Category theory

A bicategory has objects, hom-categories, composition functors and identity 1-cells. Associativity and unit laws hold through coherent invertible 2-cells, satisfying pentagon and triangle identities. A monoidal category is exactly a bicategory with a single object: its objects are the 1-cells, its morphisms the 2-cells, and its tensor product the horizontal composition.

Coreflexive pair ( $r f = r g = 1_{B}$ )

A coreflexive pair

f, g : B ⇉ A

has a common retraction

r : A \to B

with

r f = r g = 1_{B}

. It is the dual of a reflexive pair. Both

f

and

g

are split monomorphisms.

Coherent category

A coherent category has finite limits, pullback-stable regular-epi/mono image factorizations, and finite unions of subobjects stable under pullback. Each subobject lattice is distributive. These structures interpret all constructors of coherent logic.

Coherent functor

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

Coherent topology ( $J_{coh}$ )

The coherent topology on a small coherent category is generated by finite families of arrows whose image subobjects have union equal to the whole target. Pullback stability of images and unions and closure under composition make these a coverage. On a coherent syntactic category, this is the site topology used to construct the classifying topos of its coherent theory.

Unions of subobjects are pushouts ( $A \cup B = A ⨿_{A \cap B} B$ )

In a coherent category, maps

A \to Y

and

B \to Y

agreeing on the intersection define a relation from

A \cup B

Y

by taking the union of their graphs. It is total because the two subobjects cover their union, and single-valued because the maps agree on overlap. Total functional relations define morphisms, giving a unique glued map. This is the pushout universal property without assuming a general supply of pushouts.

Right lifting property ( $i ⋔ p$ )

A map

p

has the right lifting property against

i : A \to B

when every commutative square with

i

on the left and

p

on the right has a diagonal

B \to dom p

making both triangles commute. Lifting against a family of maps gives a class of right-injective maps.

Model category ( $(W, F, C)$ )

A complete and cocomplete category with designated model weak equivalences, model fibrations and model cofibrations, closed under retracts. Weak equivalences satisfy two-out-of-three; maps admit the two cofibration/fibration factorizations, and either acyclic class has the required lifting property against the opposite class. These axioms organize homotopy-theoretic calculations abstractly.

Small object argument ( $X \to Z_{\infty} \to Y$ )

Attach cells for all lifting squares at each successive stage, then take a suitable colimit. Smallness of the generating domains makes every eventual lifting square occur at an earlier stage. This factors a map into a relative cell complex followed by a map with the prescribed right lifting property.

Relative cell complex ( $X = Z_{0} \to Z_{1} \to \dots \to Z_{\infty}$ )

A composite built by successive pushouts of coproducts of prescribed generating maps. In nonnegative chain complexes, attaching a sphere-to-disk generator adds a free element with a specified existing cycle as boundary. The resulting inclusion has a degreewise free cokernel.

Sequentially small object ( $colim Hom (A, Z_{k}) ≅ Hom (A, colim Z_{k})$ )

Mapping out of the object preserves sequential colimits. A map factors through a finite stage, and two finite-stage maps agreeing in the colimit agree at a later stage. Relative sequential smallness restricts the diagrams to a stated class of morphisms.

Projective model structure on nonnegative chain complexes ( $Ch_{\geq 0} (R)$ )

The model category of nonnegative complexes of left modules has quasi-isomorphisms as model weak equivalences, positive-degree surjections as model fibrations, and degreewise injections with projective cokernel as model cofibrations. All objects are fibrant; cofibrant objects are degreewise projective.

Generating cofibrations for nonnegative chain complexes ( $I = {0 \to S^{0} (R)} \cup {S^{n - 1} (R) \to D^{n} (R) : n \geq 1}$ )

The sphere chain complex inclusions into disk chain complexes, together with the separate degree-zero cell, generate the model cofibrations by cell attachment and retracts. Their right-injective class is exactly the acyclic fibration in the projective chain-complex model structure.

