Title: Theory / Unimportant / Math / Category / wp=
Tags: fromMath mathToDef récupéré mathToSee hasSeenConjDef

Name: A zero object
Definition: An object which is both initial and terminal.
Page: wp
Variables: \(\mathbf C\) a category
Name2: A null object

Name: A strict initial object
Definition: An initial object \(0\) such that every morphism \(A\to 0\) is iso.
Page: wp
Variables: \(\mathbf C\) a category

Smaller0: \(\mathbf C\) is ccc
Greater0: \(O\) strict initial
Page: wp
Variables: \(\mathbf C\) a category,
\(O\in \mathbf C\)
Smaller1: \(O\) initial
Connect Smaller0: and

Name: A distributive category
Definition: finite products, coproduct,
Page: wp
Definition2: the canonical map \([id_A\times \iota_1,id_A\times \iota_2]:A\times B+A\times C\to A\times (B+C)\) is iso
Conjdef: Category with
Definition3: the canonical map \(0\to A\times 0\) is iso

Name: A constant morphism
Definition: A morphism \(f:X\to Y\) such that, for any \(W\in\mathbf C_0\), \(g,h:W\to X\), \(fg=fh\).
Page: wp
Variables: \(\mathbf C\) a category
Name2: Left zero morphism

Name: A coconstant morphism
Definition: A morphism \(f:X\to Y\) such that, for any \(Z\in\mathbf C_0\), \(g,h:Y\to Z\), \(gf=hf\).
Page: wp
Variables: \(\mathbf C\) a category
Name2: A right zero morphism

Name: A zero morphism
Definition: A left and right zero morphism.
Page: wp
Variables: \(\mathbf C\) a category

Name: A category with zero morphism
Definition: A category \(\mathbf C\) such that, for every two objects \(A\) and \(B\) in \(\mathbf C\), there is a fixed morphism \(0_{AB} : A\to B\) such that for all objects \(X, Y, Z\in\mathbf C_0\) and all morphisms \(f : Y\to Z, g : X\to Y\), the following diagram commutes: .
Page: wp

Smaller0: each hom-set has a zero morphism
Greater0: the category has zero morphisms
Page: wp
Variables: \(\mathbf C\) a category

Name: A difference kernel
Definition: A binary equalizer.
Page: wp

Name: A mediating morphism
Definition: The unique arrow from a cone of a diagram \(\mathbf J\) to its limit.
Page: wp

Name: The kernel pair of \(f\)
Definition: The pullback of \(f\) with itself.
Page: wp
Variables: \(f:X\to Y\) in a category

Name: A regular epi
Definition: An epi which coequalizes some parallel pair of morphism.
Page: wp

Name: A regular category
Definition: finitely complete
Page: https://en.wikipedia.org/wiki/Regular_category
Definition2: for each \(f:X\to Y\) with \((Z,p_0,p_1)\) its kernel pair, \((p_0,p_1)\) admits a coequalizer.
Conjdef: A category such that, it is all of
Definition3: for each regular epi \(f:X\to Y\), \(f':X'\to Y\), let \((W,g,g')\) the pullback of \((f,f')\), \(g'\) is a regular epi.

Name: A presheaf of \(\mathbf C\)
Definition: A functor \(F:\mathbf C^{\text{op } }\to \mathbf{Set}\).
Page: wp
Variables: \(\mathbf C\) a category

Name: The basis
Definition: \(X\).
Page: wp
Variables: \((\mathbf C,F)\) a concrete category,
\(X,Y\) sets,
\(i:X\to Y\),
\(A\) a free object on \(X\) with respect to \(i\)

Name: The canonical injection
Definition: \(i\).
Page: wp
Variables: \((\mathbf C,F)\) a concrete category,
\(X,Y\) sets,
\(i:X\to Y\),
\(A\) a free object on \(X\) with respect to \(i\)

Name: The forgetful functor
Definition: \(U\).
Page: wp
Variables: \((\mathbf C,U)\) a concrete category

Name: A bimorphism
Definition: An epi mono.
Page: wp
Variables: \(\mathbf C\) a category

Name: A reflective subcategory
Definition: A subcategory whose inclusion functor has a left adjoint.
Page: wp
Variables: \(\mathcal C\) a category

Name: A coreflective subcategory
Definition: A subcategory whose inclusion functor has a right adjoint.
Page: wp
Variables: \(\mathcal C\) a category

Name: The reflector
Definition: The left adjoint of the inclusion functor.
Page: wp
Variables: \(\mathbf C\) a category,
\(\mathbf A\) a reflective subcategory

Name: An \(\mathbf A\)-reflection of \(C\)
Definition: \((A_B,r_B)\) with \(A_B\in\mathbf A_0\), \(r_B:B\to A_B\in \mathbf C_1\) such that for each \(f:B\to A\) there exists a unique \(\overline f:A_B\to A\) with \(\overline f\circ r_B=f\).
Page: wp
Variables: \(\mathbf C\) a category,
Let \(\mathbf A\) a reflective subcategory,
Let \(C\in\mathbf C_0\)

