free

en-adj

free (comparative freer or more free, superlative freest or most free)

freer (comparative)
more free (comparative)
freest (superlative)
most free (superlative)

From Middle English free, fre, freo, from Old English frēo (“free”), from Proto-West Germanic *frī, from Proto-Germanic *frijaz (“beloved, not in bondage”), from Proto-Indo-European *priHós (“pleased, loved”), from *preyH- (“to please, love”). Related to friend.
cognates, etc
Germanic cognates include West Frisian frij (“free”), Dutch vrij (“free”), Low German free (“free”), German frei (“free”), Danish, Swedish and Norwegian fri (“free”).
Other cognates include Russian приятель (prijatelʹ, “friend”) and Sanskrit प्रिय (priyá-, “beloved”).
Germanic and Celtic are the only Indo-European language branches in which the PIE word with the meaning of "dear, beloved" acquired the additional meaning of "free" in the sense of "not in bondage". This was an extension of the idea of "characteristic of those who are dear and beloved", in other words friends and tribe members (in contrast to unfree inhabitants from other tribes and prisoners of war, many of which were among the slaves – compare the Latin use of liberi to mean both "free persons" and "children of a family").
The verb comes from Middle English freen, freoȝen, from Old English frēon, frēoġan (“to free; make free”), from Proto-West Germanic *frijōn, from Proto-Germanic *frijōną, from Proto-Indo-European *preyH-.

inh

there ain't no such thing as a free lunch

why buy the cow when you can get the milk for free

(algebra, of an algebraic structure) In any of various technical senses generic, universal.

(most generally, category theory, of an object F in a concrete category, with respect to a set X and a map i from X to the underlying set of F) Such that any map f from X to the underlying set of an object A in the same category as F induces a map ̄f from F to A which is compatible with f (i.e. such that f=̄f∘i).


adjunct

/ˈæd͡ʒ.ʌŋkt/

From Latin adiunctus, perfect passive participle of adiungō (“join to”), from ad + iungō (“join”). Doublet of adjoint.

(category theory) One of a pair of morphisms which relate to each other through a pair of adjoint functors.


dual

PIE word
 *dwóh₁
Borrowed from Latin dualis (“two”), from duo (“two”) + adjective suffix -alis.

Every category is dual to its own dual, so if a statement holds in all categories so does its dual.

1992, Colin McLarty, Elementary Categories, Elementary Toposes, Clarendon Press, →ISBN, page 77:

(category theory) Being the dual of some other category; containing the same objects but with source and target reversed for all morphisms.


sieve

From Middle English sive, syfe, from Old English sife, from Proto-West Germanic *sibi (“sieve”), from Proto-Indo-European *seyp-, *seyb- (“to pour, sieve, strain, run, drip”). Akin to German Sieb, Dutch zeef, Proto-Slavic *sito (Russian си́то (síto), сев (sev), се́ять (séjatʹ)).

(category theory) A collection of morphisms in a category whose codomain is a certain fixed object of that category, which collection is closed under precomposition by any morphism in the category.


concrete

en-adj

concrete (comparative concreter, superlative concretest)

concreter (comparative)
concretest (superlative)

Borrowed from Latin concrētus, past participle of concrescō (to curdle) from con- (with, together) + crescō (to grow, rise).

(category theory, of a category) Analogous to the categories of algebraic objects which category theory was created to generalize, in the sense of having objects which can be thought of as sets equipped with some additional structure. Formally, equipped with a faithful functor to the category of sets.


unit

Formerly unite, a later form of unity; see unity.

natural transformation
natural transformation

(category theory) In an adjunction, a natural transformation from the identity functor of the domain of the left adjoint functor to the composition of the right adjoint functor with the left adjoint functor.


lift

en-verb

lift (third-person singular simple present lifts, present participle lifting, simple past lifted or (rare, regional, obsolete) lift, past participle lifted or (rare, regional, obsolete) lift or (obsolete) yleft)

lifts (present singular third-person)
lifting (participle present)
lifted (past)
lift (obsolete past rare regional)
lifted (participle past)
lift (obsolete participle past rare regional)
yleft (obsolete participle past)

From Middle English liften, lyften, from Old Norse lypta (“to lift, air”, literally “to raise in the air”), from Proto-Germanic *luftijaną (“to raise in the air”), related to *luftuz (“roof, air”), perhaps from Proto-Indo-European *lewp- (“to peel, break off, damage”) or from a root meaning roof (see *luftuz). Cognate with Danish and Norwegian Bokmål løfte (“to lift”), Norwegian Nynorsk and Swedish lyfta (“to lift”), German lüften (“to air, lift”), Old English lyft (“air”). See above. 1851 for the noun sense "a mechanical device for vertical transport".
(To steal): For this sense Cleasby suggests perhaps a relation to the root of Gothic 𐌷𐌻𐌹𐍆𐍄𐌿𐍃 (hliftus) "thief", cognate with Latin cleptus and Greek κλέπτω (kléptō)).

der

(category theory, transitive) Given morphisms f and g with the same target: To produce a morphism which the given morphism factors through (i.e. a morphism h such that f=g∘h; c.f. lift n.18)


lift

en-noun

lift (countable and uncountable, plural lifts)

→ Cantonese: 𨋢／䢂 (lip¹)

→ Dutch: lift
→ Papiamentu: left

→ Malay: lif

desctree

→ Russian: лифт (lift)
→ Armenian: լիֆտ (lift)
→ Georgian: ლიფტი (lipṭi)
→ Kazakh: лифт (lift)
→ Mongolian: лифт (lift)
→ Uyghur: لىفت (lift)
→ Uzbek: lift
→ Yakut: лифт (lift)

→ Swahili: lifti

→ Swedish: lift

For a covering space p#x3A;#x5C;tildeX#x5C;toX a path #x5C;gamma [i.e. a continuous map #x5B;0,1#x5D;#x5C;toX] in X has a unique lift #x5C;tilde#x5C;gamma starting at a given point of p#x7B;-1#x7D;(#x5C;gamma(0))

2001, Allen Hatcher, Algebraic Topology, page 69:

(category theory) A morphism which some given morphism factors through; i.e. given a pair of morphisms f:X→Y and g:Z→Y, a morphism h such that f=g∘h. (In this case h is said to be a lift of f via Z or via g).


full

en-adj

full (comparative fuller or more full, superlative fullest or most full)

fuller (comparative)
more full (comparative)
fullest (superlative)
most full (superlative)

empty

antonym(s) of “containing the maximum possible amount”

empty

antonym(s) of “satisfied, in relation to eating”

→ Gulf Arabic: فُل (ful)

→ Japanese: フル

From Middle English ful, from Old English full (“full”), from Proto-West Germanic *full, from Proto-Germanic *fullaz (“full”), from Proto-Indo-European *pl̥h₁nós (“full”).
Germanic cognates include West Frisian fol, Low German vull, Dutch vol, German voll, Danish fuld, and Norwegian and Swedish full (the latter three via Old Norse). Proto-Indo-European cognates include English plenty (via Latin, compare plēnus), Welsh llawn, Russian по́лный (pólnyj), Lithuanian pilnas, Persian پر (por), Sanskrit पूर्ण (pūrṇá).
See also fele and Scots fou. For the "drunk, intoxicated" sense, compare also Swedish full and other Scandinavian languages.

fool (accents with FOOT-GOOSE merger or full-fool merger)

nervous as a long-tailed cat in a room full of rocking chairs

(category theory, of a functor between locally small categories) Surjective as a map of morphisms

Complete; with nothing omitted.
Surjective as a map of morphisms

(category theory, of a subcategory S of C) Including all morphisms. Formally: Such that for every pairs of objects (X, Y) in S, the hom-sets operatorname Hom_S(X,Y) and operatorname Hom_C(X,Y) are equal.


