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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Category Theory  101

This page is dedicated to the memory of  Alexander Grothendieck  (1928-2014).
The great champion of category theory passed away on November 13, 2014.  RIP
Category theory makes no sense without some fairly 
detailed knowledge of groups, rings, and vector spaces.

Peter Cameron  (2010, paraphrased).
 Gerard P. Michon  border
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 Alexander Grothendieck 
Alexander Grothendieck

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Related Links (Outside this Site)

Gentle Introduction to Computational CT   by  Maarten M. Fokkinga  (1994).
CT Lecture Notes   by  Daniele Turi  (University of Edinburgh, 1996-2001).
Category Theory   Stanford Encyclopedia of Philosophy  (1996, 2014).
Categorification  by  John C. Baez  &  James Dolan  (1998).
An introduction to CT for Software Engineers  by  Steve Easterbrook  (1999).
Distributors at Work  [= profunctors]  by  Jean Bénabou   (June 2000).
Categorical Myths and Legends  (2001).  About category theorists.
Basic Category Theory   by  Jaap van Oosten  (2002).
Objects of Categories as Complex Numbers  by  Marcelo Fiorea  &  Tom Leinster  (2002).
Categories, Quantization, and Much More  by John Baez  (April 2006).
Euler characteristic of a category  by  Tom Leinster   (Oct. 2006).
Le langage des catégories (in French)  by  Mathieu Bélanger  (Nov. 2006).
Make category theory intuitive!   by  Jocelyn Ireson-Paine  (c. 2009).
Physics, Topology, Logic and Computation: A Rosetta Stone.   Baez  &  Stay.
Categories and Homological Algebra  (2011)   by  Pierre Schapira  (1943-).
Introduction à la théorie des catégories  (2011)   by  Stéphane Dugowson.
"CT: An abstract setting for analogy and comparison"   by  Ronald Brown & Timothy Porter.
Category Theory  by  P.T. Johnstone   (Cornell, 2011).
The Magnitude of Metric Spaces  by  Tom Leinster   (2012).
Haskell / Category theory  WikiBook  (2007-2014...)
Introduction to Category Theory   Graham Hutton  (U. of Nottingham, 2014).
CT Seminar on Unifying Mathematics  by  John Baez  (Fall 2015).
Blogs, Forums, Institutions :   Bartosz Milewski's Programming Café
Why Category Theory Matters  by  Robb Seaton.
Category Theory Research Center  (Montréal / McGill)
The n-Category CaféJohn Baez, David Corfield, Alexander Hoffnung et al.
The  n-categorical point of view  lives on at  nLab.

Wikipedia :   Algebraic topology   |   Homological algebra   |   Outline of category theory   |   Category theory   |   Morphism   |   Functor   |   Functor category   |   Categorification   |   Universal property   |   Forgetful functor   |   Initial and terminal objects   |   Flat module   |   Flat morphism   |   List of categories (draft)   |   nLab
Cartesian closed category   |   Categorical logic   |   Lambda calculus   |   PCF   |   Curry-Howard isomorphism
Pontryagin duality