Name: An (extremal epi, epi, bi)reflective subcategory
Definition: A reflective subcategory \(\mathbf A\) whose \(\mathbf A\)-reflection arrows are (extremal epi, epi, bi)morphisms.
Page: wp
Variables: \(\mathbf C\) a category

Name: The \(E\)-reflective hull of \(\mathbf A\)
Definition: The smallest \(E\)-reflective subcategory containing \(\mathbf A\).
Page: wp
Variables: \(\mathbf C\) a category,
\(A\) a subcategory with embedding functor \(E\)

Name: An extremal epi
Definition: An epi which whose mono second hand factor are iso.
Page: wp
Variables: \(\mathbf C\) a category

\(\mathbf C\) is a topos or \(\mathbf C\) is Abelian or \(\mathbf C\) is the models of a variety or \(\mathbf C\) is bounded meet-semilattice implies \(\mathbf C\) is regular
Page: wp
Variables: \(\mathbf C\) a category

Name: An exact sequence
Definition: A diagram \(R\rightrightarrows X\to Y\) if it is both a coequalizer and a kernel pair.
Page: wp
Variables: \(\mathbf C\) a category
Assuming: \(\mathbf C\) regular

Name: A regular functor
Definition: A functor between regular categories which preserves finite limits and coequalizers of kernel pairs
Page: wp
Definition2: A functor between regular categories which preserves finite limits and exact sequences

Smaller0: \(C\) is terminal
Greater0: is the product of \(A\) and \(B\)
Page: wp
Variables: \(\mathbf C\) a category,
\(A,B,C\) objects
Hide middle: yes
Prefix Greater: \(A\times_CB\)

Name: A pushout
Definition: The colimit of the diagram composed of \(X,Y_1,Y_2,f_1,f_2\).
Page: wp
Variables: \(\mathbf C\) a category,
\(X_i,Y_i\) objects,
\(f_i:X_i\to Y_i\)
Notation: \(Y_1+_XY_2\)
Name2: Fibered coproduct/sum
Extra: object
Assuming: \(X_1=X_2\)
Notation2: \(Y_1\sqcup_XY_2\)
Name3: Cocartesian square
Name4: Amalgamated sum

Name: A weak limit of \(F\)
Definition: A cone \((L,\phi)\) to \(F\) such that for any other cone \((N,\Psi)\) there is morphism \(u:N\to L\) such that \(\phi_K\circ u=\Psi_X\).
Page: wp
Variables: \(\mathbf C\) a category,
\(F:\mathbf J\to\mathbf C\) a diagram,

Name: A weak colimit of \(F\)
Definition: A cocone \((L,\phi)\) to \(F\) such that for any other cocone \((N,\Psi)\) there is morphism \(u:N\to L\) such that \(u\circ\phi_K=\Psi_X\).
Page: wp
Variables: \(\mathbf C\) a category,
\(F:\mathbf J\to\mathbf C\) a diagram,

Name: A cokernel of \(f\)
Definition: A coequalizer of \(f\) and of \(0_{XY}:X\to Y\).
Page: wp
Variables: \(\mathbf C\) a category,
\(f:X\to Y\in\mathbf C_1\)
Assuming: \(\mathbf C\) has 0 morphism

Name: A normal category
Definition: A category whose mono are normal.
Page: wp

Name: A conormal category
Definition: A category whose epi are cokernel of some morphism.
Page: wp

Smaller0: \(eq(f,g)\)
Greater0: \(ker(g-f)\)
Page: wp
Variables: \(\mathbf C\) a category,
\(f,g:A\to B\)
Smaller to greater: =
Hide middle: yes
Assuming: \(f,g\) preadditive

Smaller0: \(im(f)\)
Greater0: \(ker(coker(f))\)
Page: wp
Variables: \(\mathbf C\) a category,
\(f:A\to B\)
Smaller to greater: =
Hide middle: yes
Assuming: \(f\) Abelian

\(\mathbf C\) is normal implies \(\mathbf C\) is normal and conormal
Page: wp
Variables: \(\mathbf C\) a category

Name: A normal mono
Definition: A mono which is the kernel of some morphism.
Page: wp

Q: The algebraic notion of [...] is similar to the categorical notion of [...].
A: kernel; kernel pair
Page: wp

Question: The coequalizer of the two functors from 1 to 2 is
Answer: the monoid of natural numbers under addition.
Page: wp

Smaller0: \(\mathbf C\) have a zero object
Greater0: \(\mathbf C\) has zero morphisms
Page: wp
Variables: \(\mathbf C\) a category

Name: A bifunctor
Definition: A functor whose domain is a product category.
Page: wp

Name: A Cartesian monoidal category
Definition: \((\mathbf C,\otimes, I)\) where \(\mathbf C\) is a category with finite product, \(\otimes\) is the product and \(I\) is the terminal object.
Page: wp

Name: A coCartesian monoidal category
Definition: \((\mathbf C,\otimes, I)\) where \(\mathbf C\) is a category with finite product, \(\otimes\) is the coproduct and \(I\) is the initial object.
Page: wp