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

der

(mathematics, category theory) For a category with zero morphisms: the equalizer of a given morphism and the zero morphism which is parallel to that given morphism.


graph

Clipping of graphic formula. From 1878; verb from 1889.

(category theory, of a morphism f) A morphism Γ_f from the domain of f to the product of the domain and codomain of f, such that the first projection applied to Γ_f equals the identity of the domain, and the second projection applied to Γ_f is equal to f.


intersection

/ˈɪntəɹˌsɛkʃən/

/ˌɪntəɹˈsɛkʃən/

From Middle French intersection, from Latin intersectiō.

(category theory) The pullback of a corner of monics.


iso

en-noun

iso (countable and uncountable, plural isos)

(category theory) Clipping of isomorphism.


allegory

allegory (countable and uncountable, plural allegories)

/ˈæl.ɪˌɡɔː.ɹi/

/ˈæl.əˌɡɔɹ.i/

From Middle English allegorie, from Old French allegorie, from Latin allēgoria, from Ancient Greek ἀλληγορία (allēgoría), from ἄλλος (állos, “other”) + ἀγορεύω (agoreúō, “I speak”).

(mathematics, category theory) A category that retains some of the structure of the category of binary relations between sets, representing a high-level generalisation of that category.


source

non-rhotic, without the horse–hoarse merger

From Middle English sours, from Old French sorse (“rise, beginning, spring, source”), from sors, past participle of sordre, sourdre, from Latin surgō (“to rise”), which is composed of sub- (“up from below”) + regō (“lead, rule”), ultimately from Proto-Indo-European *h₃réǵeti (“to straighten; right”), from the root *h₃reǵ-. See surge.

der

Coordinate term: target

(mathematics, category theory) The domain of a function; the object which a morphism points from.


sketch

→ German: Sketch

→ Italian: sketch

desc

→ Portuguese: esquete (Brasilian), sketch (European)

→ Turkish: skeç

From Dutch schets or German Skizze, from Italian schizzo, from Latin schedium, from Ancient Greek σχέδιος (skhédios, “made suddenly, off-hand”), from σχεδόν (skhedón, “near, nearby”), from ἔχω (ékhō, “I hold”). Compare scheme.

скеч

brief musical, dramatic or literary work etc.

mathematical structure
mathematical structure

(category theory) A formal specification of a mathematical structure or a data type described in terms of a graph and diagrams (and cones (and cocones)) on it. It can be implemented by means of “models”, which are functors which are graph homomorphisms from the formal specification to categories such that the diagrams become commutative, the cones become limiting (i.e., products), the cocones become colimiting (i.e., sums).


inverse

Recorded since 1440, from Latin inversus, the past participle of invertere (“to invert”), itself from in- (“in, on”) + vertere (“to turn”).

(category theory, of a category) Whose every element has an inverse (morphism which is both a left inverse and a right inverse).


inverse

(category theory) A morphism which is both a left inverse and a right inverse.


object

From Old French object, from Medieval Latin obiectum (“object”, literally “thrown against”), from obiectus, perfect passive participle of obiciō (“I throw against”), from ob- (“against”) + iaciō (“I throw”), as a calque of Ancient Greek ἀντικείμενον (antikeímenon). Doublet of objet.

ἀντικείμενον

massive astronomical compact halo object

massive astrophysical compact halo object

Similarly, there is a category whose objects are groups and whose arrows are the homomorphisms from one group to another.

(category theory) An instance of one of the two kinds of entities that form a category, the other kind being the arrows (also called morphisms).

objektum

category theory: instance of a kind of entities forming a category


mono

(category theory) Abbreviation of monomorphism.


replete

en-adj

replete (comparative more replete, superlative most replete)

more replete (comparative)
most replete (superlative)

From Middle English replete (adjective) and repleten (verb), from Old French replet, from Latin repletus.

(category theory, of a subcategory S of C) Isomorphism-closed: Inheriting all the isomorphisms of C. Formally: such that for any isomorphism f in C, if f 's source is in S, then f and f 's target is also in S.


section

From Middle English seccioun, from Old French section, from Latin sectiō (“cutting, cutting off, excision, amputation of diseased parts of the body, etc.”), from sectus, past participle of secāre (“to cut”). More at saw.

разрэ́з

surgery: incision or the act of making an incision

A cross-section (image that shows an object as if cut along a plane).

(generalizing the topology sense, algebra, category theory) A right inverse of a morphism in some category


formation

en-noun

formation (countable and uncountable, plural formations)

/fɔːˈmeɪʃən/

/fɔɹˈmeɪʃən/

/foːˈmæɪʃən/

From Middle English formacioun, formation, borrowed from Old French formacion, from Latin fōrmātiō, from fōrmō (“form”, verb); see form as verb.
Morphologically form + -ation

(category theory) A structure made of two categories, two functors from the first to the second category, and a transformation from one of the functors to the other.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest use of the word in English was to mean "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height and weight and blood pressure, the usual or most common values came to be referred to as "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(category theory) Being (as a morphism) or containing (as a category) only normal epimorphism(s) or monomorphism(s), that is, those which are the kernel or cokernel of some morphism, respectively.


relation

en-noun

relation (countable and uncountable, plural relations)

From Middle English relacion, relacioun, from Anglo-Norman relacioun and Old French relacion (whence French relation), from Latin relātiō, noun of process form from perfect passive participle relātus (“related”), from verb referō (“I refer, I relate”), from prefix re- (“again”) + ferō (“I bear, I carry”).
Morphologically relate + -ion

عَلَاقَة

way in which two things may be associated

(category theory) A subobject of a product of objects.


fiber

fiber (countable and uncountable, plural fibers)

From French fibre, from Old French fibre, from Latin fibra.

(category theory) The pullback of a morphism along a global element (called the fiber of the morphism over the global element).


fibre

fibre (countable and uncountable, plural fibres)

fibres (plural)
spellings (New-Zealand South-Africa)

From French fibre, from Old French, from Latin fibra.

(category theory) Said to be of a morphism over a global element: The pullback of the said morphism along the said global element.


target

→ Japanese: ターゲット (tāgetto)

From Middle French targette, targuete, diminutive of targe (“light shield”), from Old French, from Frankish *targa (“buckler”), akin to Old Norse targa (“small round shield”) (whence also Old English targe, targa (“shield”)) from Proto-Germanic *targǭ (“edge”), from Proto-Indo-European *derǵʰ- (“fenced lot”). Akin to Old High German zarga (“side wall, rim”) (German Zarge (“frame”)), Spanish tarjeta (“card”).

cog

Coordinate term: source

(mathematics, category theory) The codomain of a function; the object at which a morphism points.


limit

German: Limit

From Middle English limit, from Old French limit, from Latin līmes (“a cross-path or balk between fields, hence a boundary, boundary line or wall, any path or road, border, limit”). Displaced native Old English ġemǣre. Doublet of limes.

der

(category theory) The cone of a diagram through which any other cone of that same diagram can factor uniquely.


monad

From Latin monas (“unit”) (from Ancient Greek μονάς (monás), from μόνος (mónos), from Proto-Indo-European *men-) + -ad.

Coordinate term: comonad

(category theory) A monoid object in the category of endofunctors of a fixed category.


site

From Middle English site, from Anglo-Norman site, from Latin situs (“position, place, site”), from sinere (“to put, lay, set down, usually let, suffer, permit”). Doublet of sitio and situs.