Lifting characterization of an acyclic chain-complex fibration ( $d z = x, f (z) = y when f (x) = d y$ )

The right lifting property for each sphere-to-disk map fills a prescribed cycle and a compatible target element. The degree-zero generator supplies initial surjectivity; all other fillers force an acyclic kernel and the remaining degreewise surjections.

Acyclic fibration in the projective chain-complex model structure ( $f_{n} surjective for all n, H_{*} (f) invertible$ )

A map that is both a model fibration and a quasi-isomorphism. The homology isomorphism together with degree-one surjectivity forces surjectivity in degree zero. Equivalently it has the right lifting property against the generating cofibrations for nonnegative chain complexes.

Model cofibration ( $C$ )

A morphism in the chosen cofibration class of a model category. In the projective model structure on nonnegative chain complexes it is a degreewise split injection with degreewise projective cokernel. This usage is distinguished from the topological cofibration.

Model fibration ( $F$ )

A morphism in the chosen fibration class of a model category. In the projective model structure on nonnegative chain complexes, it is surjective in strictly positive degrees; degree zero is not required.

Model weak equivalence ( $W$ )

A morphism in the chosen weak-equivalence class of a model category. In the projective model structure on nonnegative chain complexes these are quasi-isomorphisms.

Monoidal category ( $(C, \otimes, I)$ )

A category equipped with a monoidal tensor product, a monoidal unit object, and natural associators and unitors satisfying the pentagon and triangle axioms.

Product and coproduct monoidal structures on sets

Cartesian product with singleton unit and disjoint union with empty unit each make the Category of sets monoidal. Associators and unitors simply retag or rebracket the same elements, so the pentagon and triangle identities hold elementwise. These structures differ even up to monoidal equivalence: any category equivalence preserves initial and terminal objects, so it cannot identify the empty unit with the singleton unit.

Triangle identity for a monoidal category

The associator and unitors of a monoidal category satisfy

(1_{X} \otimes λ_{Y}) a_{X, I, Y} = ρ_{X} \otimes 1_{Y} .

Thus the two ways to remove a unit between two tensor factors agree. Together with the pentagon identity for a monoidal category this implies all unit identities derived from the monoidal pentagon and triangle and the monoidal coherence theorem.

Pentagon identity for a monoidal category

For an associator

a

in a monoidal category, the two paths from

((W \otimes X) \otimes Y) \otimes Z

W \otimes (X \otimes (Y \otimes Z))

agree:

a_{W, X, Y \otimes Z} a_{W \otimes X, Y, Z} = (1_{W} \otimes a_{X, Y, Z}) a_{W, X \otimes Y, Z} (a_{W, X, Y} \otimes 1_{Z}) .

This five-vertex diagram is the basic compatibility for reassociating four tensor factors. The monoidal coherence theorem extends it to arbitrary finite tensor expressions.

Unit identities derived from the monoidal pentagon and triangle ( $λ_{I} = ρ_{I}$ )

The associator

a

and unitors of a monoidal category satisfy

ρ_{X \otimes Y} = (1_{X} \otimes ρ_{Y}) a_{X, Y, I}, λ_{X \otimes Y} a_{I, X, Y} = λ_{X} \otimes 1_{Y}, λ_{I} = ρ_{I} .

For the first identity, postcompose the pentagon on

X, Y, I, Z

1_{X} \otimes (1_{Y} \otimes λ_{Z})

, then use the triangle twice and cancel

a_{X, Y, Z}

. This gives the first equality after tensoring with

1_{Z}

; set

Z = I

and cancel using the natural invertible right unitor. Reversing tensor order proves the second identity. Naturality of the left unitor at

λ_{X}

gives

λ_{I \otimes X} = 1_{I} \otimes λ_{X}

; comparing the second identity with the triangle on

I, I, X

gives

(λ_{I} \otimes 1_{X}) = (ρ_{I} \otimes 1_{X})

and hence the last identity. These are consequences of the axioms, not extra axioms imported from the monoidal coherence theorem.

