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\begin{document}

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\section*{functor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{external_definition}{External definition}\dotfill \pageref*{external_definition} \linebreak
\noindent\hyperlink{InternalDefinition}{Internal definition}\dotfill \pageref*{InternalDefinition} \linebreak
\noindent\hyperlink{enriched_definition}{Enriched definition}\dotfill \pageref*{enriched_definition} \linebreak
\noindent\hyperlink{profunctors}{Profunctors}\dotfill \pageref*{profunctors} \linebreak
\noindent\hyperlink{higher_categorical_functors}{Higher categorical functors}\dotfill \pageref*{higher_categorical_functors} \linebreak
\noindent\hyperlink{special_properties_of_functors}{Special properties of functors}\dotfill \pageref*{special_properties_of_functors} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{morphisms_of_monoids_and_groups}{Morphisms of monoids and groups}\dotfill \pageref*{morphisms_of_monoids_and_groups} \linebreak
\noindent\hyperlink{representations}{Representations}\dotfill \pageref*{representations} \linebreak
\noindent\hyperlink{linear_maps}{Linear Maps}\dotfill \pageref*{linear_maps} \linebreak
\noindent\hyperlink{left_inverses}{Left Inverses}\dotfill \pageref*{left_inverses} \linebreak
\noindent\hyperlink{right_inverses}{Right Inverses}\dotfill \pageref*{right_inverses} \linebreak
\noindent\hyperlink{presheaves}{Presheaves}\dotfill \pageref*{presheaves} \linebreak
\noindent\hyperlink{OnGeneralizedElements}{Functors and generalized elements}\dotfill \pageref*{OnGeneralizedElements} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\begin{itemize}%
\item A functor is what goes between [[category|categories]].


\item A functor from $C$ to $D$ is an \emph{image of $C$ in $D$} (but the naive [[image]] need not be a [[subcategory]] of $D$).


\item A functor between (small) categories is a morphism of the underlying [[quivers]] that respects the composition of edges.



\end{itemize}
So a functor $F : C \to D$ is a [[morphism]] between two [[category|categories]] that

\begin{itemize}%
\item consists of a map $F_0 : Obj(C) \to Obj(D)$ of the objects of the categories


\item and a map $F_1 : Mor(C) \to Mor(D)$ of the morphisms of the categories



\end{itemize}
such that

\begin{itemize}%
\item it respects source and target: $F_1$ coincides with $F_0$ on source and target objects;


\item it respects identities: $F(1_X) = 1_{F(X)}$;


\item it respects composition: the image of the composite of two morphisms under $F$ is the composite of their images.



\end{itemize}
These last two properties are the decisive ones of a functor; they are called the \textbf{functoriality conditions}. They are a direct generalization of the notion of [[homomorphism]] (of [[monoids]], [[groups]], [[algebras]], etc.) to the case that there are more objects. As a slogan we have

The notion of functor is a [[horizontal categorification]] of that of [[homomorphism]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{external_definition}{}\subsubsection*{{External definition}}\label{external_definition}

A \textbf{functor} $F$ from a [[category]] $C$ to a category $D$ is a map sending each [[object]] $x \in C$ to an object $F(x) \in D$ and each [[morphism]] $f : x \to y$ in $C$ to morphism $F(f) : F(x) \to F(y)$ in $D$, such that

\begin{itemize}%
\item $F$ preserves [[composition]]: $F(g\circ f) = F(g)\circ F(f)$ whenever the left-hand side is well-defined,


\item $F$ preserves [[identity morphisms]]: for each object $x \in X$, $F(1_x) = 1_{F(x)}$.



\end{itemize}
Or equivalently, since compositions $g f = g\circ f$ (commuting triangles) and identities $1_x$ (commuting loops) are both simple commuting diagrams, we can combine the above conditions to the single statement

\begin{itemize}%
\item $F$ preserves commuting diagrams.

\end{itemize}
Given morphisms $f:X\to Y$, $g:Y\to Z$, and $h:X\to Z$, declaring the triangle commutes amounts to declaring

\begin{displaymath}
h = g\circ f.
\end{displaymath}
In this case, for $F:C\to D$ to preserve the commutative triangle means

\begin{displaymath}
F(h) = F(g)\circ F(f)
\end{displaymath}
as depicted below



Preserving commuting triangles means $F$ preserves compositions.