Grothendieck topology
Grothendieck topology

(category theory) A category together with a choice of Grothendieck topology.


diagram

From French diagramme, from Italian diagramma, from Ancient Greek διάγραμμα (diágramma).

(category theory) A functor from an index category to another category. The objects and morphisms of the index category need not have any internal substance, but rather merely outline the connective structure of at least some part of the diagram's codomain. If the index category is J and the codomain is C, then the diagram is said to be "of type J in C".


cone

From Middle English cone (“corner, angle”) and conoun (“cone”), from Medieval Latin cōnus, cōnon (“cone, wedge, peak”), from Ancient Greek κῶνος (kônos, “cone, spinning top, pine cone”). Reinforced by Middle French cone, from the same Graeco-Latin source.

A cone is an object (the apex) and a natural transformation from a constant functor (whose image is the apex of the cone and its identity morphism) to a diagram functor. Its components are projections from the apex to the objects of the diagram and it has a “naturality triangle” for each morphism in the diagram. (A “naturality triangle” is just a naturality square which is degenerate at its apex side.)

(category theory) An object V together with an arrow going from V to each object of a diagram such that for any arrow A in the diagram, the pair of arrows from V which subtend A also commute with it. (Then V can be said to be the cone’s vertex and the diagram which the cone subtends can be said to be its base.)


projection

en-noun

projection (countable and uncountable, plural projections)

/pɹəˈd͡ʒɛkʃən/

From either the Middle French projection or its etymon, the Classical Latin prōiectiō (stem: prōiectiōn-), from prōiciō, equivalent to project + -ion. Compare the Modern French projection, the German Projektion, and the Italian proiezione.

(category theory) A morphism from a categorical product to one of its (two) components.


Ob

(category theory) The set of objects of a category.


isomorphism

isomorphism (countable and uncountable, plural isomorphisms)

Isomorphism on Encyclopedia of Mathematics

Natural Isomorphism on Wolfram MathWorld

(category theory) A morphism which has a two-sided inverse; the composition of the morphism and such an inverse yields either one of two identity morphisms (depending on the order of composition).


embedding

(mathematics) A map which, in any of several technical senses, represents the containment of one structure inside another.

(category theory) A injective morphism in a concrete category which is also initial.


injection

injection (countable and uncountable, plural injections)

Nicolas Bourbaki
Éléments de mathématique

Borrowed from Middle French injection, from Latin iniectio. The mathematical sense is from French injection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

(category theory) A morphism from either one of the two components of a coproduct to that coproduct.


equalizer

(category theory) A morphism whose codomain is the domain of a parallel pair of morphisms and which forms part of the limit of that parallel pair. Equivalently, a morphism which equalizes a parallel pair of morphisms in a limiting way, which is to say that any other morphism which equalizes that parallel pair factors through this limiting morphism; and moreover such factorization is unique.


projective

en-adj

projective (comparative more projective, superlative most projective)

more projective (comparative)
most projective (superlative)

Formed by the suffixation of prōiect- (the perfect passive participial stem of the Classical Latin prōiciō, whence the English verb project) with the English -ive, forming project + -ive; however, compare the post-Classical Latin prōiectīvus (“relating to purging”).

cog

(algebra) In a technical sense, general (but not necessarily so general as to be free); involving such objects:

(category theory, most generally, of an object P in a category) Such that, given an epimorphism e:E→X and morphism f:P→X, f factors through e; that is, there exists a morphism ̃f:P→E with e∘̃f


subcategory

(category theory) A subclass of a category which is itself a category, whose arrows are a restriction of the arrows of the parent category, and whose composition rule is a restriction of the parent category's.


injective

This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

(category theory, most generally, of an object Q in a category C) Loosely, having a property analogous to that which characterizes injective modules (see above). Formally, such that, given a monomorphism f:X→Y in C, for every morphism g:X→Q there exists a morphism h:Y→Q such that h∘f=g; see Injective object.


adjoint

Adjoint representation
Adjoint representation of a Lie algebra
adjoint

/ˈæd͡ʒ.ɔɪnt/

From French adjoindre (“to join”), from late 19th C; see also adjoin. Doublet of adjunct.
In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.

(mathematics, category theory, of a functor) That is related to another functor by an adjunction.


adjoint

/ˈæd͡ʒ.ɔɪnt/

(mathematics, category theory) A functor related to another functor by an adjunction.


monomorphism

(category theory) A morphism n such that for any other morphisms f and g, if n∘f=n∘g then f = g.


monic

If a composition (of morphisms) is monic then, by theorem, the pre-composite of it is monic as well.

It is often convenient to view a monic arrow i:A↣C, defined in Chapter 1, as showing that A is a copy of a part of C, and that i maps the copy on to that part. For example, in Set, the monic arrows are the one-to-one functions and, clearly, if i is one-to-one then A is a copy of a subset of C, namely of the image of i.

1995, Colin McLarty, “Introduction”, in Elementary Categories, Elementary Toposes, page 5

(category theory) Of a morphism: that it is a monomorphism.


functor

In the category of categories, #x5C;mathbb#x7B;CAT#x7D;, the objects are categories and the morphisms are functors.

1991, Natalie Wadhwa (translator), Yu. A. Brudnyǐ, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, Volume I, Elsevier (North-Holland), page 143,
Choosing for U the operation of closure, regularization or relative completion, we obtain from a given functor ℱ∈𝒥ℱ the functors
◌̅F: overrightarrow X→◌̅F( overrightarrow X),F⁰: overrightarrow X→F( overrightarrow X)⁰,Fᶜ: overrightarrow X→F( overrightarrow X)ᶜ.

Given a homotopical category X and a functor u#x3A;A#x5C;rightarrowB, a homotopical u-colimit (resp. u-limit) functor on X will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor Xᵘ#x3A;Xᴮ#x5C;rightarrowXᴬ (47.1).

2004, William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, American Mathematical Society, page 165:

2009, Benoit Fresse, Modules Over Operads and Functors, Springer, Lecture Notes in Mathematics: 1967, page 35,
In this chapter, we recall the definition of the category of Σ_*-objects and we review the relationship between Σ_*-objects and functors. In short, a Σ_*-object (in English words, a symmetric sequence of objects, or simply a symmetric object) is the coefficient sequence of a generalized symmetric functor S(M):X→S(M,X), defined by a formula of the form
S(M,X)=⨁ ᪲ᵣ₌₀(M(r)⊗X)_(Σᵣ).

(category theory) A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows (either covariantly or contravariantly), in such a way as to preserve morphism composition and identities.


topos

From Ancient Greek τόπος (tópos, “place”). Compare topic.


equalize

en-verb

equalize (third-person singular simple present equalizes, present participle equalizing, simple past and past participle equalized)

equalizes (present singular third-person)
equalizing (participle present)
equalized (participle past)
equalized (past)

(category theory) Said of a morphism: to pre-compose with each of a parallel pair of morphisms so as to yield the same composite morphism.


morphism

1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. Čencov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,
1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second.

1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198,
In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems.

The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaks up into two parts: a reduction to prepared morphisms and then a proof of toroidalization of prepared morphisms from n-folds to [surfaces] in [CK].

2007 November, Steven Dale Cutkosky, “Toroidalization of Dominant Morphisms of 3-Folds”, in Memoirs of the, volume 190, number 890, American Mathematical Society, page 3:

(mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.


cokernel

(category theory) For a category with zero morphisms: the coequalizer between a given morphism and the zero morphism which is parallel to that given morphism.


direct product

(category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects.


contravariant

(category theory, of a functor) Which reverses the order of morphism composition: formally, which associates each morphism f:X→Y to a morphism F(f):F(Y)→F(X).


covariant

(category theory, of a functor) Which preserves the order of morphism composition: formally, which associates each morphism f:X→Y to a morphism F(f):F(X)→F(Y).


epimorphism

(category theory) A morphism p such that for any other pair of morphisms f and g, if f∘p=g∘p, then f = g.

