				1B	1B

Beyond introductory material  like the above, the loops corresponding to identity arrows are almost never displayed. They're just always understood to be there.  Their "trivial" presencecan neither impede nor facilitate the commuting of a diagram.

All the diagrams we've drawn so far have been commuting ones,but a diagram can be drawn and discussed even when it's notknown a priori  to commute (one purpose of such a discussion might be to show a posteriorithat the displayed diagram does commute).

(2015-01-11)  
The simplest structures satisfying the category axioms.

Arguably, the simplest conceptual example of a category is the Set category  (seenext section). Unfortunately, it can be an intimidating example because its objects are so numerous thatthey don't even form a set  (there's no such thing as a set of all sets). Familiarity with the basic concepts described so far can be gained with a few categorieswhich have only finitely many objects and finitely many arrows...

The simplest category is the empty category or zero category,  denoted 0  (bold zero).  It has no objects and no arrows.

The categories 1  (one) consists of one object and one identity arrow. Likewise, the category 2 (two)  has two objects and a total of three arrows:

	A		A		f			B
		1A	1A						1B

Both of those diagrams show all  objects and arrows. Beyond this point, we'll no longer show the identities.  The category 3  (three)  has three objects and six arrows but we only showthe three that are not identities:

A		f		B
					g
		gf
				C

(2019-01-11) 
An object which is both initial and terminal is called a zero object.

In a given category C,  an object  I  is said to be an initial object  when,  for any object  X there is a unique morphism from  I  to  X.

Likewise,  an object  T  is said to be a terminal object  when,  for any object  X there is a unique morphism from  X  to  T.

When an object is both initial and terminal, it's called a zero object  (or null object). A category with such an object is called a pointed category.

(2014-11-25)  
The objects  are sets.  The morphisms  are thefunctions between sets.

The abstract algebra of total functions thus defines the category ofsets. The notion of cardinality emerges  (whereas membership is obfuscated).

The composition of twofunctions f  and g,denoted f  g (and commonlypronounced"f  after g ")  is the function defined by the equation:

f  g (x)   =  f ( g (x) )

As a total function is vacuously defined from the empty set to any set, the empty set is an initial object (or universal object)  of Set. The singletons are terminal objects.  There are no zero objects.

(2014-11-26)  
The objects are sets and the arrows are relations  between them.

By definition,  a relation  between two sets is a part of theircartesianproduct.  The composition of two relations is the relation whichcontains  (x,z)  if and only if there is an element  y such that  (x,y)  is in the first relation and  (y,z) in the second.

Sets and relations form a large category, denoted Rel, of which theprevious category of sets and functions (Set)  is just a subcategory.

Rel  is a pointed category whose zero object is the empty set (which is the only initial object and the only terminal object).

(2014-11-24)  
Categories, large and small, abstract or concrete.

A category is said to be small  when its objects and its arrowsboth form sets.  Conversely, when either type of constituents form a properclass, a category is said to be a large  category. For example,  the Set  category is large, becausethe collection of all sets is a proper class  (not a set).

A large category is said to be locally small  when, for any pairof its objects  X  and  Y, the morphisms from  X  to  Y form a set; hom (X,Y).

A large category can be neither a member of a class nor a component (i.e., object or arrow)  of a category.

To prevent foundational queezes, we could only considersmall categories and a finite number  of large ones, includingthe well-established large categories listed below (with a standard name, normally capitalized and printed in bold type). Feel free to add your own...

A Selection of Large Categories

Symbol	Meaning	Objects	Arrows
Set	category of all sets	all sets	all [total]functions
Rel	relations	all sets	allrelations
Sym	symmetric relations	all sets	allsymmetric relations
Gunk	poset of all sets	all sets	inclusions between sets
FinSet	all finite sets	finite sets	functions
Smgrp	all semigroups	semigroups	semigroup homomorphisms
Mon	all monoids	monoids	monoid homomorphisms
CMon	all commutative monoids	monoids	monoid homomorphisms
Grp	all groups	groups	group homomorphisms
LieGrp	Lie groups	groups	smoothhomomorphisms
Ab	allAbelian groups	groups	group homomorphisms
Rng	allunital rings	rings	ring homomorphisms
CRng	commutative unital rings	rings	ring homomorphisms
Grph	all directed graphs	graphs	graph homomorphisms
Pos	all partially ordered sets	posets	monotone maps
JPos	all partially ordered sets	posets	join-preserving maps
Lat	all lattices	lattices	lattice homomorphisms
DLat	alldistributive lattices	d. lattices	lattice homomorphisms
Bool	allBoolean algebras	b. lattices	b.homomorphisms
Heyt	allHeyting algebras	h. lattices	Heyting morphisms
Top	topological spaces	top. spaces	continuous maps
		c. H. spaces	continuous maps
hTop	topological spaces	top. spaces	homotopy classes
Unif	alluniform spaces	spaces	uniform maps
Met	allmetric spaces	metric spaces	nonexpansive maps
Metu	allmetric spaces	metric spaces	uniformly continuous maps
Diff	differentiable manifolds	manifolds	smooth maps
Euclid	all Euclidean spaces	Euclidean spaces	orthogonal maps
Hilb	all Hilbert spaces	Hilbert spaces	unitary maps
VectK	vector spaces overK	linear spaces	linear maps
Vec	vector spaces over	vector spaces	linear maps
	finite-dimensional spaces	linear spaces	linear maps
TVect	topological vector spaces	linear spaces	continuous linear maps
TVectK	top. v. spaces over K	linear spaces	continuous linear maps
FinVec	finite vector spaces	Galois spaces	linear maps
LCA	Locally compact Abelian groups		group homomorphisms
Cat	allsmall categories	categories	functors
Adj	all small categories	categories	adjunctions