Yang–Baxter operator ( $y : X \otimes X \to X \otimes X$ )

An invertible morphism

y : X \otimes X \to X \otimes X

satisfying

(y \otimes 1) (1 \otimes y) (y \otimes 1) = (1 \otimes y) (y \otimes 1) (1 \otimes y)

with coherent parentheses. It produces representations of the braid groups; a self-braiding is an example.

Dual pair in a monoidal category ( $(X, Y, e, n)$ )

Objects

X, Y

with an evaluation morphism

e : X \otimes Y \to I

and a coevaluation morphism

n : I \to Y \otimes X

satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

Quantum dimension ( $dim_{q} (V) = tr_{q} (1_{V})$ )

Closing an identity strand using compatible evaluation morphisms and coevaluation morphisms gives the categorical dimension in a pivotal or ribbon setting. For the standard two-dimensional quantum enveloping algebra of sl2 module, the weighted contraction has diagonal coefficients

q^{- 1}, q

, giving quantum dimension

q + q^{- 1} = [2]_{q}

. The chosen duality and framing conventions matter.

Snake identity

The two triangular identities

(e \otimes 1_{X}) (1_{X} \otimes n) = 1_{X}

and

(1_{Y} \otimes e) (n \otimes 1_{Y}) = 1_{Y}

, with canonical associators and unitors inserted, for a dual pair in a monoidal category.

Coevaluation morphism ( $n : I \to Y \otimes X$ )

The unit-valued duality map

n : I \to Y \otimes X

of a dual pair in a monoidal category, constrained together with evaluation by both snake identities.

Evaluation morphism ( $e : X \otimes Y \to I$ )

For a dual pair in a monoidal category, the contraction

e : X \otimes Y \to I

. In a left closed monoidal category, evaluation also denotes the adjunction map

[M, N] \otimes M \to N

Left closed monoidal category ( $- \otimes M ⊣ [M, -]$ )

Using the convention

- \otimes M ⊣ [M, -]

, every tensoring-on-the-right functor has a right adjoint functor. Evaluation is

[M, N] \otimes M \to N

. Naming left and right closure varies in the literature, so the adjunction fixes the convention.

Comonoid ( $(C, Δ, ε)$ )

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

Counit ( $ε$ )

The structure map

ε : C \to I

of a comonoid, with

(ε \otimes 1) Δ

and

(1 \otimes ε) Δ

equal to the inverse unitors. This is distinct from the counit of an adjunction.

Comultiplication ( $Δ$ )

The structure map

Δ : C \to C \otimes C

of a comonoid; it is coassociative up to the ambient associator.

Comonoid morphism

A morphism

f : C \to D

satisfying

Δ_{D} f = (f \otimes f) Δ_{C}

and

ε_{D} f = ε_{C}

Monoid object ( $(A, m, j)$ )

An object with multiplication

m : A \otimes A \to A

and unit

j : I \to A

satisfying associativity and unit diagrams using the ambient associators and unitors.

Frobenius monoid ( $(A, m, j, ε)$ )

A monoid object

A

with a morphism

ε : A \to I

such that

ε m

and a suitable coevaluation make

(A, A)

a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

Bimonoid

An object that is both a monoid object and a comonoid in a braided monoidal category, with

Δ, ε

preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

Coquasitriangular structure ( $γ : H \otimes H \to I$ )

In a symmetric monoidal category, a convolution-invertible scalar pairing

γ : H \otimes H \to I

on a bimonoid inducing the comodule braiding

u \otimes v \mapsto \sum v_{(0)} \otimes u_{(0)} γ (u_{(1)}, v_{(1)})

. Its multiplicativity, normalization and commutation axioms express the hexagon, unit and colinearity conditions. The notation abbreviates morphisms and ambient symmetries, not a requirement for elements.

Monoid morphism

A morphism

f : A \to B

satisfying

f m_{A} = m_{B} (f \otimes f)

and

f j_{A} = j_{B}

Braided monoidal category

A monoidal category with natural invertible braidings

c_{X, Y} : X \otimes Y \to Y \otimes X

satisfying the two hexagon axioms. The equation

c_{Y, X} c_{X, Y} = 1

is an additional symmetric condition, not a braided axiom.