Textbooks  &  Monographs   [ discussion ]
"Homological Algebra"  by  Henri Cartan  &  Samuel Eilenberg  (1956).
"Abelian Categories"  by  Peter Freyd  (1964).
"Categories for the working mathematician"   Saunders Mac Lane  (1971, 1998).
"Category Theory"  by  Horst Herrlich  &  George E. Strecker  (1973).
"Arrows, Structures and Functors"   Michael A. Arbib,  Ernest G. Manes  (1975).
"Topoi:  The Categorial Analysis of Logic"   Robert Goldblatt  (1979,1983,2006).
"Toposes, Triples and Theories"   Michael Barr, Charles Wells  (1985, 2005).
"Abstract and Concrete Categories"  J. Adámek, Horst Herrlich, George E. Strecker  (1990, 2004).
"Basic Category Theory for Computer Scientists"  by  Benjamin C. Pierce  (1991).
"An Introduction to Homological Algebra"  by  Charles A. Weibel  (1995).
"Category Theory for Computing Science"  by  Michael Barr & Charles Wells  (1998).
"Categories & Sheaves"   by  Masaki Kashiwara  &  Pierre Schapira  (2006).
"Category Theory"  by  Steve Awodey  (2006, 2010).  [review, June 2007]
"Tool & Object"  (history/philosophy of CT)   by  Ralf Krömer  (2007).
"Handbook of Categorical Algebra"  by  Francis Borceux  (V. 1, 2008) (V. 2, 1995) (v. 3, 1994).
"Conceptual Mathematics"   F. William Lawvere & Stephen H. Schanuel  (2009).
"Algebra: Chapter 0"  by  Paolo Aluffi  (2009).
"An Introduction to Category Theory"  by  Harold Simmons  (2011).
"Category Theory for Scientists"  by  David I. Spivak  (2013, 2014) [review, 2013]
"Basic Category Theory"   by  Tom Leinster  (2007 LN, 2014-09-22, 2016-12-30).
"Categorical Homotopy Theory"   by  Emily Riehl  (Cambridge, 2014)  pdf.
"Category Theory in Context"   by  Emily Riehl  (Aurora, 2016)  pdf.

Videos :   Marvels of Mathematics  by  Saunders Mac Lane  (1989).
Motivational clip  by  Richard Garner  (2011).
Category Theory Foundations (2012)  by Steve Awodey :   1 | 2 | 3 | 4
The Catsters  (Eugenia Cheng  &  Simon Willerton)  |  Catsters Guide  by  Brent Yorgey.
A categorical framework for the quantum harmonic oscillator  by  Jamie Vicary  (January 2008).
Using categories to understand programming  by  Alain Prouté  (2010-12-16).
Categories  &  string diagrams   by  Dominic Verity  (Sydney, 2011-06-06).
Free groups defined using CT  by  Michael L. Baker  (2012-04-05).
Greenhorn category theory  by  Tom LaGatta  (Meetup, 2014-03-11).
Category theory:  A framework for reasoning.   Michael L. Baker  (2015-01-31).
Categorification of Fourier Theory  by  Jacob Lurie  (Harvard, 2015-04-24).
Category Theory Lulz   by  Ken Scambler   (2015-05-06).
Lesson 1 (June 2015)  Martin J.M. Codrington   1 | 2 | 3 | 4 | 5 | 6   [ Ex. & Sol. ]
Introduction to Category Theory   by  Steven Roman   1 | 2 | 3 | 4 | 5 | 6 | ...

   Saunders Mac Lane
Saunders Mac Lane

Abstract Nonsense  &  Honest Proofs

Following  Robert Goldblatt  and others, "categorial" here means pertaining to Category theory  ("categorical" is for Aristotelian logic).

   Sammy Eilenberg
Samuel Eilenberg
Category theory originated
in two papers (1942,1945)
by Mac Lane & Eilenberg.

(2014-11-24)   Category Theory
Describing objects  externally.

Many interesting mathematical patterns are based on relationships between objects rather than whatever concrete meaning is found  inside  those objects.

Category theory  focuses on this  external  viewpoint, illustrated by the description of sets solely in terms of the functions between them  (such is the prototypical example of a category,  called  Set,  presented below).

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

Definition of a category :

category  consists of two classes  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 A to 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 and every arrow  g  from B to C, their composition  g o f  (pronounce  "g  after  f ")  is an arrow  from A to C.
A f B
  | se-arrow-butt | g
g o f  se-arrow-point
  right-arrow C
  • The composition of arrows is associative:   h o (g o f )   =   (h o go f 
A f B
| se-arrow-butt | se-arrow-butt  h o g
g o f  se-arrow-point gse-arrow-point
C right-arrow D
  • For any object B, there's an identity arrow  1B  from B to B  such that:
    • 1B o f  =  f   for every arrow  f  of target B.
    • g o 1B  =  g   for every arrow  g  of source B.
    Such an identity is necessarily unique  (the proof is an easy exercise).

