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Book by Saunders Mac Lane
Categories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who founded the subject
Categories for the Working Mathematician
Categories_for_the_Working_Mathematician
Collection of objects and morphisms
text on category theory is Categories for the Working Mathematician by Saunders Mac Lane. Other references are given in the References below. The basic
Category_(mathematics)
Property of operations
(1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer. ISBN 1441931236. OCLC 851741862. Wikibooks has more on the topic
Idempotence
Category theory constructs
In Categories for the Working Mathematician, Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that The notion
Kan_extension
fibration Topos Descent theory Indexed category 2-category Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics
Beck–Chevalley_condition
Category theory
Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli. Let ⟨T, η, μ⟩ be a monad over a category C. The Kleisli category of
Kleisli_category
Category in which all small limits exist
and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts
Complete_category
Structure-preserving map between two algebraic structures of the same type
Sankappanavar. ISBN 978-0-9880552-0-9. Mac Lane, Saunders (1971). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. Springer. Exercise
Homomorphism
British mathematician
MR 2003030. Thomas, R. P (2000). "Derived categories for the working mathematician". Proceedings of the Winter School on Mirror Symmetry, Vector Bundles
Richard Thomas (mathematician)
Richard_Thomas_(mathematician)
Relationship between two functors abstracting many common constructions
from Latin via French. In the classic text Categories for the Working Mathematician, Mac Lane makes a distinction between the two. Given a family φ c d
Adjoint_functors
Relation of categories in category theory
is again isomorphic to C. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag
Isomorphism_of_categories
General theory of mathematical structures
ISBN 978-0-691-14049-0. MR 2522659. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag
Category_theory
Mathematical concept
Type of category in category theory Limits and colimits in an ∞-category Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate
Limit_(category_theory)
Category whose objects are rings and whose morphisms are ring homomorphisms
just the polynomial ring Z[x]. Early references (including Mac Lane's Categories for the Working Mathematician) used Rng to denote the category of (unital)
Category_of_rings
Generalized object in category theory
Lawvere metrics, and closed categories)". Annoying Precision. Lane, S. Mac (1988). Categories for the working mathematician (1st ed.). New York: Springer-Verlag
Product_(category_theory)
Category whose objects are R-modules and whose morphisms are module homomorphisms
Abstract Algebra. Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (second ed.). Springer
Category_of_modules
Mathematical concept in category theory
acting on sets Section VII.3 in Mac Lane, Saunders (1988). Categories for the working mathematician (4th corr. print. ed.). New York: Springer-Verlag. ISBN 0-387-90035-7
Monoid_(category_theory)
Embedding of categories into functor categories
(Contravariant Yoneda lemma). Mac Lane, Saunders (1998). Categories for the working mathematician. Graduate Texts in Mathematics. Vol. 5 (2 ed.). New York
Yoneda_lemma
(1945) The first paper on category theory. Mac Lane later wrote in Categories for the Working Mathematician that he and Eilenberg introduced categories so
List of publications in mathematics
List_of_publications_in_mathematics
Study of abstract machines and automata
Algebras in Categories. Kluwer Academic Publishers:Dordrecht and Prague Mac Lane, Saunders (1971). Categories for the Working Mathematician. New York:
Automata_theory
Categorical generalization of a function space in set theory
finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still
Exponential_object
Abstract mathematics relationship
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories
Equivalence_of_categories
Set with operations obeying given axioms
Springer-Verlag, ISBN 978-3-540-90578-3 Category theory Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag
Algebraic_structure
Standard that diagrams must satisfy up to isomorphism
Mac Lane, Saunders (1971). "7. Monoids §2 Coherence". Categories for the working mathematician. Graduate texts in mathematics. Vol. 4. Springer. pp. 161–165
Coherency_(homotopy_theory)
Target set of a mathematical function
Press, ISBN 978-0-521-53361-4 Mac Lane, Saunders (1998), Categories for the working mathematician (2nd ed.), Springer, ISBN 978-0-387-98403-2 Scott, Dana
Codomain
Set of all things that may be the input of a mathematical function