Epimorphismus

mathematics: map f such that whenever g composed with f equals h composed with f, then g=h


pullback

(category theory) The limit of a cospan: a Cartesian square or “pullback square”.

(category theory) Within a Cartesian square (which has a pair of divergent morphisms and a pair of convergent morphisms) the divergent morphism which is directly opposite to a given one of the convergent morphisms, said to be “along” the convergent morphism which is between that pair of opposite morphisms. (The pullback is said to be “of” the given morphism.)


fiber bundle

Coined as fibre bundle by American mathematician Norman Steenrod in 1951, The Topology of Fibre Bundles. The related usages fiber and fiber space probably derive (as calques respectively of German Faser and gefaserter Räume) from 1933, Herbert Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Mathematica, 60, (1933), 147-238.

A Möbius strip is a fiber bundle which looks locally (i.e., over a connected proper subset of its base space) like the corresponding part of a cylinder S¹#x5C;times#x5B;0,1#x5D; (a Möbius strip and a cylinder have isomorphic base spaces). A Klein bottle is a fiber bundle which looks locally like the corresponding part of a torus S¹#x5C;timesS¹ (again they could be thought of as sharing the same base space S¹; cutting out even a single point of that base space makes the cut Klein bottle isomorphic to the cut torus).

In general, a fiber bundle consists of a set of mutually disjoint fibers “over” a base space, which indexes the fibers; there is a copy of some fiber on top of, or projecting (“canonically”) onto each point of the base space.

In the 1960s, some physicists including E. Lubkin and A. Trautman recognized that interaction potentials can be represented by connections on principal fiber bundles. In 1975, T. T. Wu and C. N. Yang used the fiber bundle method to solve a problem on magnetic monopoles.

1995, Sunny Y. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, page 214:

One proposed general model for high-level scientific data uses fiber bundles, which Butler and Pendley [22] proposed in 1989.[…] A fiber bundle is the Cartesian product of the fibers and the base space; in other words, it is the collection of valid data ranges for the base space.

2001, John M. May, Parallel I/O for High Performance Computing, Morgan Kaufmann Publishers, page 236:

Finding the right notion of fiber bundle for diffeology [Igl85] has been a question raised by the study of the irrational torus T#x5F;#x5C;alpha [Dolg85].

2013, Patrick Iglesias-Zemmour, Diffeology, American Mathematical Society, page 229:

(American spelling, topology, category theory) An abstract object in topology where copies of one object are "attached" to every point of another, as hairs or fibers are attached to a hairbrush. Formally, a topological space E (called the total space), together with a topological space B (called the base space), a topological space F (called the fiber), and surjective map π from E to B (called the projection or submersion), such that every point of B has a neighborhood U with π⁻¹(U) homeomorphic to the product space U × F (that is, E looks locally the same as the product space B × F, although its global structure may be quite different).

Herbert Seifert
Norman Steenrod
Project Euclid
fiber bundle


categorical dual

en-noun

categorical dual (plural categorical duals)

(category theory) For a given category C, its dual category Cᵒᵖ is obtained by reversing the direction of each one of the arrows of C.


initial object

In the category of sets, the initial object is the empty set, since there exists a unique function, namely the empty function, from the empty set into any other given set.

(category theory) An object within a category which sends out arrows to all other objects in that category, and such that each of these arrows is unique.


pushout

(category theory) The colimit of a pair of morphisms which share the same domain.


terminal object

terminal object (plural terminal objects)

In the category of sets, a terminal object is any singleton, since there exists a unique function, namely the constant function, from any given set into that singleton.

The terminal object is the limit over the empty diagram.

(category theory) An object within a category which receives arrows from all other objects in that category, and such that each of these arrows is unique.


exponential object

exponential object (plural exponential objects)

An exponential object generalizes its interpretation in category #x5C;mathbf#x7B;Set#x7D;; namely, that of as a function set or internal hom-set.

The pair Z^Y,#x5C;mbox#x7B;eval#x7D;#x3A;Z^Y#x5C;timesY#x5C;rightarrowZ is the terminal object of the comma category (-#x5C;timesY)#x5C;downarrowZ. Therefore the exponential object is a kind of universal morphism.

(category theory) An object which indexes a family of arrows between two given objects in a universal way, meaning that any other indexed family of arrows between the same given pair of objects must factor uniquely through this universally-indexed family of arrows.

currying ∼ exportation_(logic)

an instance of which is the transpositioning λ in the figure

modus ponens

homologous to the universal evaluation morphism, eval


contravariance

contravariance (countable and uncountable, plural contravariances)

(category theory) A functor which reverses composition.


groupoid

[...] A category in which every morphism is an isomorphism is called a groupoid.
 This means that a group is a groupoid with only one object. [...]

(algebra and category theory) A set with a partial binary operation that is associative and has identities and inverses.

groupoïde

a set with a partial binary operation...


categorification

categorification (countable and uncountable, plural categorifications)

2008, E. Krenkel, Ramifications of the Geometric Langlands Program, Michael Cowling, Edward Frenkel, Masaki Kashiwara, Alain Valette, David A. Vogan, Jr., Nolan R. Wallach (editors), Representation Theory and Complex Analysis: Lectures Given at the C.I.M.E. Summer School, Springer, Lecture Notes in Mathematics 1931, page 83,
In Section 3.4 we have already discussed the question of categorification of the algebra of functions on a homogeneous space like G(F)/K.

2009, Volodymyr Mazorchuk, Lectures on sl₂( C )-Modules, Imperial College Press, page 221,
Show that Φ⊕Φ is also a (naïve) homomorphism of naïve categorifications.

Khovanov's categorification of the Jones polynomial by Khovanov homology is the subject of Section 11.6.

2010, Kishore Marathe, Topics in Physical Mathematics, Springer, page 351:

2011, Robert Wisbauer, Categorical aspects of Hopf algebras, Matilde Marcolli, Deepak Parashar (editors), Quantum Groups and Noncommutative Spaces: Perspectives on Quantum Geometry, Springer Science+Business (Vieweg+Teubner), page 146,
Since Lawvere's categorification of general algebra, algebras and coalgebras are used as basic notions in universal algebra, logic, and theoretical computer science, for example (e.g. [AdPo], [Gu], [TuPl]).

(category theory) A procedure that defines theorems in terms of category theory by mapping concepts from set theory to category theory.


representable functor

representable functor (plural representable functors)

If a representable functor is naturally isomorphic to a hom functor mbox Hom(C,-) or mbox Hom(-,C), then the object C is said to represent that representable functor.

(category theory) A functor from some category to the category of sets (Set) which is naturally isomorphic to a hom functor.


contravariant functor

contravariant functor (plural contravariant functors)

A contravariant functor F#x3A;#x5C;mathcal#x7B;C#x7D;#x5C;rightarrow#x5C;mathcal#x7B;D#x7D; is the same as a covariant functor F#x3A;#x5C;mathcal#x7B;C#x7D;#x7B;op#x7D;#x5C;rightarrow#x5C;mathcal#x7B;D#x7D;.