Free braided monoidal category on one object ( $F b r$ )

The braided monoidal category generated by one object: objects are parenthesized tensor expressions in that object and the unit, and morphisms are structural isomorphisms and crossings subject to precisely the monoidal category and braiding axioms.

Braid category ( $B$ )

The braided monoidal category with objects

n \geq 0

, endomorphism groups

B_{n}

, no arrows between unequal objects, tensor given by juxtaposition, and block-crossing braiding. Its tensor is strict.

Braided monoidal functor

A monoidal functor between braided monoidal categories compatible with their braidings:

F (c_{X, Y}) F_{2} (X, Y) = F_{2} (Y, X) c_{FX, F Y}

Braiding ( $c_{X, Y}$ )

The natural tensor interchange isomorphism in a braided monoidal category, constrained by the two hexagon axioms.

Opmonoidal functor

A functor with natural maps

F_{2} : F (X \otimes Y) \to FX \otimes F Y

and

F_{0} : F I \to I

satisfying the reversed monoidal functor associativity and unit diagrams. No invertibility is required.

Opmonoidal natural transformation ( $τ : F \Rightarrow G$ )

A natural transformation

τ : F \Rightarrow G

satisfying

G_{2} τ_{X \otimes Y} = (τ_{X} \otimes τ_{Y}) F_{2}

and

G_{0} τ_{I} = F_{0}

Monoidal functor

Here the unqualified term allows lax comparison maps

F_{2} : FX \otimes F Y \to F (X \otimes Y)

and

F_{0} : I \to F I

, natural and compatible with associators and unitors. Invertible comparison maps give a strong monoidal functor. The opposite direction gives an opmonoidal functor.

Frobenius monoidal functor

A lax monoidal functor

φ

and opmonoidal functor

ψ

on the same underlying functor, satisfying

ψ_{A \otimes B, C} φ_{A, B \otimes C} = (φ_{A, B} \otimes 1) (1 \otimes ψ_{B, C})

and

ψ_{A, B \otimes C} φ_{A \otimes B, C} = (1 \otimes φ_{B, C}) (ψ_{A, B} \otimes 1)

in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

Strong monoidal functor

A monoidal functor whose tensor and unit comparison maps are isomorphisms.

Strict monoidal functor

A strong monoidal functor whose comparison maps are identities. It preserves the tensor, unit and structural constraints exactly; its source need not be a strict monoidal category.

Strict monoidal category

A monoidal category whose associators and unitors are identities; the tensor and unit laws are then equalities of objects and morphisms.

Monoidal coherence theorem

Every diagram formed solely from the canonical associators, unitors and their inverses commutes. Consequently structural reparenthesizations can be suppressed in calculations; this statement does not make distinct braidings equal.

Termination proof of monoidal coherence

Orient associativity towards right association and unit maps towards deletion of unit symbols. A lexicographic measure consisting of the number of leaves and the sum of the sizes of left subtrees decreases at every step. All overlapping reductions have commuting completions: the essential cases are the pentagon identity for a monoidal category, the triangle identity for a monoidal category, and the derived left and right unit identities. Induction on the decreasing measure makes every normalization arrow equal, proving that every structural arrow is determined by its endpoints.

Word normalization proof of monoidal coherence

Associate to a finite ordered word

w

the right-associated tensor

N (w)

, retaining a terminal unit:

N (\emptyset) = I

and

N (A w) = A \otimes N (w)

. Define concatenation maps recursively by

c_{\emptyset, v} = λ_{N (v)}, c_{A u, v} = (1_{A} \otimes c_{u, v}) a_{A, N (u), N (v)} .

The unit identities derived from the monoidal pentagon and triangle give

c_{u, \emptyset} = ρ_{N (u)}

. Induction on

u

, using the pentagon in the induction step and left-unitor compatibility in the base case, gives

c_{uv, w} (c_{u, v} \otimes 1) = c_{u, v w} (1 \otimes c_{v, w}) a_{N (u), N (v), N (w)} .