ISBN 978-0-521-59718-0. Mac Lane, Saunders (25 September 1998). Categories for the Working Mathematician. Springer Science & Business Media. ISBN 978-0-387-98403-2
Domain_of_a_function
Mapping between categories
ISBN 0-7167-1933-9. Mac Lane, Saunders (1998) [1971]. Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (second ed.). New
Functor
Category whose objects are sets and whose morphisms are functions
1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. Springer. ISBN 0-387-98403-8. Pareigis, Bodo (1970), Categories and
Category_of_sets
Mathematical category formed by reversing morphisms
for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 33. ISBN 1441931236. OCLC 851741862. Awodey, Steve (2010). Category theory
Opposite_category
Right inverse of a morphism
Saunders (1978). Categories for the working mathematician (2nd ed.). Springer Verlag. Barry, Mitchell (1965). Theory of categories. Academic Press. Splitting
Section_(category_theory)
Category with direct sums and certain types of kernels and cokernels
abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are regular and they satisfy the snake
Abelian_category
In mathematics, invariant of square matrices
Springer, ISBN 9789401799447 Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nd ed.), Springer-Verlag
Determinant
Operation in algebra and mathematics
Bibcode:2012arXiv1209.3606L MacLane, Saunders (1978), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, doi:10.1007/978-1-4757-4721-8
Monad_(category_theory)
Functors which are surjective and injective on hom-sets
(2016), p. 31 Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8. Jacobson, Nathan
Full_and_faithful_functors
Algebraic object with geometric applications
the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult... MacLane, Saunders (2013). Categories for the Working Mathematician
Tensor
Functor mapping hom objects to an underlying category
(September 1998). Categories for the Working Mathematician (Second ed.). Springer. ISBN 0-387-98403-8. Goldblatt, Robert (2006) [1984]. Topoi, the Categorial
Hom_functor
Category admitting tensor products
commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be
Monoidal_category
Generalization of natural transformations
transformation Natural transformation Mac Lane, Saunders (2013). Categories for the working mathematician. Springer Science & Business Media. p. 218. Fosco, Loregian
Dinatural_transformation
Type of abstraction in science, mathematics, and philosophy
and the Foundations of Modern Mathematics". CNRS News. Retrieved 2025-01-26. Mac Lane, Saunders (2010). Categories for the working mathematician. Graduate
Abstract_structure
Construction in category theory
Inverse limit#Cones – Construction in category theory Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). New York: Springer. ISBN 0-387-98403-8
Cone_(category_theory)
Concept in mathematical category theory
indexing". The Grothendieck construction generalizes this to categories. For each category C {\displaystyle {\mathcal {C}}} and each family of categories { F
Category_of_elements
Type of category in category theory
Lane (1978). Categories for the Working Mathematician (2nd ed.). Springer. ISBN 1441931236. OCLC 851741862. "cartesian closed category in nLab". ncatlab
Cartesian_closed_category
Product of two categories, in category theory
a family of categories is defined exactly the same way. Just like for sets, a product of a family of categories is characterized by the following universal
Product_category
Correspondence between properties of a category and its opposite
Mathematics, EMS Press, 2001 [1994] Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 33. ISBN 1441931236
Dual_(category_theory)
Algebraic structure in linear algebra
Sons, ISBN 978-0-471-18117-0 Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
Vector_space
Extended physical object in string theory
(1998). Categories for the Working Mathematician. ISBN 978-0-387-98403-2. Moore, Gregory (2005). "What is ... a Brane?" (PDF). Notices of the AMS. 52:
Brane
Category whose objects are sets and whose morphisms are binary relations
Lane, S. (1988). Categories for the Working Mathematician (1st ed.). Springer. p. 26. ISBN 0-387-90035-7. Pareigis, Bodo (1970). Categories and Functors.
Category_of_relations
Functor type
and Computing Categories Topos. CRC Press. p. 28. ISBN 978-1482231502. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts
Representable_functor
American mathematician (1909–2005)
undergraduates using an English text. His Categories for the Working Mathematician remains the definitive introduction to category theory.[citation needed] 1997 (1941)
Saunders_Mac_Lane
Mathematical concept in category theory
Lane & Moerdijk 1994, p. 32. Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (2nd ed.), New York
Subobject
Concept in mathematics regarding sets operating on groups
Chapters 1–3. Springer-Verlag. ISBN 3-540-64243-9. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag. ISBN 0-387-98403-8.