(category theory) A functor which maps a morphism f:X → Y to a morphism F(f):F(Y) → F(X), such that if h=g∘f, then F(h)=F(f)∘F(g).


natural transformation

natural transformation (plural natural transformations)

(category theory) A morphism between a pair of parallel functors such that if each object of the shared domain category subtends a correlated arrow — called a component — in the shared codomain (which arrow represents the difference between applying the second functor and the first functor to the correlated object) then each arrow of the shared domain subtends a commuting square — called a naturality square — between two components (correlated to the domain and codomain of the arrow).


Yoneda lemma

Lemma named after the Japanese mathematician Nobuo Yoneda (1930–1996).

As a corollary of the Yoneda lemma, given a pair of contravariant hom functors #x5C;mbox#x7B;Hom#x7D;(-,A) and #x5C;mbox#x7B;Hom#x7D;(-,B), then any natural transformation #x5C;alpha from #x5C;mbox#x7B;Hom#x7D;(-,A) to #x5C;mbox#x7B;Hom#x7D;(-,B) is determined by the choice of some function f#x3A;A#x5C;rightarrowB to map the identity #x5C;mbox#x7B;id#x7D;#x5F;A#x3A;A#x5C;rightarrowA to, by the component #x5C;alpha#x5F;A#x3A;#x5C;mbox#x7B;Hom#x7D;(A,A)#x5C;rightarrow#x5C;mbox#x7B;Hom#x7D;(A,B) of #x5C;alpha. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.

• Yoneda Lemma: Nat(Hom(A,–), F) ≅ F(A)
∴ Nat(Hom(A,–), Hom(B,–)) ≅ Hom(B,A)
∴ A ≅ B iff Hom(A,–) ≅ Hom(B,–)
i.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.

And then there's the Yoneda Lemma embodied in the classical Gaussian row reduction operation, that a given row reduction operation (on matrices with say k rows) being a "natural" operation (in the sense of natural transformations) is just multiplication (on the appropriate side) by the effect of that operation on the k-by-k identity matrix.
And dually for column-reduction operations :-)

2020, Emily Riehl, quoting Fred E. J. Linton, The Yoneda lemma in the category of Matrices:

(category theory) Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 𝒞 to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation α from H to F is determined by what α_A( mbox id_A) is.)

米田引理

theorem which states that there is a natural isomorphism...


Yoneda embedding

Yoneda embedding (plural Yoneda embeddings)

Named after the Japanese mathematician Nobuo Yoneda (1930–1996).

(category theory) An embedding of a category 𝒞 within the category of functors from 𝒞 to the category of sets. Such an embedding is effected by a Yoneda functor.


adjunction

adjunction (countable and uncountable, plural adjunctions)

From Latin adjunctio, from adjungere: compare French adjonction, and see adjunct.

(category theory, loosely) A relationship between a pair of categories that makes the pair, in a weak sense, equivalent.

adiunkcja

a form of similarity between a pair of categories mapped to each other by dual morphisms

(category theory, strictly) A natural isomorphism between a pair of functors satisfying certain conditions, whose existence implies a close relationship between the functors and between their (co)domains; the natural isomorphism, functors, and their (co)domains thought of as a single object.

Meronyms: adjoint, left adjoint, right adjoint

(formally, given two categories 𝒞 and 𝒟 and (covariant) functors F:𝒞→𝒟 and G:𝒟→𝒞) A natural isomorphism Φ: operatorname Hom_( mathcal )C(G·,·)→ operatorname Hom_( mathcal )D(·,F·) (where the hom-functors are understood as bifunctors from 𝒟^( operatorname )op×𝒞 to mathbf Set). See Adjoint functors on Wikipedia.Wikipedia.


additive group

additiivinen ryhmä

group or algebraic object regarded as a group with operation called addition

(mathematics) Any group, or algebraic object that can be regarded as a group, whose binary operation is called addition and denoted +; specifically:

(category theory) Any group object (such as a group functor, group scheme, etc.), whose binary operation is called addition.


epic

From epi-, from Ancient Greek ἐπί (epí, “on top of”).

(category theory, of a morphism) That is an epimorphism.


isogeny

isogeny (countable and uncountable, plural isogenies)

It is clear that if G#x5F;2 is connected, then two of the defining properties of an isogeny imply the third.

2000, Marc Hindry, Joseph H. Silverman, Diophantine Geometry: An Introduction, Springer, page 95:

2002, Mireille Fouquet, François Morain, Isogeny Volcanoes and the SEA Algorithm, Claus Fieker, David R. Kohel (editors), Algorithmic Number Theory: 5th International Symposium, Proceedings, Springer, LNCS 2369, page 279,
Lemma 2.2 Let E be an elliptic curve such that Z [π] is maximal at 𝓁. If there exists an 𝓁-isogeny of E, then this 𝓁-isogeny is an horizontal 𝓁-isogeny.

2005, Fred Diamond, Jerry Shurman, A First Course in Modular Forms, Springer, page 29,
The dual isogeny of an isomorphism is its inverse. The dual of a composition of isogenies is the composition of the duals in the reverse order. If φ is an isogeny and ̂φ is its dual then the formulas φ(z+Λ)=mz+Λ', φ(z'+Λ')=( operatorname φ/m)z'+Λ show that also
φ∘̂φ=[ operatorname deg(φ)]=[ operatorname deg(̂φ)],
so that φ is in turn the dual isogeny of its dual ̂φ. Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms.

(algebraic geometry, category theory) An epimorphism of group schemes that is surjective and has a finite kernel.


functorial

en-adj

functorial (comparative more functorial, superlative most functorial)

more functorial (comparative)
most functorial (superlative)

(category theory) Of or pertaining to a functor

náttúrlegur

category theory: pertaining to a functor


comonad

(mathematics, category theory) A monad of the opposite category.


colimit

(category theory) The cocone of a diagram through which any other cocone of that same diagram can factor uniquely.


endofunctor

(category theory) A functor that maps a category to itself.


left inverse

(category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that s∘f= mbox id_X.


subobject

equivalence relation
equivalence relation

(category theory) An object and a monomorphism from it to another object, which monomorphism is interpreted as an inclusion. Actually it is an equivalence class of monomorphisms to the same object, where the equivalence relation is the ability of a pair of monomorphisms to factor through each other.


coterminal

(category theory) Describing an object in a category, such that there is precisely one morphism that maps that object to every object in the category.


right inverse

(category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that f∘s= mbox id_Y.


presheaf

Let X be a topological space. (Following tradition, I will switch from my previous convention of using X to denote an object of a topos.) Write Open(X) for its poset of open subsets. A presheaf on X is a functor F#x3A;#x5C;mathbf#x7B;Open#x7D;(X)#x5C;mbox#x7B;op#x7D;#x5C;rightarrow#x5C;mathbf#x7B;Set#x7D;. It assigns to each open subset U a set F(U), whose elements are called sections over U (for reasons to be explained). It also assigns to each open V#x5C;subseteqU a function F(U)#x5C;rightarrowF(V), called restriction from U to V and denoted by s#x5C;rightarrows#x7C;#x5F;V. I will write Psh(X) for the category of presheaves on X.
Examples 3.1 i. Let F(U) = {continuous functions U#x5C;rightarrow#x5C;mathbb#x7B;R#x7D;}; restriction is restriction.