(2)

Normalize a letter by

ρ_{A}^{- 1}

, a formal unit by

1_{I}

, and a tensor expression recursively using

c

. The displayed identities show that normalization commutes with each generating associator or unitor, and therefore also with inverses, tensor products and composites of them. Every structural arrow from expression

S

to expression

T

is consequently

ν_{T}^{- 1} ν_{S}

, proving monoidal coherence without presupposing it.

Unitor ( $λ, ρ$ )

The natural isomorphisms

λ_{X} : I \otimes X \to X

and

ρ_{X} : X \otimes I \to X

in a monoidal category. Together with the associator they satisfy the triangle axiom.

Associator ( $α$ )

The natural isomorphism

α_{X, Y, Z} : (X \otimes Y) \otimes Z \to X \otimes (Y \otimes Z)

in a monoidal category. It satisfies the pentagon axiom.

Monoidal unit object ( $I$ )

The distinguished object

I

in a monoidal category, with the natural unitors

I \otimes X ≅ X ≅ X \otimes I

Monoidal tensor product ( $\otimes$ )

The bifunctor

\otimes : C \times C \to C

in a monoidal category. Its associativity is expressed by the associator, rather than equality in general.

Category of groups ( $Grp$ )

The category of groups has groups as objects and group homomorphisms as morphisms. Its product is the direct product of groups, and its coproduct is the free product, as expressed by the universal property of a free product.

Injective object

An object

I

is injective when every morphism

A \to I

extends across every monomorphism

A ↪ B

to a morphism

B \to I

. This is the categorical extension property of an injective module or an injective sheaf; it is dual to the lifting property of a projective object in a category.

Universal property

A universal property characterizes an object by a prescribed family of morphisms and a unique factorization of every competing family. It determines that object up to a unique compatible isomorphism. Examples include the pushout in a category and the universal property of the tensor product of modules.

Category ( $C$ )

A category specifies objects of a category, categorical morphisms between them, and composition in a category. An arrow

f : A \to B

can be followed by

g : B \to C

, giving

g \circ f : A \to C

. Every object

A

has an identity morphism

1_{A} : A \to A

, and the rules satisfy

h \circ (g \circ f) = (h \circ g) \circ f, f \circ 1_{A} = f = 1_{B} \circ f .

Associativity means that grouping a chain differently does not change its composite. It does not assert commutativity, and the sources and targets must match before a composite is defined.

In the Category of sets, arrows are functions. Take two functions on the real numbers,

f (x) = x + 1

and

g (x) = 2 x

. Then

(g \circ f) (x) = 2 x + 2

, whereas

(f \circ g) (x) = 2 x + 1

. At

x = 3

, the two orders give 8 and 7, respectively. Both orders are defined here because these maps share their source and target; for arbitrary arrows the reverse composite need not exist.

Figure 1.
Function composition remembers order, while a naturality square gives equal results along both routes
. **Details**.

A partially ordered set supplies a different example: its elements are the objects of a category, with exactly one arrow

a \to b

when

a \leq b

, and none otherwise. Transitivity gives composition in a category, and reflexivity gives identity morphisms. For subsets ordered by inclusion, the two chains

\emptyset \subseteq {a} \subseteq {a, b}

and

\emptyset \subseteq {b} \subseteq {a, b}

have the same composite arrow. Thus arrows need not be maps of elements: they can record a relation.

Duality (category theory)

Reversing the arrows in a categorical statement produces its dual statement. A theorem holding for all categories therefore has a dual theorem obtained by applying it to the opposite category; for example, categorical limits are exchanged with colimits.

Morphism in a category ( $f : A \to B$ )

A categorical morphism is an arrow with a specified source and target in a category. An arrow

f : A \to B

composes with

g : B \to C

to give

g f : A \to C

. Composition is associative, and every object has an identity arrow acting as a unit. A functor preserves the sources, targets, identities and compositions of these arrows. In a locally small category, the arrows between fixed objects form a hom-set.