A category is a subcategory  of another whenall the objects  (resp. arrows)  of the formerare objects  (resp. arrows)  of the latter. For example, Set  is apropersubcategory of Rel  (since all functions are relations). So is Sym.

By definition,  a concrete  category is a subcategory of Set. Neither Rel  nor Sym  are concrete categories.

(2019-04-21)  
Direct product, arrow category, opposite of a category.

The direct product  of two categories is composedof objects which are ordered pairs of objects and arrowswhich are ordered pairs of arrows. In either case, the first component is from the first category andthe second component is from the second one.

The arrow category  of a category C is the category whose objects are arrows of C and whose arrows are commuting squares in C.

(2014-11-26)  
Homomorphisms between categories.

A functor from a category to another maps objects and arrows of the firstrespectively to objects and arrows of the second, while preservingdomains and codomains, identities and composition of arrows.

(2014-11-26)  
Category of all small  categories and functors.

Cat  is the category whose objects are small categories and whosearrows  (morphisms)  are functors between them. This is not a small category  (otherwise it would be an object of itself).

(2014-11-26)     (Eilenberg  &  Mac Lane, 1942)
Natural transformations  are homomorphisms between functors.

I did not invent category theory to talk aboutfunctors.
I invented it to talk about natural transformations.
 Saunders Mac Lane (1909-2005) 

The category of functors  from C to D,  written as Fun(C, D), Funct(C,D) or D^C is defined as the category havingas objects the covariant functors from C to D, and as arrows the natural transformations.

(2014-11-28)  
Opposite of a category.

The opposite of a category C  is the category C^op whose objects are the same as C  and whose arrows are oppositesof the arrows of C  (the opposite of an arrow from A to B is an arrowfrom B to A).

(2014-11-26)  
Isomorphisms are invertible arrows.  In a groupoid,  that's all there is.

In a category, an arrow  (morphism) f from A to B is said to be an isomorphism  if thereis an arrow g  from B to A such that:

g  f   =   1_A
f  g   =   1_B

For example, in theSet category, the isomorphismsare thebijections.

In amonoid categoryM (corresponding to a single-object category whose morphisms are labeled with the elements ofthe monoid)  the isomorphisms aresimply theinvertible elements (which form the groupM*).

A category whose arrows are all isomorphisms is called a groupoid. (Some authors use the word groupoid to denote a magma.  I don't.)

(2014-11-27)     (Mac Lane, 1949)
A categorial  construction defined "up to isomorphism" among objects.

Historically, this was the first example of a universal mapping property, characterizing a unique kind of equivalent objectsin term of all possible morphisms between objects in a given category...

The object  X  is a product  of two objects A₁  and  A₂  when there are two morphisms (called canonical projections) p₁  and  p₂  of source  X and of respective targets  A₁  and  A₂  such that,for any domain Y and any two morphisms of source Y and respective targets A₁  and  A₂ ,   there is a unique morphism f  from Y to X which makes this diagram commute:

			Y
	f1				f2
				f
A1			X			A2
	p1			p2

In this, the convention is used that a dotted  line indicates uniqueness (i.e.,  no other arrow has the same source and target as a dotted arrow).

When an object  X  is a product, we observe that  1_X must be the only  arrow from  X  to  X. (:  Consider  Y = X.)

As is usual with constructions based on such a universal mapping property, the above product  X  may not be uniquely defined, but if there is anothersatisfactory object  X'  with the same property, then there's an isomorphism  between  X  and  X'. Indeed, consider the counterpart of the above for  X', which is a commuting diagram valid for any choice of  Z, g₁  and g₂ :

			Z
	g1				g2
				g
A1			X'			A2
	p1'			p2'

We may choose  Z = X, g₁ =p₁  and g₂ =p₂  which establishes g as the  unique  arrow from  X  to  X'. Likewise, in the previous diagram, choosing  Y = X', f₁ =p₁'  and f₂ =p₂'  establishes f as the unique  arrow from  X'  to  X. Our preliminaryremark  then implies that:

   g  f   =   1_X'      and      f  g   =   1_X   

Conversely, in a category where  X  is a product of A₁  and  A₂  so is  X'  whenever there isa unique  isomorphism from  X  to  X'. (The straightforward proof is left to the reader.)