Group_with_operators
Category where each homset contains at most one morphism
ISBN 978-3-319-41916-9. Mac Lane, Saunders (1998). Categories for the working mathematician. Graduate Texts in Mathematics. Vol. 5 (2 ed.). New York
Thin_category
Verlag, 1972. Exposé I, 2.7. Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
Filtered_category
Object that is both a product and coproduct
in abelian categories. Borceux, 4-5 Saunders Mac Lane, Categories for the Working Mathematician, Second Edition, page 194. Borceux, 8 Borceux, 7 H.D. Macedo
Biproduct
Mathematical concept
39–42 Dummit; Foote. Abstract Algebra. Mac Lane, Saunders. Categories for the Working Mathematician. Fraenkel (1954). "Abstract Set Theory". Nature. 173 (4412):
Inverse_function
North-Holland Publishing Company. p. 5. Mac Lane, Saunders (1998), Categories for the Working Mathematician, 2nd edition, Springer-Verlag, New York, NY. *Putnam, Hilary
Philosophy_of_mathematics
Type of quotient object in mathematics
the "first isomorphism theorem" for categories. Monoids and groups may be regarded as categories with one object. In this case the quotient category coincides
Quotient_category
Category whose objects are abelian groups and whose morphisms are group homomorphisms
ISBN 978-0-387-95385-4, MR 1878556 Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer
Category_of_abelian_groups
Mathematical concept
Mac Lane (2013). Mac Lane, Saunders (2013). Categories For the Working Mathematician. Springer Science & Business Media. pp. 222–226. Loregian, Fosco
End_(category_theory)
Saunders (1978). "Symmetry and Braidings in Monoidal Categories". Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. pp. 251–266
Strictification
Quotient space of a codomain of a linear map by the map's image
= im(T). Saunders Mac Lane: Categories for the Working Mathematician, Second Edition, 1978, p. 64 Emily Riehl: Category Theory in Context, Aurora Modern
Cokernel
Repeated application of an operation to a sequence
Binary function Ternary operation Saunders MacLane (1971). Categories for the Working Mathematician. New York: Springer-Verlag. p. 142. ISBN 0387900357. Bulk
Iterated_binary_operation
Concept in category theory
functors Functors Projection (set theory) Mac Lane, Saunders. Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin
Forgetful_functor
1998, Ch III, § 7, Theorem 1. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). New York
Density theorem (category theory)
Density_theorem_(category_theory)
All-encompassing set or class
Retrieved September 21, 2022. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag New York, Inc. "Universe", Encyclopedia
Universe_(mathematics)
Category whose objects are representations and whose morphisms are equivariant maps
arXiv:math/0412266. Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 41. ISBN 1441931236
Category_of_representations
Category-theoretic construction
Lawvere metrics, and closed categories)". Annoying Precision. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics
Coproduct
Central object of study in category theory
so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently
Natural_transformation
Homological construction
algebraic geometry S. Mac Lane, Categories for the Working Mathematician. Gabriel, Peter; Zisman, M. (6 December 2012). "1.2 The Calculus of Fractions: Proposition
Derived_category
Type of object in category theory
1215/ijm/1255454110, MR 0121775 Mac Lane, Saunders (1978), Categories for the Working Mathematician (Second ed.), New York, NY: Springer New York, p. 114,
Projective_object
equivalence of categories. Mac Lane (1998), Theorem IV.4.1 Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.).