Small projective object ( $E (A, -) preserves all small colimits$ )

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a presheaf category, every representable presheaf has this property by the Yoneda lemma and pointwise computation of colimits. Conversely the canonical colimit presentation of a presheaf shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has equalizers, a representable retract is again representable by the representable retract criterion.

Category of small categories ( $Cat$ )

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

Indiscrete category

The indiscrete category on a set has exactly one morphism between each ordered pair of objects. Every morphism is invertible. A functor into it is determined by its object assignment.

Discrete category

The discrete category on a set has that set as its objects and only identity morphisms. Functors from it are exactly assignments of objects in the target category.

Free category on independent arrows ( $L (S) = ∐_{s \in S} [1]$ )

Take two distinct objects and one nonidentity arrow for each element of

S

, without arrows between different copies. A functor from this category to

C

is exactly a choice of one arbitrary arrow of

C

for each element of

S

. Consequently this construction is left adjoint to the functor taking the set of all arrows of a small category. Its free object has three arrows per generator, including the two identity arrows.

Preadditive category

A category is preadditive if every hom-set is an abelian group and composition is additive in each variable. Zero morphisms and differences of parallel morphisms are therefore defined. A zero object or finite biproducts need not exist; these are extra requirements in an additive category.

Coreflective subcategory ( $I ⊣ q$ )

A full subcategory is coreflective when its inclusion

I

has a right adjoint

q

. The counit

I qX \to X

is universal for maps to

X

from objects of the subcategory. The comonad

I q

is idempotent. For a finite-limit-closed coreflective subcategory of a topos, its coalgebra construction can prove that the subcategory is itself a topos.

Generating set in a category ( $(P_{i})_{i \in I}$ )

A generating set is a set-indexed family

(P_{i})

such that unequal parallel morphisms

f, g : A \to B

are distinguished by some

h : P_{i} \to A

, with

f h \neq = g h

. It is dual to a cogenerating set.

Composition in a category ( $g \circ f$ )

Given morphisms

f : A \to B

and

g : B \to C

, their composite is

g f : A \to C

. Composition is associative and has the identity morphisms as units.

Object of a category

The objects are the entities between which a category specifies morphisms. Each object has an identity morphism.

Terminal category ( $1$ )

The category with one object and only its identity morphism. A functor from it selects an object of the target category.

Nested partial unary operation category ( $C_{n}$ )

An object of

C_{n}

is a set with a finite list of partial unary operations. The first is total; the domain of each later operation consists exactly of the defined fixed points of the immediately preceding operation. Morphisms preserve every defined operation. Take

C_{0}

to be the Category of sets. This tower gives examples of arbitrary finite monadic length even though its long free–forgetful composites induce the same first monad.

Free extension of nested partial unary operations ( $F_{n} ⊣ G_{n}$ )

For an object of the nested partial unary operation category

C_{n}

, adjoining the next operation freely gives one new value at each defined fixed point of the previous operation. Each new value starts an infinite free chain under the first unary operation; the other old operations are undefined on those new points. The new top operation consequently has no fixed points, so subsequent free extensions add no points. This gives a left adjoint to forgetting the last operation; extensions of maps along the chains are forced by the first target operation.

Split-coequalizer lifting for a fixed-point-domain operation ( $β_{Q} (z) = q β_{B} (sz)$ )

In the nested partial unary operation category, suppose the lower-operation diagram

A ⇉ B q Q

has a split coequalizer with

q s = 1

f t = 1

and

g t = s q

. The section

s

takes eligible fixed points to eligible fixed points, so the displayed next operation is defined on exactly the required domain. For eligible

x \in B

, the equations give

q β_{B} (x) = q f β_{A} (t x) = q g β_{A} (t x) = q β_{B} (s q x) .

Thus

q

preserves the next operation. Every factored morphism preserves it by evaluating at

sz

, giving creation of the specified coequalizer. Together with reflection of isomorphisms, the Beck monadicity theorem proves one-step monadicity.