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 target is actually the composition 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
| se-arrow-butt | 1B 1B | se-arrow-butt g
f  se-arrow-point se-arrow-point
B B right-arrow C

More generally, a diagram is said to  commute  when the compositions of displayed arrows along two distinct paths sharing 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
right-arrow right-arrow
  | 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" presence can neither impede nor facilitate the commuting of a diagram.

The founders of category theory  (Mac Lane and Eilenberg)  have stressed that the entire structure of a category resides in its arrows  (every object being adequately represented by its identity arrow).  There's actually no need to display both an object and its identity.

All the diagrams we've drawn so far have been commuting ones, but a diagram can be drawn and discussed even when it's not known  a priori  to commute  (one purpose of such a discussion might be to show  a posteriori that the displayed diagram does commute).

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

Arguably, the simplest conceptual example of a category is the  Set  category  (see next section).  Unfortunately, it can be an intimidating example because its objects are so numerous that they don't even form a set  (there no such thing as a set of  all sets).  Familiarity with the basic concepts described so far can be gained with a few categories which 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 show the three that are not identities:

A f B
  | se-arrow-butt | g
g o f  se-arrow-point
  right-arrow C

The Zeta Function of a Finite Category (2013)  by  Kazunori Noguchi.

(2014-11-25)   The  Set  category:
The  objects  are sets.  The  morphisms  are all the functions between sets.

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

The  composition  of two functions  f  and  g, denoted  f o g  (and commonly pronounced "f  after g ")  is the function defined by the equation:

f o g (x)   =   f ( g (x) )

 Come back later, we're
 still working on this one...

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

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

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

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

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

A large category is said to be  locally small  when, for any pair of its objects  X  and  Y,  the morphisms from  X  to  Y  form a  set,  denoted  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.

Some militant category theorists are satisfied to consider categories as "useful fictions" and cannot be bothered with fundamental considerations or logical paradoxes.  They will gladly consider monsters like the category of all categories where objects and/or arrows don't even form classes.  I beg to differ here.

To prevent foundational queezes, we could only consider small categories and a  finite number  of large ones, including the 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
Setcategory of all setsall setsall [total] functions
Relrelationsall setsall relations
Symsymmetric relationsall setsall symmetric relations
 poset of all setsall setsinclusions between sets
FinSetall finite setsfinite setsfunctions
Smgrpall semigroupssemigroupssemigroup homomorphisms
Monall monoidsmonoidsmonoid homomorphisms
CMonall commutative monoidsmonoidsmonoid homomorphisms
Grpall groupsgroupsgroup homomorphisms
Aball Abelian groupsgroupsgroup homomorphisms
Rngall unital ringsringsring homomorphisms
CRngcommutative unital ringsringsring homomorphisms
Grphall directed graphsgraphsgraph homomorphisms
Posall partially ordered setsposetsmonotone maps
JPosall partially ordered setsposetsjoin-preserving maps
Latall latticeslatticeslattice homomorphisms
DLatall distributive latticesd. latticeslattice homomorphisms
Boolall Boolean algebrasb. lattices b. homomorphisms
Heytall Heyting algebrash. lattices Heyting morphisms
Toptopological spacestop. spacescontinuous maps
HCompcompact Hausdorff spacesc. H. spacescontinuous maps
hToptopological spacestop. spaceshomotopy classes
Unifall uniform spacesspacesuniform maps
Metall metric spacesmetric spacesnonexpansive maps
Metuall metric spacesmetric spacesuniformly continuous maps
Diffdifferentiable manifoldsmanifoldssmooth maps
Euclidall Euclidean spacesEuclidean spacesorthogonal maps
Hilball Hilbert spacesHilbert spacesunitary maps
VectKvector spaces over Klinear spaceslinear maps
Vecvector spaces over R vector spaceslinear maps
 finite-dimensional spaceslinear spaceslinear maps
TVecttopological vector spaces linear spacescontinuous linear maps
TVectKtop. v. spaces over  Klinear spacescontinuous linear maps
FinVecfinite vector spacesGalois spaceslinear maps
LCALocally compact Abelian groupsgroup homomorphisms
Catall small categoriescategoriesfunctors
Adjall small categoriescategoriesadjunctions

A category is a  subcategory  of another when all the objects  (resp. arrows)  of the former are objects  (resp. arrows)  of the latter.  For example,  Set  is a proper subcategory 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.

