Glossary of category theory

${\displaystyle \operatorname {Hom} _{D}(F(X),Y)\simeq \operatorname {Hom} _{C}(X,G(Y))}$;

${\displaystyle \operatorname {Hom} _{D}(F(-),-)\simeq \operatorname {Hom} _{C}(-,G(-))}$

1. A class of objects,
2. For each pair of objects X, Y, a set ${\displaystyle \operatorname {Hom} (X,Y)}$, whose elements are called morphisms from X to Y,
3. For each triple of objects X, Y, Z, a map (called composition)

   ${\displaystyle \circ$ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f} ,

4. For each object X, an identity morphism ${\displaystyle \operatorname {id} _{X}\in \operatorname {Hom} (X,X)}$

subject to the conditions: for any morphisms ${\displaystyle f:X\to Y}$, ${\displaystyle g:Y\to Z}$ and ${\displaystyle h:Z\to W}$,

• ${\displaystyle (h\circ g)\circ f=h\circ (g\circ f)}$ and ${\displaystyle \operatorname {id} _{Y}\circ f=f\circ \operatorname {id} _{X}=f}$.

${\displaystyle x\to y}$

${\displaystyle x\leq y}$

${\displaystyle F:C^{\text{op}}\times C\to X}$

${\displaystyle \int ^{c\in C}F(c,c)}$.

For example, if R is a ring, M a right R-module and N a left R-module, then the tensor product of M and N is

${\displaystyle M\otimes _{R}N=\int ^{R}M\otimes _{\mathbb {Z} }N}$

${\displaystyle \varinjlim }$

${\displaystyle \Delta :C\to \operatorname {Fct} (I,C)}$

${\displaystyle f:I\to C}$

${\displaystyle \operatorname {Hom} (\varinjlim f,X)\simeq \operatorname {Hom} (f,\Delta _{X}),}$

${\displaystyle {\mathfrak {P}}(S)}$

${\displaystyle f:S\to T}$

${\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(T)\to {\mathfrak {P}}(S)}$

${\displaystyle f^{-1}(A)}$

${\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} }$

${\displaystyle \Delta :C\to \mathbf {Fct} (I,C),\,A\mapsto \Delta _{A}}$

${\displaystyle f:A\to B}$

${\displaystyle \Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B}$

${\displaystyle F:C^{\text{op}}\times C\to X}$

${\displaystyle \int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)}$

where ${\displaystyle C^{\#}}$ is the category (called the subdivision category of C) whose objects are symbols ${\displaystyle c^{\#},u^{\#}}$ for all objects c and all morphisms u in C and whose morphisms are ${\displaystyle b^{\#}\to u^{\#}}$ and ${\displaystyle u^{\#}\to c^{\#}}$ if ${\displaystyle u:b\to c}$ and where ${\displaystyle F^{\#}}$ is induced by F so that ${\displaystyle c^{\#}}$ would go to ${\displaystyle F(c,c)}$ and ${\displaystyle u^{\#},u:b\to c}$ would go to ${\displaystyle F(b,c)}$. For example, for functors ${\displaystyle F,G:C\to X}$,

${\displaystyle \int _{c\in C}\operatorname {Hom} (F(c),G(c))}$

1. A class of objects,
2. For each pair of objects X, Y in D, an object ${\displaystyle \operatorname {Map} _{D}(X,Y)}$ in C, called the mapping object from X to Y,
3. For each triple of objects X, Y, Z in D, a morphism in C,

   ${\displaystyle \circ$ :\operatorname {Map} _{D}(Y,Z)\otimes \operatorname {Map} _{D}(X,Y)\to \operatorname {Map} _{D}(X,Z)} ,

   called the composition,

4. For each object X in D, a morphism ${\displaystyle 1_{X}:1\to \operatorname {Map} _{D}(X,X)}$ in C, called the unit morphism of X

subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

${\displaystyle \mathbf {Fct} (C,D)\to D,\,\,F\mapsto F(A).}$

${\displaystyle F(f\circ g)=F(f)\circ F(g)}$

${\displaystyle f\circ g}$

${\displaystyle F(\operatorname {id} _{x})=\operatorname {id} _{F(x)}}$

${\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ,\,S\mapsto {\mathfrak {P}}(S)}$,

${\displaystyle {\mathfrak {P}}(S)}$

${\displaystyle f:S\to T}$

${\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(S)\to {\mathfrak {P}}(T)}$

${\displaystyle {\mathfrak {P}}(f)(A)=f(A)}$

• every map of simplicial sets ${\displaystyle f:\Lambda _{i}^{n}\to C}$ extends to an n-simplex ${\displaystyle f:\Delta ^{n}\to C}$

${\displaystyle \Lambda _{i}^{n}}$

${\displaystyle f:I\to J}$

${\displaystyle f^{*}=-\circ f:\operatorname {Fct} (J,C)\to \operatorname {Fct} (I,C)}$

${\displaystyle f_{!}}$

${\displaystyle \alpha :I\to C}$

${\displaystyle f_{!}\alpha :J\to C}$

${\displaystyle (f_{!}\alpha )(j)=\varinjlim _{f(i)\to j}\alpha (i)}$

${\displaystyle f(i)\to j}$

${\displaystyle f:I^{\text{op}}\to \mathbf {Set} }$

${\displaystyle \varprojlim _{i\in I}f(i)=\{(x_{i}|i)\in \prod _{i}f(i)|f(s)(x_{j})=x_{i}{\text{ for any }}s:i\to j\}.}$

${\displaystyle T(X)=G\times X}$

${\displaystyle \mu :T\circ T\to T}$

${\displaystyle \mu _{X}:G\times (G\times X)\to G\times X,\,\,(g,(h,x))\mapsto (gh,x)}$

${\displaystyle F:X\rightleftarrows A:G}$

${\displaystyle T=G\circ F}$

${\displaystyle \{\phi _{x}:F(x)\to G(x)\mid x\in \operatorname {Ob} (C)\}}$

${\displaystyle \phi _{y}\circ F(f)=G(f)\circ \phi _{x}}$

${\displaystyle GL_{n}(R)}$

${\displaystyle GL_{n}}$

${\displaystyle R\mapsto R^{*}}$

${\displaystyle GL_{n}}$

${\displaystyle F:C^{\text{op}}\to B}$

${\displaystyle G:C\to B}$

${\displaystyle F\otimes _{C}G=\int ^{c\in C}F(c)\otimes G(c).}$

${\displaystyle X\to f(u_{X})}$

${\displaystyle \operatorname {Hom} _{C}(u_{X},c)\to \operatorname {Hom} _{D}(X,f(c)),\,\alpha \mapsto (X\to f(u_{x}){\overset {f(\alpha )}{\to }}f(c))}$

${\displaystyle \operatorname {Hom} _{C}(u_{X},-)\simeq \operatorname {Hom} _{D}(X,f(-))}$

${\displaystyle \operatorname {Hom} _{D}(X,f(-))}$

${\displaystyle F(X)\simeq \operatorname {Nat} (\operatorname {Hom} _{C}(-,X),F)}$

where Nat means the set of natural transformations. In particular, the functor

${\displaystyle y:C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} ),\,X\mapsto \operatorname {Hom} _{C}(-,X)}$