Given morphisms $f:X\to Y$, $g:Y\to Z$, and $h:Z\to X$, declaring the loop commutes amounts to declaring

\begin{displaymath}
1_X = h\circ g\circ f.
\end{displaymath}
In this case, for $F:C\to D$ to preserve the commutative loop means

\begin{displaymath}
F(1_X) = F(h)\circ F(g)\circ F(f)
\end{displaymath}
as depicted below



However, it means more than that. Since any commutative loop is equal to the identity morphism, we must also have

\begin{displaymath}
1_{F(X)} = F(h)\circ F(g)\circ F(f)
\end{displaymath}
implying

\begin{displaymath}
F(1_X) = 1_{F(X)}.
\end{displaymath}
Preserving commuting loops means $F$ preserves identity morphisms.

Another equivalent way to say this is that a functor $F : C \to D$ is precisely a morphism of [[simplicial sets]] $N(F) : N(C) \to N(D)$ between the [[nerves]] of these categories

\begin{itemize}%
\item the [[objects]] of $C$ and $D$ are the 0-cells of $N(C)$ and $N(D)$, so $N(F)_0 : N(F)_0 \to N(F)_0$ maps objects of $C$ to objects of $D$;


\item the [[morphisms]] of $C$ and $D$ are the 1-cells of $N(C)$ and $N(D)$, so $N(F)_1 : N(F)_1 \to N(F)_1$ maps morphisms of $C$ to objects of $D$;


\item the identity morphisms of $C$ and $D$ are the degenerate 1-cells of $N(C)$ and $N(D)$, so the fact that $N(F)$ respects degeneracy maps means that $F$ respects identities;


\item the commuting triangles of $C$ and $D$ are the 2-cells of $N(C)$ and $N(D)$, so the fact that $N(F)$ maps 2-cells to 2-cells means that it respects commuting triangles, hence that it respects composition.



\end{itemize}
See [[nerve]] for more details on this.

The functors between two categories $C$ and $D$ form themselves a category, the [[functor category]] $[C,D]$, whose morphisms are [[natural transformations]]. Equipped with these functor categories as [[hom-objects]], we have a $2$-[[2-category|category]] [[Cat]] of categories, functors and natural transformations. In other words, functors are [[morphisms]] in $Cat$.

\hypertarget{InternalDefinition}{}\subsubsection*{{Internal definition}}\label{InternalDefinition}

If $C$ and $D$ are [[internal categories]] in some ambient category $A$, then an \textbf{internal functor} $F : C \to D$ is

\begin{itemize}%
\item a morphism of objects $F_0 : C_0 \to D_0$ in $A$;


\item a morphisms of morphisms $F_1 : C_1 \to D_1$ in $A$;


\item such that the following diagrams commute

\begin{itemize}%
\item respect for the source map: $\itexarray{
C_1 &\stackrel{f_1}{\to}& D_1
\\
\downarrow^s && \downarrow^s
\\
C_0 
&\stackrel{f_0}{\to}&
D_0
}$;


\item respect for the target map: $\itexarray{
C_1 &\stackrel{f_1}{\to}& D_1
\\
\downarrow^t && \downarrow^t
\\
C_0 
&\stackrel{f_0}{\to}&
D_0
}$;


\item respect for identities $\itexarray{
C_0 &\stackrel{f_0}{\to}&
D_0
\\
\downarrow^i && \downarrow^i
\\
C_1 &\stackrel{f_1}{\to}& D_1
}$;


\item respect for composition $\itexarray{
C_1 \times_{t,s} 
C_1 &\stackrel{f_1\times_{t,s} f_1}{\to}&
D_1 \times_{t,s} D_1
\\
\downarrow^{\circ} && \downarrow^{\circ}
\\
C_1 &\stackrel{f_1}{\to}& D_1
}$.



\end{itemize}


\end{itemize}
This reproduces the external definition of functors above for [[small categories]], which are categories [[internalization|internal to]] [[Set]]

In many cases, this notion is too restrictive, and we should use internal [[anafunctors]] instead.