Some small categories capture just one instance of a mathematical structure
any monoidonly one objectelements of the monoid
a partially ordered setelements of the posetat most one per pair of objects
a formal theoryformulasderivations (concatenated)

Mac Lane's Punchline
DC all functors from C to Dfunctorsnatural transformations

"Natural isomorphisms in group theory" by Eilenberg and Mac Lane (1942) Proc. Nat. Acad. Sci. U.S.A. v.28
General Theory of Natural Equivalences  by  Samuel Eilenberg & Saunders Mac Lane  (1945).
Rudolf Carnap's  "Logical Syntax of Language"  (1934).   Steve Awodey  (2005-10-11).
Category theory, yesterday, today and tomorrow. Colloquium Jean Bénabou (June 2011)
In French :   Jacques Riguet (1921-2013)  by  Stéphane Dugowson  (Dec. 2012).
Jean Bénabou (1932-, ENS 1952)  interviewed by  Stéphane Dugowson  (2012-09-07).
Andrée Ehresmann (né Bastiani, 1935-)  interviewed by  Stéphane Dugowson  (Dec. 2012).

(2014-11-26)   Functors
Homomorphisms between categories.

A functor from a category to another maps objects and arrows of the first respectively to objects and arrows of the second, while preserving domains and codomains, identities and composition of arrows.

 Come back later, we're
 still working on this one...

Wikipedia :   Functor

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

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

 Come back later, we're
 still working on this one...

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

I did not invent category theory to talk about functors.
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 DC is defined as the category having as objects the covariant functors from C to D, and as arrows the natural transformations.

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 still working on this one...

Wikipedia :   natural transformations

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

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

 Come back later, we're
 still working on this one...

(2014-11-26)   Isomorphisms  (or  equivalences )  in a category
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 there is an arrow  g  from B to A such that:

g o f   =   1A
f o g   =   1B

For example, in the Set category, the isomorphisms are the bijections.

In a monoid category M  (corresponding to a single-object category whose morphisms are labeled with the elements of the monoid)  the isomorphisms are simply the invertible elements  (which form the group M*).

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)   Categorial product of two objects   (Mac Lane, 1949)
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 objects in term of all possible morphims 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  A,   there is a  unique  morphism  f  from Y to X which makes this diagram commute:

| f1 sw-arrow-butt | se-arrow-butt f2
sw-arrow-point  fse-arrow-point
A1 left-arrow X right-arrow 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  1X  must be the  only  arrow from  X  to  X.  (HINT:  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 another satisfactory 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,  g and  g:

| g1 sw-arrow-butt | se-arrow-butt g2
sw-arrow-point  gse-arrow-point
A1 left-arrow X' right-arrow 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 preliminary remark  then implies that:

    g o f   =   1X'       and       f o g   =   1X     QED

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

Coproducts :

Coproducts are products defined in the  opposite  category.

 Come back later, we're
 still working on this one...

Wikipedia :   Product of two objects   |   Cartesian closed category   |   Exponential object

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

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

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 still working on this one...

Wikipedia :   Subcategory   |   Skeleton of a category

(2015-01-20)   Exponential Object   (map object)
Building on  products of objects  and  products of arrows.

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

Product of arrows :

 Come back later, we're
 still working on this one...

Exponential object

(2015-01-20)   Cartesian-closed categories   (CCC)

A category is said to be  cartesian-closed  when:

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

 Come back later, we're
 still working on this one...

Cartesian-closed category

(2014-12-15)   Category enriched over another

 Come back later, we're
 still working on this one...