2011 June 27, Tom Leinster, “An informal introduction to topos theory”, in arXiv.org, Cornell University Library, retrieved 2018-03-18:

contravariant functor
contravariant functor

(category theory, sheaf theory) An abstract mathematical construct which associates data to the open sets of a topological space, generalizing the situation of functions, fiber bundles, manifold structure, etc. on a topological space (but not necessarily in such a way as to make the local and global data compatible, as in a sheaf). Formally, A contravariant functor ℱ whose domain is a category whose objects are open sets of a topological space (called the base space or underlying space) and whose morphisms are inclusion mappings. The image of each open set under ℱ is an object whose elements are called sections, and are which are said to be over the given open set; the image of each inclusion map A→B under ℱ is a morphism ℱ(B)→ℱ(A), called the restriction from B to A and denoted operatorname res_(B,A) or |_(B,A).


slice category

en-noun

slice category (plural slice categories)

(category theory) A category whose objects are morphisms (of some given category) with a common codomain, and whose morphisms are commuting triangles where two morphisms of each of such triangles share the said common codomain.


localic

(category theory) Relating to the category of locales.


coinversion

(category theory) The inversion operation of a cogroup object. A coinversion must satisfy the dual versions of the axioms for group objects.


coidentity

(category theory) The identity element of a cogroup.


null object

(category theory) An object that is both an initial and terminal object.


cocone

(category theory) An object V together with an arrow going from each object of a diagram to V such that for any arrow A in the diagram, the pair of arrows to V which form a triangle with A also commute with it.


bicartesian closed category

bicartesian closed category (plural bicartesian closed categories)

Cartesian closed category
Cartesian closed category

(category theory) A Cartesian closed category which also has an initial object and such that for any pair of objects, A and B, in the category, the category has another object which is their coproduct, A∐B.


Cartesian closed category

Cartesian closed category (plural Cartesian closed categories)

Named after René Descartes (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.

In any event, Lambek showed that every typed lambda-theory gives a cartesian closed category — and conversely, every cartesian closed category gives a typed lambda-theory. This discovery led to a rich line of research blending category theory and computer science.

2009 February 3, John C. Baez with Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, page 54:

(category theory) A category which has a terminal object and which for every two objects A and B has a product A × B and an exponential object Bᴬ.

Cartesian closed category
René Descartes


zero object

(category theory) An object which is both an initial object and a terminal object.


balanced category

balanced category (plural balanced categories)

(category theory) A category in which every bimorphism is an isomorphism.


bimorphism

(category theory) A morphism which is both a monomorphism and an epimorphism.


generic element

generic element (plural generic elements)

(category theory) The identity element of an object when thought of as a generalized element of that object.


generalized element

generalized element (plural generalized elements)

Thinking of x∈_AB as a generalized element, for any f:B→C, we may write f(x) for the composite f∘x. In this notation the first domain–codomain axiom reads as follows: for any x∈_AB and f:B→C there is a well-defined f(x)∈_AC; that is, at each stage A, f takes A-elements of B to A-elements of C.

1995, Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, page 17

(category theory) A morphism whose codomain is some specified object.


split epimorphism

split epimorphism (plural split epimorphisms)

(category theory) A morphism which has a right inverse.


factor through

en-verb

factor through (third-person singular simple present factors through, present participle factoring through, simple past and past participle factored through)

factors through (present singular third-person)
factoring through (participle present)
factored through (participle past)
factored through (past)

LEMMA 16.3 An arrow f#x3A;A#x5C;rightarrowB is epic iff 1#x5F;B is the smallest sub-object of B that f factors through.

1995, Colin McLarty, Elementary Categories, Elementary Toposes, page 142:

(category theory) Said of a morphism: to be equal to a composition of two morphisms, one of which is the given morphism (that is being said to be "factored through").


subset classifier

en-noun

subset classifier (plural subset classifiers)

subobject classifier
subobject classifier

(category theory, uncountable) The subobject classifier restricted to the category Set.


epi

(category theory) Abbreviation of epimorphism.


prestack

(category theory) The category c is called a prestack over a category C with a Grothendieck topology if it is fibered over C and for any object U of C and objects x, y of c with image U, the functor from objects over U to sets taking F:V→U to Hom(F*x,F*y) is a sheaf.


Grothendieck topology

Grothendieck topology (plural Grothendieck topologies)

Named after the mathematician Alexander Grothendieck.

(category theory) A structure on a category which makes the category's objects act like the open sets of a topological space.

Alexander Grothendieck
Grothendieck topology


forgetful functor

forgetful functor (plural forgetful functors)

(category theory) a functor that forgets or drops some or all of its input's structure or properties before mapping to the output

gleymskuvarpi

category theory: a functor that forgets its inputs's structure


autoequivalence

(mathematics, category theory) An equivalence between an object and itself.


copower

(category theory) The dual object of a power.


coequalizer

(category theory) An arrow whose domain is the codomain of a parallel pair of arrows and which forms part of the colimit of that parallel pair.


Kan extension

Named after Jewish and Dutch mathematician Daniel M. Kan (1927–2013), who constructed certain (Kan) extensions using limits in 1960.

2010, Matthew Ando, Andrew J. Blumberg, David Gepner, Twists of K-Theory and TMF, Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and C*-algebras, American Mathematical Society, page 34,
Moreover, f∗ admits both a left adjoint f_! and a right adjoint f_∗, given by left and right Kan extension along the map SingY→SingX, respectively. Note that this is left and right Kan extension in the ∞-categorical sense, which amounts to homotopy left and right Kan extension on the level of simplicial categories or model categories.

2012, Rolf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, Jeremy Gibbons, Pablo Nogueira (editors), Mathematics of Program Construction: 11th International Conference, MPC 2012, Proceedings, Springer, Lecture Notes in Computer Science: 7342, page 336,
We can specialise Kan extensions to the preorder setting, if we equip a preorder with a monoidal structure: an associative operation that is monotone and that has a neutral element.

We are going to introduce the direct and inverse image functor for presheaves as special Kan extensions and show that they behave well with respect to global and local model structures on simplicial model categories.

2013, Franz Vogler, Derived Manifolds from Functors of Points, Logos Verlag, page 5:

(category theory) A construct that generalizes the notion of extending a function's domain of definition.

extension de Kan

construct that generalizes the extending of a function's domain of definition


hom-set

Short for "homomorphism set", because in some categories the morphisms are homomorphisms.

(category theory) The set or collection of all morphisms from A to B for some given ordered pair (A, B) of objects from some given category.


functor category

functor category (plural functor categories)

(category theory) A category whose objects are functors (from some fixed given domain category to some fixed given codomain category) and whose morphisms are natural transformations.


natural isomorphism

natural isomorphism (plural natural isomorphisms)

(category theory) A natural transformation whose every component is an isomorphism.


identity functor

identity functor (plural identity functors)

(category theory) A functor from a category to itself which maps each object of that category to itself and each morphism of that category to itself.


identity morphism

identity morphism (plural identity morphisms)

(category theory) A unique morphism corresponding to each object of a category, which has its domain equal to its codomain, and which composed with any morphism (with which it is composable) gives that same morphism.


universal morphism

universal morphism (plural universal morphisms)

(category theory) The terminal object of a comma category from a functor to a fixed object; or, dually, the initial object of a comma category from a fixed object to a functor.


coslice category

coslice category (plural coslice categories)

(category theory) The category opposite to that of a slice category.


comma category

comma category (plural comma categories)

Given a pair of functors S:𝒜→𝒞 and T:ℬ→𝒞, objects of the comma category S↓T are arrows h:S(A)→T(B) parametrized by triples (A, B, h), and given morphisms f:A→A' and g:B→B', then a morphism of the said comma category is a commuting square parametrized by the pair (f, g) and spanning the area from h to h':S(A')→T(B') and from S(f) to T(g).

(category theory) A category built out of a pair of functors that have the same codomain.


adjoint functor

adjoint functor (plural adjoint functors)

(category theory) One of a pair of functors such that the domain and codomain of one of them are identical to the codomain and domain of the other one, respectively, and such that there is a pair of natural transformations which turns the pair of functors into an adjunction.