Coseparator

A coseparator

S

in a category distinguishes parallel arrows by postcomposition: if

u \neq = v : X \to B

, some

t : B \to S

satisfies

t u \neq = t v

. Equivalently, the contravariant functor

C (-, S)

is faithful. In a complete category that is a locally small category, the evaluation arrow

B \to \prod_{t \in C (B, S)} S

is a monomorphism.

Cogenerating set ( $(Q_{i})_{i \in I}$ )

A cogenerating set is a set-indexed family

(Q_{i})_{i \in I}

such that whenever

f \neq = g : A \to B

, there is a morphism

q : B \to Q_{i}

with

q f \neq = q g

. It generalizes a single coseparator. In a locally small category with the needed products in a category, it gives an evaluation embedding into cogenerator products.

Evaluation embedding into cogenerator products

For a small cogenerating family

(Q_{i})

in a locally small category, the evaluation arrow

C \to \prod_{i, q \in C (C, Q_{i})} Q_{i}

is a monomorphism whenever that small product in a category exists. Equality after all its projections is equality after every map to a cogenerator, hence equality of the original parallel morphisms.

Equalizer presentation by powers of a coseparator ( $B ⟶ S^{I} ⇉ S^{J}$ )

In a complete category that is a locally small category with a coseparator

S

, suppose every monomorphism is a regular monomorphism. The evaluation embedding

B \to S^{I}

I = C (B, S)

, is an equalizer of some pair into

Q

. Composing that pair with the evaluation embedding

Q \to S^{J}

leaves the equalizer unchanged. Here powers mean products in a category indexed by sets, without assuming an exponential object exists.

Category of rings ( $Rng$ )

The category of unital rings has identity-preserving ring homomorphisms as morphisms. The zero ring, allowing

0 = 1

, is terminal, and

Z

is initial. Nonzero rings admit no maps from the zero ring. Monomorphisms are injective: maps from

Z [x]

probe individual elements. It is a regular category, with surjective homomorphisms as regular epimorphisms; these are distinct from arbitrary categorical epimorphisms.

Skeletal category

A skeletal category has no distinct isomorphic objects. A full and faithful functor with essential surjectivity between two skeletal categories is an isomorphism of categories: surjectivity on isomorphism classes becomes surjectivity on objects, and faithfulness and fullness reflect an equality of image objects to an isomorphism, hence an equality, of source objects.

Skeleton of a category

A skeleton of

C

is a full subcategory with exactly one object from each isomorphism class. The axiom of choice supplies skeletons of small categories. Conversely, if every small category has a skeletal equivalent, apply this to the groupoid on

∐_{i} A_{i}

with exactly one morphism between objects in the same nonempty fibre. A quasi-inverse from the skeletal category selects one object per fibre, giving a choice function. This converse uses an equivalence of categories with specified quasi-inverse functors, not just the existence of a full essentially surjective functor in one direction.

Subcategory

A subcategory

S

of a category

C

specifies some objects and some morphisms between them, containing their identities and closed under composition. It is full if it contains every ambient morphism between its selected objects. A skeleton of a category is a full subcategory containing one object from each isomorphism class.

A groupoid is a category in which every morphism is invertible. A group is a groupoid with one object, while an equivalence relation defines a groupoid with exactly one arrow between any related pair of objects. Every functor sends invertible arrows to invertible arrows.

Functor ( $F : C \to D$ )

A functor assigns an object of a category

F (A)

to each object

A

and a categorical morphism

F (f) : F (A) \to F (B)

to each arrow

f : A \to B

. It preserves the rules for composing arrows:

F (1_{A}) = 1_{F (A)}, F (g \circ f) = F (g) \circ F (f) .

For example, the forgetful functor from the category of groups to the Category of sets keeps the underlying sets and maps of elements while forgetting the group operations. A group homomorphism remains a function, and composing the underlying functions gives the underlying map of the composite group homomorphism.

