functorial aspects of the GNS representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Remark

This is a small subcollection of my ([[Tom Mainiero]]'s) personal notes on some functorial aspects of the [[Gelfand-Naimark-Segal construction]] (GNS construction) that have been sitting around for a while.
Please feel free to contribute!
The ideas here underlie some of the constructs of [Homtools](#Mainiero2019): specifically, one can think of a ``multipartite'' state as a presheaf valued in the category of states (defined below) over a collection of subsystems.
The GNS representation can then be thought of as a functor that takes us into a "representation-theoretic" category where we can, e.g. compute an associated (co)homology using [[Čech methods]].

## Baby's first GNS functor {#baby}
For the purposes of this section, $R$ is a fixed a unital ring and $Q$ is a unital $R$-algebra.
Although $R$ will not appear explicitly in any of the discussion in this section (so that $Q$ could effectively just be thought of as a unital ring), we keep it around as the case where $Q$ is a finite-dimensional (semisimple) $\mathbb{C}$-algebra is most relevant to later discussions involving the GNS construction.

\begin{definition}
Let ${}_{Q}\mathbf{Mod}^{\bullet}$ denote the category with:

* objects given by *[[pointed module|pointed left $Q$-modules]]*: pairs $(M,m)$ of a left $Q$-module $M$ with an element $m \in M$;

* morphisms given by point preserving intertwiners.

${}_{Q}\mathbf{Mod}^{\odot}$ is the [[full subcategory]] of ${}_{Q}\mathbf{Mod}^{\bullet}$ whose objects are *cyclic* pointed modules: $(M,m)$ such that $Q m = M$.
\end{definition}

\begin{remark}
For those that find the notion of a [[pointed module]] somewhat awkward, note that: ${}_{Q}\mathbf{Mod}^{\bullet}$ is equivalent to the [[coslice category]] ${}_{Q}{Q}/{}_{Q}\mathbf{Mod}$, where ${}_{Q}\mathbf{Mod}$ is the [[category of modules|category of left $Q$-modules]].
\end{remark}

\begin{remark}\label{rem:mod_reflective}
${}_{Q}\mathbf{Mod}^{\odot}$ is a [[reflective subcategory]] of ${}_{Q}\mathbf{Mod}^{\bullet}$: the right adjoint of the inclusion functor
\[
    \iota \colon {}_{Q}\mathbf{Mod}^{\odot} \longrightarrow {}_{Q}\mathbf{Mod}^{\bullet}
\]
is given by the "cyclification" functor:
\[
    \mathsf{Cyc} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}
\]
that acts on objects by assigning to any pointed module $(M,m)$ the cyclic pointed submodule generated by the orbit of $m$, i.e. $\mathsf{Cyc} \colon (M,m) \to (Q \cdot m, m)$ and acts on morphisms by restriction to cyclic submodules.
\end{remark}

\begin{notation}
For any left $Q$-module $M$, we denote the annihilator of a point $m \in M$ as:
\[
	\mathrm{Ann}_{Q}(m) \coloneqq \{ q \in Q : q \cdot m = 0 \}.
\]
\end{notation}

The following lemma, which will become useful later, describes the structure of the hom-sets ${}_{Q}\mathbf{Mod}^{\bullet}((M,m), (N,n))$ when $(M,m)$ is a cyclic pointed module.

\begin{lemma}\label{lem:cyclic_thinness}
	Let $(M,m)$ and $(N,n)$ be pointed $Q$-modules and suppose $(M,m)$ is cyclic.
	Then there exists a morphism from $(M,m)$ to $(N,n)$ if and only if $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$.
	Moreover, this morphism is unique when it exists.
\end{lemma}
\begin{proof}
	This follows quickly from the fact that a morphism $f: (M,m) \rightarrow (N,n)$ must satisfy $f(q m) =q n$ for all $q \in Q$.
\end{proof}

There is an immediate corollary of this Lemma: ${}_{Q}\mathbf{Mod}^{\odot}$ is equivalent to the poset of left-ideals of $Q$ (thought of as a category).
This equivalence of categories is actually part of the restriction of a larger adjunction.
We now spell out the definitions behind this claim.

\begin{definition}
$\mathbf{L}(Q)$ is the [[thin category]] given by the poset of left ideals of $Q$:

* objects are left ideals of $Q$, and

* there is a unique morphism $I \overset{!}{\to} J$ if and only if $I \subseteq J$.

\end{definition}

The process of taking annihilators of distinguished points of left $Q$-modules defines a functor:
\[
  \mathsf{Ann}_{Q} \colon {}_{Q}\mathbf{Mod}^{\bullet} \longrightarrow \mathbf{L}(Q).
\]
Explicitly: it acts on objects by:
\[
   \mathsf{Ann}_{Q} \colon (M,m) \mapsto \mathrm{Ann}_{Q}(m),
\]
and its action on morphisms is deduced from the following straightforward remark.

\begin{remark}
 If $f \colon (M,m) \to (N,n)$ is a morphism of pointed modules, then $q \cdot n = q \cdot f(m) = f(q \cdot m)$; so we must have $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$.
\end{remark}

Thus, for $f \colon (M,m) \to (N,n)$ a $\mathbf{Mod}^{\bullet}_{Q}$-morphism, we can define $\mathsf{Ann}_{Q}(f) \colon \mathsf{Ann}_{Q}(m) \to \mathsf{Ann}_{Q}(n)$ as the unique $\mathbf{L}(Q)$-morphism $\mathsf{Ann}_{Q}(m) \overset{!}{\to} \mathsf{Ann}_{Q}(n)$ given by the inclusion of ideals $\mathrm{Ann}_{Q}(m) \subseteq \mathrm{Ann}_{Q}(n)$.