Grothendieck universe

Grothendieck universe (plural Grothendieck universes)

Named after Alexander Grothendieck (1928–2014), a German-born French mathematician.

(mathematics, category theory) A kind of universal set whose elements follow the rules of Zermelo–Fraenkel set theory, and for which, with respect to an arbitrary set, an instance of its kind which has that set as a member may be posited to exist through an additional Tarski–Grothendieck axiom (which is not part of ZF but augments it, yielding Tarski–Grothendieck set theory).


subobject classifier

subobject classifier (plural subobject classifiers)

classifying morphism
classifying morphism

(category theory) An object which serves as the codomain of a classifying morphism, together with a "true" global element of the said object.


kernel pair

(category theory) A couple of morphisms which constitute the pullback of a given morphism along itself.


classifying morphism

classifying morphism (plural classifying morphisms)

(category theory) A morphism from an object to the subobject classifier which corresponds to a unique subobject of the said object, which subobject is the pullback, along this morphism, of the "true" global element of the subobject classifier.


global element

(category theory) A morphism from the terminal object to a given object (to which it is said to belong).


twist morphism

(category theory) An isomorphism between a pair of products which have the same components but in swapped order, which isomorphism commutes with the two associated product diagrams.


Cartesian natural transformation

Cartesian natural transformation (plural Cartesian natural transformations)

Cartesian natural transformations (plural)

So called because its naturality squares are Cartesian squares.

(category theory) A natural transformation whose naturality squares are pullbacks.


diagonal morphism

diagonal morphism (plural diagonal morphisms)

(category theory) A morphism from an object to the product of that object with itself, which morphism is induced by a pair of identity morphisms of the said object.


copresheaf

(category theory) The dual of a presheaf.


biproduct

(category theory) A biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct.


subfunctor

Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated.

2015, Serge Bouc, Jacques Thévenaz, “The representation theory of finite sets and correspondences”, in arXiv:

(category theory) A functor such that all of the objects it maps are mapped by the parent functor, and for any arrow it maps the parent functor includes the same mapping (although it may also map arrows from the same domain to additional images outside the image of the subfunctor).


semiorthogonal

For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles..

2015, Asher Auel, Marcello Bernardara, “Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields”, in arXiv:

(mathematics, category theory) Having the property that every subcategory is in the set of orthogonal subcategories of every higher order subcategory.


schemoid

We also apply the main result to a schemoid constructed from a Hamming scheme..

2016, Yasuhide Numata, “On functors between categories with colored morphisms”, in arXiv:

(category theory) A generalization of an association scheme from the point of view of small categories.


overcategory

(category theory) Any functor into a specified base category.


overfunctor

(category theory) A morphism of an overcategory.


faithful functor

faithful functor (plural faithful functors)

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is injective into the corresponding target hom-set.


full functor

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is surjective into the corresponding target hom-set.


monoidal category

monoidal category (plural monoidal categories)

(category theory) A category 𝒞 with a bifunctor ⊗:𝒞×𝒞→𝒞 which may be called tensor product, an associativity isomorphism α_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C), an object I which may be called tensor unit, a left unit natural isomorphism λ_A:I⊗A≃A, a right unit natural isomorphism ρ_A:A⊗I≃A, and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms).


Galois connection

Galois connection (plural Galois connections)

From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.

1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,
Define maps G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind.

Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of "polarities," and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.

2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:

Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #x5C;left#x5C;langle#x5C;uparrow,#x5C;downarrow#x5C;right#x5C;rangle of mappings #x5C;uparrow assigning subcollections of Y to subcollections of X, and #x5C;downarrow assigning subcollections of X to subcollections of Y.

2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:

partially ordered set
partially ordered set

(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.


group object

If H is another group object in C, a morphism f#x3A;G#x5C;rightarrowH is a morphism of group objects if it commutes with the three structure morphisms; as is standard for sets and true generally (again by Lemma 7.7), it is enough to check #x5C;mu. Thus we form the category #x5C;textit#x7B;Gp#x7D;(C) of all group objects in C; one important example is #x5C;textit#x7B;Gp#x7D;(#x5C;textit#x7B;Ho#x7D;).

1995, J. Michael Boardman, “Chapter 14: Stable Operations in Generalized Cohomology”, in I.M. James, editor, Handbook of Algebraic Topology, Elsevier (North-Holland), page 617:

2005, Angelo Vistoli, Part 1: Grothendieck typologies, fibered categories, and descent theory, Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained, American Mathematical Society, page 20,
The identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by Grp(C).

(category theory) Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element.


group scheme

It is instructive to give a few explicit examples of such^([affine]) group schemes:

1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:

We will tend to use boldface characters, such as #x5C;mathbf#x7B;G#x7D;, for algebraic groups over k, as also for group schemes over #x5C;mathfrak#x7B;X#x7D; that are obtained from groups over k. A quintessential example is #x5C;mathbf#x7B;G#x7D;#x5F;#x5C;mathfrak#x7B;X#x7D;#x3D;#x5C;mathbf#x7B;G#x7D;#x5C;times#x5F;#x7B;k#x7D;#x5C;mathfrak#x7B;X#x7D;. For arbitrary group schemes, or more generally group functors, over #x5C;mathfrak#x7B;X#x7D; we shall tend to use german characters, such as #x5C;mathfrak#x7B;G#x7D;.

2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:

Every finite flat group scheme of order p over S is of the form Gᴸ#x5F;#x7B;a,b#x7D; for some triple (L,a,b), and G#x7B;L#x7D;#x5F;#x7B;a,b#x7D; is isomorphic to G#x7B;L#x5F;1#x7D;#x5F;#x7B;a#x5F;1,b#x5F;1#x7D; if and only if there exists an isomorphism from L to L#x5F;1 carrying a to a#x5F;1 and b to b#x5F;1.

2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:

(category theory, scheme theory) A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.


group functor

In order to be able to efficiently state the main results of this chapter (in §4) we formulate here a group-functor version of the R-nilpotent tower lemma (Ch.III, 6.4).

1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106:

A direct product of k–group functors is again a k–group functor; so is a fibre product if the morphisms used in its construction are homomorphisms of k–group functors.

2003, Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, American Mathematical Society, page 20:

By a group functor we mean a functor from small k-algebras to groups. A homomorphism of group functors is a natural transformation. A subgroup functor of a group functor G is a subfunctor H such that H(R) is a subgroup of G(R) for all R; it is normal if H(R) is normal in G(R) for all R.

2017, J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, page 118:

(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.


vector bundle

Let E and F be two vector bundles on a compact algebraic set X.

1995, Mauro Beltrametti, Andrew John Sommese, The Adjunction Theory of Complex Projective Varieties, Walter de Gruyter, page 8:

2012, Igor Burban, Bernd Kreussler, Vector Bundles on Degenerations of Elliptic Curves and Yang-Baxter Equations, American Mathematical Society, page 5,
This solutions actually depend on the choice of a trivialisation ξ of the universal family 𝒫(n,d) of stable vector bundles of rank n and degree d.

Definition 6.1.5 A non-zero vector bundle #x5C;mathcal#x7B;F#x7D; on X is said to be decomposable if it can be written as #x5C;mathcal#x7B;F#x7D;#x5F;1#x5C;oplus#x5C;mathcal#x7B;F#x7D;#x5F;2 with #x5C;mathcal#x7B;F#x7D;#x5F;1 and #x5C;mathcal#x7B;F#x7D;#x5F;2 non-zero vector bundles. Otherwise it is said to be indecomposable.