Enriched categories

(2014-12-15)   Category of matrices

 Come back later, we're
 still working on this one...

Category of matrices  [ 1 | 2 | 3 | 4 ]  by  John Armstrong  ("The Unapologetic Mathematician").

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

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

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 still working on this one...

Wikipedia :   Functor category   |   Yoneda lemma   |   Nobuo Yoneda (1930-1996)

(2015-01-12)   Adjoint functors   (Daniel Kan, 1958)

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 still working on this one...

Wikipedia :   Adjoint functors   |   Daniel Kan (1927-2013)

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

 Come back later, we're
 still working on this one...

"Topologie Algébrique et Théorie des Faisceaux" (Hermann 1958, 1960)   by   Roger Godement (b. 1921).
Wikipedia :   Monads   |   Monads in functional programming

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

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 still working on this one...

Slice and Coslice

The  slice  of a category  C  with respect to one 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

Wikipedia :   Comma category   |   F.W. Lawvere (b. 1937)

(2014-12-02)   Limits and Colimits

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

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 still working on this one...

Wikipedia :   Limit   |   Pullback (cartesian square)   |   Cone

(2014-12-03)   The plural of  topos  is  topoi or  toposes  (Grothendieck)
Categories whose properties make them similar to the  Set  category.

An  elementary topos  is a Cartesian-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 set and to logical quantifiers.

 Come back later, we're
 still working on this one...

Topos Theory in a Nutshell  by  John Baez  (April 2006).
Wikipedia :   History of topos theory   |   Topos

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

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 still working on this one...

Wikipedia :   Higher category theory   |   2-category
Weak n-category   |   Bicategory (Bénabou, 1967)   |   Tricategory
Profunctor (Bénabou, 1973)

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

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 still working on this one...

Reflective subcategory

(2015-01-22)   Localization of a Category
Calculus of fractions.

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 still working on this one...

Spans and cospans   |   Localization of a Category

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

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

 Come back later, we're
 still working on this one...

Abelian category (Wikipedia)   |   Alex Heller (1925-2008)   |   David A. Buchsbaum (b. 1929)

(2014-12-04)   Category Theory  vs.  Set Theory
Which one would provide the best foundation for Mathematics?

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

The situation hasn't changed much since  Bénabou  published the above words...  When consistency isn't vouched for by the  "safe"  framework of set theory, categorial arguments are only convincing for those who have acquired the expertise Bénabou refers to.  Since the value of any mathematical argument resides in its ability to convince nonexperts, it would be highly desirable to have a consistent logical foundation for full-blown category theory  (covering the actual practice).  Otherwise, the belief cannot be dismissed that category theory is just a discovery tool, not a language for expressing ultimate proofs.  Other theories which are now fully accepted once had a similar status  (infinitesimal calculus being just one example).

Besides Jean Bénabou, one of the few opponents of set-based  Bourbakism  within French Academia was  Roger Apéry  (1916-1994)  who was 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 through its 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 and jeopardizing the entire project by diverting the energies of the participants.

In a 2014 videoPierre Cartier (1932-, Ulm 1950)  reveals how that internal Bourbaki debate was doused for pragmatic reasons, against the wishes of the radical "idealists" led by Grothendieck,  Lang, Chevalley and Sergent.

"Sergent" presumably refers to  Pierre Sergent  (Ulm, 1949).  Cartier describes him as "active" in the Bourbaki group at the time, implying that he was never made a full member.

Cartier doesn't say which side of the fence Eilenberg was on, possibly because Eilenberg was rarely 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 record as stating that the Bourbaki project had already lost much of its original appeal by that time).

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 still working on this one...

Fibered Categories and the Foundations of Naive Category Theory  (1985)  by  Jean Bénabou.
Théorie des Ensembles & Théorie des Catégories  (French)  Jean-Yves Béziau (2002).
Debate (in French) :   Saab Abou-Jaoudé  vs.  Jean Bénabou  (December 17, 2014).
The Interaction Between Category Theory and Set Theory  Lectures Audio Mathematics  (2015-09-17).

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