\hypertarget{enriched_definition}{}\subsubsection*{{Enriched definition}}\label{enriched_definition}

In [[enriched category theory]] a functor maps not [[hom-sets]] but the given [[hom-objects]] to each other, in a way that respects their composition. This is described at

\begin{itemize}%
\item [[enriched functor]].

\end{itemize}
\hypertarget{profunctors}{}\subsubsection*{{Profunctors}}\label{profunctors}

A generalization of the notion of [[enriched functor]] is the notion of [[profunctor]].

\hypertarget{higher_categorical_functors}{}\subsubsection*{{Higher categorical functors}}\label{higher_categorical_functors}

In [[higher category theory]] there are corresponding higher notions of functor, such as

\begin{itemize}%
\item [[2-functor]]

\begin{itemize}%
\item [[strict 2-functor]]


\item [[pseudofunctor]]


\item [[lax functor]]



\end{itemize}

\item [[(∞,1)-functor]]


\item See also an informal discussion about an [[experimental alternative definition of functor]].



\end{itemize}
\hypertarget{special_properties_of_functors}{}\subsection*{{Special properties of functors}}\label{special_properties_of_functors}

Functors with special properties are important in applications. See for instance

\begin{itemize}%
\item [[essentially surjective functor]]


\item [[full functor]]


\item [[faithful functor]]


\item [[full and faithful functor]]



\end{itemize}
And for more background on this see [[stuff, structure, property]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{morphisms_of_monoids_and_groups}{}\subsubsection*{{Morphisms of monoids and groups}}\label{morphisms_of_monoids_and_groups}

For $A,B$ [[monoids]] or $G, H$ [[groups]], let $\mathbf{B}A, \mathbf{B}B$, $\mathbf{B}G$, $\mathbf{B}H$ be the corresponding one-object [[category|categories]] (as described at [[delooping]]). Then functors

\begin{displaymath}
\mathbf{B}A \to \mathbf{B}B
\end{displaymath}
are canonically in bijection with monoid homomorphisms $A \to B$ and accordingly functors

\begin{displaymath}
\mathbf{B}G \to \mathbf{B}H
\end{displaymath}
are canonically in bijection with group homomorphisms $G \to H$.

\hypertarget{representations}{}\subsubsection*{{Representations}}\label{representations}

With $\mathbf{B}G$ as above, functors on $\mathbf{B}G$ with values in [[Vect]] are the same as linear [[representations]] of the [[group]] $G$. In fact, we have a canonical isomorphism of categories

\begin{displaymath}
Funct(\mathbf{B}G, Vect) \simeq Rep(G)
\end{displaymath}
of the [[functor category]] with the representation category.

\hypertarget{linear_maps}{}\subsubsection*{{Linear Maps}}\label{linear_maps}

Let $\mathbf{B} End(U)$ and $\mathbf{B} End(V)$ be one-object categories whose objects are each finite-dimensional vector space and whose morphisms are all of the linear [[endomorphisms]] on that space, i.e. one-object [[full subcategory|full subcategories]] of $Fin Vect$.

\hypertarget{left_inverses}{}\paragraph*{{Left Inverses}}\label{left_inverses}

If the linear map $F:U\to V$ has a left inverse, i.e.

\begin{displaymath}
F^*\circ F = 1_U,
\end{displaymath}
where $F^*$ is the [[preimage]], then we can construct a functor

\begin{displaymath}
F_*:\mathbf{B} End(U)\to\mathbf{B} End(V)
\end{displaymath}
by defining its action on objects by

\begin{displaymath}
F_*(U) = F_* U
\end{displaymath}
where $F_*$ on the right-hand side is the [[image]] and its action on endomorphisms by

\begin{displaymath}
F_*(f) = F f F^*.
\end{displaymath}
Composition follows immediately

\begin{displaymath}
F_*(f g) = F f g F^* = F f F^* F g F^* = F_*(f) F_*(g).
\end{displaymath}
Identity morphisms are preserved since for any vector $y$ in $F_*U\subset V$, then

\begin{displaymath}
y = F x
\end{displaymath}
for some vector $x$ in $U$ and we have

\begin{displaymath}
F_*(1_U)y = F\circ F^* y = F\circ F^*\circ F x = F x = y
\end{displaymath}
so that

\begin{displaymath}
F_*(1_U) = 1_{F_*(U)}
\end{displaymath}
as required. Hence, $F_*$ is a functor.