Q: [...].
A: A 6-tuple \((O,C,I,\circ)\) with:
-\(O\) a class of object,
-\(C:O\times O\to \mathbf M\),
-\(I:O\to \mathbf M\) such that \(I(o)\in C(o,o)\),
-\(\circ\) sends elements of \(C(b,c)\otimes C(a,b)\) to \(C(a,c)\), is associative when defined and \(I\) its is identity
Page: wp
Variables: M a monoidal category
Name: An enriched category over M
-Enriched category over \((\mathbf{Set},\otimes,\{.\})\) are categories,
-Enriched category over \((\mathbf{SmSet},\otimes,\{.\})\) are locally small categories,
-Enriched category over \((\mathbf{Set_{fin} },\otimes,\{.\})\) are locally finite categories,
-Enriched category over \((\mathbf{Cat },\otimes,\{.\})\) are 2-categories,
-Enriched category over \((\mathbf{2 },\otimes,\{.\})\) are category of preordered sets,
-Enriched category over \((\mathbf{Ab},\otimes)\) are preadditive categories,
-Enriched category over \((\mathbf{2 },\otimes,\{.\})\) are category of preordered sets,

Name: A preadditive category
Definition: An enriched category over Ab.
Page: wp
Example: Ab,
Name2: Rings, considered as a single-object category, with composition being addition and with the monoidal operation being multiplication,
Abbreviation2: Category of left \(R\)-module,

Name: Its endomorphism ring
Definition: The ring whose group is \(Hom(A,A)\) and whose multiplication is composition.
Page: wp
Variables: \(\mathbf C\) a preadditive category,
\(A\) an object

Name: Their smash product
Definition: The quotient of the cartesian product by \((x,y_0)\sim (x_0,y)\), and base point \((x_0,y_0)\).
Page: wp
Variables: \((X,x)\) and \((Y,y)\) two pointed sets
Notation: \((X,x)\land (Y,y)\)

Name: Their wedge product
Definition: The quotient of the disjoint union by the equivalent closure of \(\{(x_0,y_0)\}\). Its distinguished point is \(x_0=y_0\).
Page: wp
Variables: \((X,x)\) and \((Y,y)\) two pointed sets
Notation: \((X,x)\lor (Y,y)\)

Q: [...].
A: The tuple \((X,(p_A)_{A\in\mathcal A},(i_A)_{A\in\mathcal A})\) such that:
-for each \(A\in \mathcal A\), \(p_A\circ i_A=1_A\),
-for each \(A,B\in \mathcal A\), \(i_B\circ p_A=0_{A,B}\),
-\((X,(p_A)_{A\in\mathcal A})\) is a product for \(\mathcal A\),
-\((X,(i_A)_{A\in\mathcal A})\) is a coproduct for \(\mathcal A\).
Page: wp
Name: Let \(\mathbf C\) a category,
\(\mathcal A\) a class of objects
Q: For \(\mathcal A=[...]\), [...].
A: \emptyset; the zero object
Notation: Their biproduct
Assuming: \(\mathbf C\) has zero objects
Types: \(\bigoplus \mathcal A\)

Q: Equivalently:
-[...]
-[...].
A: A preadditive category with all finite biproduct; A category with 0 object and an iso \(A\Pi B\to A\sqcup B \)
Page: wp
Name: An additive category

Name: Endofunctor
Definition: A functor from a category to itself.
Page: wp

Name: Constant functor
Definition: A functor whose image on object is a singleton and whose image on arrows is its identity arrow.
Page: wp
Name2: Selection functor

Name: Diagonal functor
Definition: A functor \(D\to D^C\) sending each object in \(D\) to the constant functor of that object.
Page: wp

Name: A \(R\)-linear category
Definition: A category enriched over the monoidal category of \(R\)-module, with all finite biproduct.
Page: wp
Variables: \(R\) a ring
Assuming: \(R\) commutative

Name: The free object on \(X\) with respect to \(i\)
Definition: \(F(A)=Y\)
Page: wp
Variables: \((\mathbf C,F)\) a concrete category,
\(X,Y\) sets,
\(i:X\to Y\)
Definition2: For any object \(B\)
Conjdef: both
Definition3: for any map between sets \(f:X\to F(B)\),
Definition4: there exists a unique morphism \(g:A\to B\)
Definition5: such that \(f=F(g)\circ i\).
DefType: An object \(A\in\mathbf C_0\)

Question: The free functor that builds the free object \(A\) from the set \(X\) becomes
Answer: left adjoint to the forgetful functor.
Page: wp
Variables: \((\mathbf C,F)\) a concrete category,
\(X,Y\) sets,
\(i:X\to Y\),

Name: A kernel of \(f\)
Definition: An equaliser of \(f\) and the zero morphism \(0_{XY}:X\to Y\),
Page: wp
Variables: \(\mathbf C\) a category,
\(f:X\to Y\in\mathbf X_0\)
Definition2: A pair \((K,k)\) with \(K\in\mathbf C_0\) and \(k:K\to X\in\mathbf C_1\) with the universal property:
--\(0_{KY}=k\circ f\).
Conjdef: both
Abbreviation: \(\text{ker}(f)\)
Denoted by: \((K,k)\)
Assuming: \(\mathbf C\) has zero morphisms
DefType: A subobject and a mono