Another example sends a set

X

to its power set

P (X)

. It sends a function

f : X \to Y

to the function

P (f) : P (X) \to P (Y)

defined by

S \mapsto f (S)

, the image of the subset

S

. For

f (x) = x + 1

and

g (x) = 2 x

, the subset

{0, 2}

goes to

{1, 3}

and then

{2, 6}

; applying the composite

g \circ f

directly also gives

{2, 6}

. The equality of these routes expresses preservation of composition in a category.

Bifunctor ( $F : C \times D \to E$ )

A bifunctor is a functor on the product of two categories. It assigns

F (c, d)

to each pair of objects and

F (u, v)

to a pair of morphisms, preserving identity morphisms and composition in a category. Equivalently, it is functorial separately in each argument, with the two actions commuting:

F (1, v) F (u, 1) = F (u, 1) F (1, v)

whenever the expressions have the corresponding sources and targets. The product category has pairs as objects, pairs of morphisms as arrows, and componentwise composition in a category. A monoidal tensor product is a bifunctor from a category times itself.

Discrete opfibration ( $F : C \to D$ )

A functor with a unique lift, from each specified object

c

, of every arrow

F c \to d

. The lifted arrow's target is part of the uniqueness requirement. This is dual to a discrete fibration. Lifting a cocone of a finite connected category gives a cocone upstairs: lifts of its legs have a common target because uniqueness identifies targets along every diagram arrow. Thus a discrete opfibration over a weakly filtered category has weakly filtered domain, including the possibility of an empty domain.

Connected components of discrete-opfibration pullbacks ( $π_{0} (A \times_{D} B) ≅ π_{0} (A) \times_{π_{0} (D)} π_{0} (B)$ )

For discrete opfibrations

A \to D \leftarrow B

with weakly filtered categories, the map taking a pair to its connected components of a category is a bijection. For surjectivity, send two base objects in the same component to a common target and lift both arrows. For injectivity, choose common targets upstairs for two representatives; a cocone over the resulting finite connected diagram downstairs makes their lifted composites agree. Hence the representatives have a common target in the pullback in a category. Over a filtered category, the component functor on the slice of discrete opfibrations preserves all finite categorical limits: it preserves pullbacks and takes the terminal object to a singleton.

Full functor

A functor

F : C \to D

is full when each function

C (A, B) \to D (F A, FB)

is surjective. It is a faithful functor when each is injective. A full and faithful functor identifies its source, up to equivalence, with a full subcategory of the target. Fullness only concerns morphisms between image objects; it does not require every target object to lie in the image.

Flat functor ( $F : C \to E$ )

A covariant functor into a Grothendieck topos is flat when its tensor extension from presheaves preserves finite limits. For set values, its category of elements with arrows carrying source elements to target elements is cofiltered; equivalently the functor is a filtered colimit of covariant representables. If the indexing category has finite limits, flatness is equivalent to preservation of finite limits. For a site, cover-to-joint-epimorphism continuity adds the condition needed by the Diaconescu equivalence for geometric morphisms.

Flat functor and finite-limit preservation ( $C finitely complete ⟹ (F flat ⟺ F left exact)$ )

For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.

Kan extension ( $Lan_{f}, Ran_{f}$ )

A left Kan extension is left adjoint to precomposition, when it exists; a Right Kan extension is right adjoint. For small indexing categories and set-valued functors, both exist by comma-category colimit and limit formulas. These two adjoints give the essential geometric morphism induced by a functor between presheaf indexing categories.

Finite-limit-preserving functor

A finite-limit-preserving functor preserves a terminal object and pullbacks, equivalently all finite limits. It is also called left exact. Such functors from a Cartesian syntactic category to a finite-limit category correspond to models of the associated Cartesian theory.

Regular functor

A functor between regular categories is regular if it preserves finite limits and regular epimorphisms. It then preserves image factorizations, since finite-limit preservation preserves monomorphisms. A representable functor

C (P, -)

is regular exactly when

P

is projective with respect to regular epimorphisms. Regularity of a functor does not alone imply faithfulness or reflection of isomorphisms.

Conservative functor