Coproducts :

Coproducts are products defined in the opposite  category.

(2019-04-11)  
Construction of universal maps.

(2015-01-12)  
In a skeletal  category, isomorphic objects are identical.

A small category is said to be acyclic  when its identitiesare the only isomorphisms and also the only morphisms from an object to itself.

(2015-01-20)  
Building on products of objects  and products of arrows.

Exponential objects are the counterparts of a function spaces  in set theory.

Product of arrows :

(2015-01-20)  

A category is said to be cartesian-closed  when:

• Two objects A and B always have aproduct AB 
• Two objects A and B always have an exponential B^A.
• There is a unique terminal object  1
  (i.e.,  for any object A,  there's a unique  arrow from A to 1 ).

(2014-12-15)  

(2014-12-15)  

(2014-11-26)    (Yoneda, 1954)
Every category can be embedded in a functor category.

That's to categories what Cayley's group theorem is to groups...

(2015-01-12)     (Daniel Kan, 1958)

(2015-10-30)     (Roger Godement, 1958)

(2014-11-27)     (Lawvere, 1963)
Examples:  Arrow category.  Slice or coslice with respect to an object.

Slice and Coslice

The slice  of a category C  with respect toone of its objects  A  is the category denoted C/A whose objects are the arrows of C  whose targets are equal to  A and whose arrows are

(2014-12-02)  

Left-adjoints preserve colimits, right-adjoints preserve limits.

(2014-12-03)    (Grothendieck)
Categories whose properties make them similar to the Set  category.

An elementary topos  is aCartesian-closed category  with[well-defined coproduct and]

• a terminal object and pullbacks,
• an initial object  (0)  and pushouts,
• a subobject classifier.

In such a structure, equivalents can be found to the characteristic function of a setand to logical quantifiers.

(2014-11-28)  
Higher-order categories.

(2014-12-15)  
Examples include completions, quotient extensions and modifications.

(2015-01-22)  
Calculus of fractions.

(2019-04-19) 
A monomorphism  is to a morphism  what an injection is to a function.

In Set Theory,  it's the existence of an injectionfrom one set to another which establishes the fundamental hierarchy underlying the cardinalities  of sets.

(2019-04-11) 
Definitions:

• A morphism f  is a constant morphism (or left-zero morphism) when  g h  f  g  = f  h   [whenever both sides make sense].
• A morphism f  is a coconstant morphism (or right-zero morphism) when  g h  g  f  = h  f   [whenever both sides make sense].
• A zero morphism  is both constant and coconstant.

(2019-04-18) 
A monomorphism  is a morphism with a trivial  kernel.

(2014-12-29)    (1955, 1957)
Categories resembling Ab (the category ofAbelian groups).

The name of the concept and/or early attempts at defining it can be tracedto  Saunders Mac Lane  (1948)  and to two doctoralstudents ofEilenberg, namely Alex Heller  (1950)  and David Buchsbaum  (1954). This was neatly finalized by Grothendieck  in 1957.

In an abelian category where f g  is zero, the cohomology object  is the kernel of f modulo the image of g.

(2020-05-24) 

(2020-05-24) 
How exact sequences  can be defined without kernels or co-kernels.

(2020-05-23) 
The image of one morphism is the kernel of the next.

(2020-05-23) 
On the degree to which a sequence of morphisms fails to be exact.

(2020-05-25) 

Tannaka-Krein duality  generalizes Pontryagin duality to noncommutative compact groups. 

(2020-05-25) 
Theory of non-linear differential equations.

(2014-12-04)  
Which one would provide the best foundation for Mathematics?

Categoricians have, in their everyday work, a clear view of what could lead tocontradiction, and [they] know how to build ad hoc safeguards.
Jean Bénabou  (1932-2022) 1985.

BesidesJean Bénabou(1932-2022)  one of the few opponents of set-based Bourbakism  within French Academia was Roger Apéry  (1916-1994) who was also an early advocate of category theory.

As sets are just the objects of a specific category, it can be tempting to view categories as more fundamental than sets themselves.

In the late 1950's, the Bourbaki group pondered that fact, halfway throughits monumental work of describing much of mathematics in terms of set theory. They considered the possibility of adopting the categorial viewpoint instead,at the great cost of rewriting previously published work andjeopardizing the entire project by diverting the energies of the participants.

In a2014 video, Pierre Cartier (1932-1924, Ulm 1950) reveals how that internal Bourbaki debate was doused for pragmatic reasons,against the wishes of the radical "idealists" led byGrothendieck, Lang,Chevalley and Sergent.

Cartier doesn't say which side of the fenceEilenberg was on, possibly because Eilenberg wasrarely in France at the time...

Charles Ehresmann  (who developed his own flavor of category theory after 1957) had left Bourbaki in 1950, for obscure reasons (his second wife,  the categorician Andrée Ehresmann,  is on recordas stating that the Bourbaki project had already lost much of its original appeal by that time).

(2021-09-01)  
On the foundations of physics in terms of category theory.