Essentially surjective functor
Essentially_surjective_functor
Mathematical object in category theory
Press. ISBN 0-12-387850-0. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). New York
Subobject_classifier
Saunders (1998), Categories for the Working Mathematician, Springer. Monastyrsky, Michael (2001), "Some Trends in Modern Mathematics and the Fields Medal"
Glossary of mathematical jargon
Glossary_of_mathematical_jargon
Particular correspondence between two partially ordered sets
(antitone) definition: Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (Second ed.). Springer. ISBN 0-387-98403-8. Thomas Scott
Galois_connection
Aspect of category theory
in Computer Science. p. 278. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag
Coequalizer
Mathematics construct
comma categories, such as the special case of a slice category. Comma categories also guarantee the existence of some limits and colimits. The name comes
Comma_category
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
(1998). Categories for the Working Mathematician. Springer. ISBN 978-0-387-98403-2. Moore, Gregory (2005). "What is ... a Brane?" (PDF). Notices of the AMS
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
Construction in category theory
OCLC 40551485 Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0-387-98403-8 Mitchell, Barry
Inverse_limit
Set with associative invertible operation
Addison-Wesley, ISBN 978-0-201-70970-4. Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
Group_(mathematics)
The following commutative diagram shows an example of a modification and its inner workings. Mac Lane, Saunders (2010). Categories for the working mathematician
Modification_(mathematics)
Category whose hom sets have algebraic structure
Theory and Applications of Categories. Vol. 10. Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics
Enriched_category
Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 49–51. ISBN 1441931236. OCLC 851741862. free category at the nLab
Free_category
Polish mathematician (1904–1956)
1090/S0002-9904-1942-07723-8. Lane, Saunders Mac (1971). Categories for the Working Mathematician. Graduate Texts in Mathematics. p. 29. doi:10.1007/978-1-4612-9839-7
Witold_Hurewicz
General concept and operation in mathematics
x+190, hdl:2027/uc1.b4250788 Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Springer-Verlag, ISBN 978-0-387-98403-2 Mazur
Duality_(mathematics)
History of maths
Ross Street; An Australian conspectus of higher categories Elaine Landry, Jean-Pierre Marquis; Categories in context: historical, foundational, and philosophical
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
In mathematics, collection of classes
1007/BFb0059147. ISBN 978-3-540-04625-7. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (Second ed.). Springer
Conglomerate_(mathematics)
Property of topological spaces
On the foundations of k-group theory, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk Mac Lane, Saunders (1998). Categories for the Working Mathematician
Compactly_generated_space
called the connected components of J. Each connected component is a full subcategory of J. Mac Lane, Saunders (1998). Categories for the Working Mathematician
Connected_category
Generalization of a small category
category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal categories consist
Internal_category
Characterizing property of mathematical constructions
Holland. ISBN 90-277-1213-1. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer. ISBN 0-387-98403-8
Universal_property
Boolean algebra generated by a set with no relations beyond Boolean laws
Mathematical Association of America. Saunders Mac Lane (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag
Free_Boolean_algebra
Special case of colimit in category theory
Paris: Hermann, MR 0237342 Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (2nd ed.), Springer-Verlag
Direct_limit
Type of Abelian category (in category theory in mathematics)
Categories for the Working Mathematician (2nd ed.). Springer. p. 130. Popescu, Nicolae; Gabriel, Pierre (1964). "Caractérisation des catégories abéliennes avec
Grothendieck_category
Australian mathematician working in algebraic K-theory, algebraic geometry, topology, and homological algebra. He is professor emeritus at the Australian
Amnon_Neeman
Special objects used in (mathematical) category theory
ISBN 0-521-83414-7. Zbl 1034.18001. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag
Initial_and_terminal_objects
Basic concept in set theory
Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298. ISBN 978-0-521-89485-2. Mac Lane, Saunders (1998). Categories for the Working Mathematician
Pointed_set
Series of mathematics textbooks
ISBN 978-0-387-90033-9; 1997, 2nd ed., ISBN 978-0-387-94823-2) Categories for the Working Mathematician, Saunders Mac Lane (1971 1st ed. ISBN 978-0-387-90036-0;
Graduate_Texts_in_Mathematics
Topological space with a distinguished point
ISBN 0-486-40680-6. Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8. mathoverflow
Pointed_space
Concept in mathematical theory of categories
subgroup. The categories of elementary abelian groups, abelian p-groups, and p-groups are all reflective subcategories of the category of groups, and the kernels
Reflective_subcategory
Theorem in category theory
proof using the formalism of 2-categories: a monoidal category is the same data as a pseudomonoid in the 2-category of categories. Coherence for pseudomonoids
Mac_Lane's_coherence_theorem
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN
CATEGORIES FOR-THE-WORKING-MATHEMATICIAN