2017, David J. Benson, Representations of Elementary Abelian p-Groups and Vector Bundles, Cambridge University Press, page 130:

(topology, category theory) A fiber bundle for which the fiber is a vector space.


linear algebraic group

linear algebraic group (plural linear algebraic groups)

Geometric invariant theory arises in an attempt to construct a quotient of an algebraic variety X by an algebraic action of a linear algebraic group G.

2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii:

The differential Galois group of a homogeneous linear differential equation has a structure of linear algebraic group; hence it is endowed, in particular, with the Zariski topology.[…]Kolchin used the differential algebra developed by Ritt and also built the foundations of the theory of linear algebraic groups.

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page xi:

2015, Willem A. de Graaf, Orbit Closures of Linear Algebraic Groups, Jaime Gutierrez, Josef Schicho, Martin Weimann (editors), Computer Algebra and Polynomials: Applications of Algebra and Number Theory, Springer, LNCS: 8942, page 76,
Actions of linear algebraic groups appear in many contexts. […] Throughout we assume that the base field is algebraically closed and of characteristic 0, as many constructions that we use (e.g., the correspondence between a linear algebraic group and its Lie algebra) only work well in characteristic 0.

general linear group
general linear group

(algebraic geometry, category theory) An algebraic group that is isomorphic to a subgroup of some general linear group.


cofilament

(category theory) A diagram in a category that is indexed by a directed set.


profinite group

profinite group (plural profinite groups)

This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring A as a coefficient ring, like the p-adic integer ring #x5C;Z#x5F;p.

2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:

We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a "quasi-free profinite group" (Section 10.6).

2011, Moshe Jarden, Algebraic Patching, Springer, page 207:

2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,
Recall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite.

(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space.

groupe profini

topological group isomorphic to the inverse limit of an inverse system of discrete finite groups


Kleisli category

Kleisli category (plural Kleisli categories)

Named after the Swiss mathematician Heinrich Kleisli (1930–2011).

(category theory) A category naturally associated to any monad T, and equivalent to the category of free T-algebras.


coconstant morphism

coconstant morphism (plural coconstant morphisms)

(category theory) A morphism which yields the same morphism when pre-composed with either one of a parallel pair of morphisms.


zero morphism

The composition of a zero morphism with any morphism (that it is composable with) yields another zero morphism.

(category theory) A morphism which is both a constant morphism and a coconstant morphism.


constant morphism

constant morphism (plural constant morphisms)

Any morphism to a zero object must be constant.

(category theory) A morphism which yields the same morphism when post-composed with either one of a parallel pair of morphisms.


well-pointed

(category theory) Said of a category or topos: that it has a terminal object (or that it has no zero object, if it is a topos) and in which any distinct pair of parallel morphisms can be distinguished by their distinct compositions with a global element of their domain; i.e. any pair of parallel morphisms is distinct if and only if there is a global element in their domain that does not equalize them.


split monomorphism

split monomorphism (plural split monomorphisms)

(category theory) A morphism which has a left inverse.


Yoneda functor

(category theory) A functor from a given category to the category of functors from that given category to Set (the category of sets) which maps any object of the given category to a hom functor represented by that object and any morphism to a natural isomorphism induced uniquely by that morphism according to the Yoneda lemma.


elementary topos

elementary topos (plural elementary topoi or elementary toposes)

elementary topoi (plural)
elementary toposes (plural)

(category theory) A Cartesian closed category which has a subobject classifier.


equivalence of categories

equivalence of categories (plural equivalences of categories)

There is an equivalence of categories between the category of simply typed lambda calculi and the category of cartesian closed categories; this was shown by Lambek and Scott.

The defining feature of algebraic geometry is the remarkable fact that not only does the geometry determine the algebra, but conversely, the algebra determines the geometry. That is, given any finitely generated #x5C;mathbb#x7B;C#x7D;-algebra R without nilpotent elements, there exists an affine algebraic variety V, uniquely defined up to isomorphism, such that R is isomorphic to the coordinate ring of V. Moreover, any homomorphism between such #x5C;mathbb#x7B;C#x7D;-algebras uniquely defines a morphism of the corresponding varieties. In fancy language, there is an equivalence of categories between the category of affine algebraic varieties and finitely generated, reduced #x5C;mathbb#x7B;C#x7D;-algebras.

©2000, Karen E. Smith with Lauri Kahanpää, Pekka Kekäläinen, and William Traves, edited by S. Axler, F.W. Gehring, and K.A. Ribet, An Invitation to Algebraic Geometry (Universitext), New York: Springer, →ISBN, →OCLC, §2.5, page 24:

(category theory) An adjunction whose unit and counit are both natural isomorphisms.


free category

(category theory) A category that is induced by a multidigraph thus: it has as its objects the vertices of the multidigraph and its morphisms are paths in the multidigraph; composition of morphisms is concatenation of paths, as long as the end of one path coincides with the beginning of the other path; an identity morphism of an object is an “empty path” at that vertex.


small category

small category (plural small categories)

(category theory) A category such that all of its objects form a set and all of its morphisms form a set.


underlying functor

underlying functor (plural underlying functors)

3.1.10 Example If you forget [that] you can compose arrows in a category and you forget which arrows are the identities, then you have remembered only that the category is a graph. This gives an underlying functor U#x3A;#x5C;mathbf#x7B;Cat#x7D;#x5C;rightarrow#x5C;mathbf#x7B;Grf#x7D;, since every functor is a graph homomorphism although not vice versa.

1995, Michael Barr with Charles Wells, Category Theory for Computing Science, 2nd edition, University Press, Cambridge, Great Britain: Prentice Hall, §3.1.10, page 57:


discrete category

discrete category (plural discrete categories)

(category theory) A category whose morphisms are all identity morphisms.


natural numbers object

natural numbers object (plural natural numbers objects)

(category theory) An object which has a distinguished global element (which may be called z, for “zero”) and a distinguished endomorphism (which may be called s, for “successor”) such that iterated compositions of s upon z (i.e., sⁿ∘z) yields other global elements of the same object which correspond to the natural numbers (sⁿ∘z↔n). Such object has the universal property that for any other object with a distinguished global element (call it z’) and a distinguished endomorphism (call it s’), there is a unique morphism (call it φ) from the given object to the other object which maps z to z’ (ϕ∘z=z') and which commutes with s; i.e., ϕ∘s=s'∘ϕ.


Witt group

Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.

groupe de Witt

group of equivalence classes found within a field, variety or category

metabolic quadratic space
metabolic quadratic space

(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;

Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
(algebraic geometry, by extension) given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces
(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.


Cartesian square

Cartesian square (plural Cartesian squares)


pregroupoid

This implies that the so-called internal Galois pregroupoid of an extension is an internal groupoid..

2015, Marino Gran, Diana Rodelo, “Beck-Chevalley condition and Goursat categories”, in arXiv:

(category theory) A category together with its inversion.


bilimit

(category theory) A generalization of the concept of a limit that is an equivalence class rather than a unique value.


strictification

(mathematics, category theory) ??Replacement of coherent isomorphisms by equalities.


hom class

(category theory) A (possibly empty) class composed of the morphisms from A to B, where A and B are objects in some category, denoted operatorname Hom(A,B).


locally small

(category theory, of a category) Such that the hom class between any two objects is a set (and not a proper class).


finitely generated

From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.

(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:

(most generally, category theory, of an object X in a locally small category 𝒞 that admits filtered colimits of monomorphism) Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.


strictly full

(category theory) A subcategory that is isomorphism-closed and full.


fully faithful

(category theory) Of a functor, injective and surjective on hom-sets.


child category

child category (plural child categories)

(category theory) Synonym of subcategory