\hypertarget{right_inverses}{}\paragraph*{{Right Inverses}}\label{right_inverses}

If the linear map $F:U\to V$ has a right inverse, i.e.

\begin{displaymath}
F\circ F^* = 1_V,
\end{displaymath}
where $F^*$ is the preimage, then we can construct a functor

\begin{displaymath}
F^*:\mathbf{B} End(V)\to\mathbf{B} End(U)
\end{displaymath}
by defining its action on objects by

\begin{displaymath}
F^*(V) = F^*V
\end{displaymath}
where $F^*$ on the right-hand side is preimage, and its action on endomorphisms by

\begin{displaymath}
F^*(f) = F^* f F.
\end{displaymath}
Composition follows immediately

\begin{displaymath}
F^*(f g) = F^* f g F = F^* f F F^* g F = F^*(f) F^*(g).
\end{displaymath}
Identity morphisms are preserved since for any vector $x$ in $F^*V\subset U$, then

\begin{displaymath}
x = F^*y
\end{displaymath}
for some vector $y$ in $V$ and we have

\begin{displaymath}
F^*(1_V)x = F^*\circ F x = F^*\circ F \circ F^* y = F^* y = x
\end{displaymath}
so that

\begin{displaymath}
F^*(1_V) = 1_{F^*(V)}
\end{displaymath}
as required. Hence, $F^*$ is a functor.

\hypertarget{presheaves}{}\subsubsection*{{Presheaves}}\label{presheaves}

Functors $F : C \to Set$ with values in [[Set]] are also called [[presheaf|presheaves]]. As such one calls them presheaves on the [[opposite category]] $C^{op}$ of $C$. See [[presheaf]] for more on this.

\hypertarget{OnGeneralizedElements}{}\subsubsection*{{Functors and generalized elements}}\label{OnGeneralizedElements}

For $C$ a [[category]], and $X \in C$ an [[object]], and $U$ any other object, a [[morphism]] $x : U \to X$ may be regarded as a [[generalized element]] of $X$, written $x \in X$ (For this language applied to the category [[Set]] of sets see [[ETCS]]. For the general case see [[type theory]]).

The \emph{[[set]]} of generalized elements of an object $X \in C$ is thus the union of [[hom-sets]] $\coprod_{U \in C} Hom_C(U,X)$.

While a [[morphism]] $f : X \to Y$ in an arbitrary category $C$ need not at all come to us as a function of sets, it always induces a function of sets of \emph{generalized elements} : it sends the generalized element $x : U \to X$ of $X$ to the generalized element

\begin{displaymath}
f(x) : U \stackrel{x}{\to} X \stackrel{f}{\to} Y
\end{displaymath}
of $Y$, using the composition of the morphism $f$ with the morphism $x$ in $C$.

In terms of this notation, the functoriality condition on a functor $F : C \to D$, which is

\begin{displaymath}
F(U \stackrel{x}{\to} X \stackrel{f}{\to} Y)
  =
  F(U) \stackrel{F(x)}{\to} F(X) \stackrel{F(f)}{\to} F(Y)
\end{displaymath}
appears as

\begin{displaymath}
F(f(x)) = F(f)(F(x))
  \,.
\end{displaymath}
This can be illustrated in the following diagram



which provides an alternative expression of the functoriality condition as simply a statement that commuting diagrams in $C$ map to commuting diagrams in $D$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[endofunctor]]


\item [[bifunctor]]


\item [[profunctor]]


\item [[multifunctor]]


\item [[function]]


\item \textbf{functor}

\begin{itemize}%
\item [[endofunctor]]


\item [[polynomial functor]]



\end{itemize}

\item [[2-functor]] / [[pseudofunctor]]

\begin{itemize}%
\item [[monoidal functor]]

\end{itemize}

\item [[n-functor]]


\item [[(∞,1)-functor]]


\item [[(∞,n)-functor]]



\end{itemize}
[[!redirects functors]] [[!redirects covariant functor]] [[!redirects covariant functors]]

[[!redirects internal functor]] [[!redirects internal functors]]

[[!redirects functorial]]



\end{document}