Alternatively, given any left-ideal $I$ of some $R$-algebra $Q$, one can produce an $Q$-module $Q/I$.
This module has a distinguished point $[1_{Q}] \coloneqq 1_{Q} + I$, where $1_{Q}$ is the identity of $Q$.
Moreover, $Q \cdot [1_{Q}] = Q/I$; so, so the pointed module $(Q/I, [1])$ is *cyclic*.
This process defines a functor:
\[
	\mathsf{Quot} \colon \mathbf{L}(Q) \longrightarrow {}_{Q}\mathbf{Mod}^{\odot}
\]
whose action on objects is as described above:
\[
	\mathsf{Quot} \colon I \mapsto (Q/I, [1_{Q}])
\]
and action on morphisms is given by:
\[
	\mathsf{Quot} \colon (I \overset{!}{\to} J) \mapsto \tau^{I}_{J}
\]
where $\tau^{I}_{J}$ is the tautological map of pointed modules acting by:
\[
	\tau^{I}_{J}: q + I \mapsto q + J
\]
for any $q \in Q$; this is well-defined as $J \subseteq I$.

\begin{proposition}
There is an adjunction:
\begin{center}
  \begin{tikzcd}
    \mathbf{L}(Q)
	\arrow[r, shift right=6pt, "\widehat{\mathsf{Quot}}_{Q}"', "\bot"]
    &
    {}_{Q}\mathbf{Mod}^{\bullet}
     \arrow[l, shift right=6pt, "\mathsf{Ann}_{Q}"']
  \end{tikzcd}
\end{center}
restricting to an equivalence
\begin{center}
  \begin{tikzcd}
    \mathbf{L}(Q)
	\arrow[r, shift right=6pt, "\mathsf{Quot}_{Q}"']
    &
    {}_{Q}\mathbf{Mod}^{\odot}
     \arrow[l, shift right=6pt, "\mathsf{Ann}_{Q}"']
  \end{tikzcd}
\end{center}
where $\widehat{\mathsf{Quot}}_{Q} = \iota \circ \mathsf{Quot}_{Q}$ for $\iota$ the inclusion functor of cyclic pointed modules into pointed modules.
\end{proposition}
\begin{proof}
We will define a unit
\[
	\eta \colon \mathrm{id}_{\mathbf{L}(Q)} \longrightarrow \mathsf{Ann}_{Q} \circ \mathsf{Quot}
\]
and counit
\[
	\epsilon \colon \mathsf{Quot} \circ \mathsf{Ann}_{Q} \longrightarrow \mathrm{id}_{{}_{Q}\mathbf{Mod}^{\bullet}}.
\]
satisfying the [[triangle identities]].
Note that $\eta$ is given by assigning the identity morphism to each object of $\mathbf{L}(Q)$: indeed, for any left-ideal we have $\mathsf{Ann}_{Q} \circ \mathsf{Quot}(I) = \mathsf{Ann}_{Q} \left[(Q/I,[1]) \right] = I$. The counit $\epsilon$ is a bit more interesting: let $\mathsf{M} = (M,m)$ be a pointed-module and define $(K,k) \coloneqq \mathsf{Quot} \circ \mathsf{Ann}_{Q}(\mathsf{M}) = (Q/\mathrm{Ann}_{Q}(m),[1])$. We have $\mathrm{Ann}_{Q}(k) = \mathrm{Ann}_{Q}(m)$; so by Lemma \ref{lem:cyclic_thinness} there is a unique morphism $(K,k) \rightarrow (M,m)$; this defines the morphism $\epsilon_{(M,m)}$. The triangle identities are trivial.
\end{proof}

### Connection to the GNS construction
The above observations are simple shadows of the functorality of the GNS construction for [[C-star-algebra|C*-algebras]] and [[von Neumann algebra|W*-algebras]].
The *opposite* of the category of ideals $\mathbf{L}(Q)$ can be thought of as a stand-in for the category of states (to every $C^\ast$-algebraic state $\rho \colon E \to \mathbb{C}$ on a $C^{\ast}$-algebra $E$ we can associate the left ideal given by the "left kernel" $\{e \in E: \rho(e^{*} e) = 0 \}$).
The quotient functor $\mathsf{Quot}$ is a toy-model of the GNS construction.

## Notation/Terminology
In the following the letters $E, F$ and $G$ will denote $C^\ast$-algebras that are not necessarily $\W^\ast$-algebras and $A, B, C$ will denote $W^\ast$-algebras.

The term (normal) state will be taken to mean *(normal) positive linear functional*
Note that this is at odds with some definitions in the physics literature, where "state" typically means "normalized" positive linear functional (if $f(1_{E}) = 1$ for $f: E \to \mathbb{C}$ a positive linear functional).
Using the definition of "state" here, the zero map $0: E \to \mathbb{C}$ is a perfectly valid state.

# The GNS construction as a functor on a fixed C$^\ast$-algebra

With an appropriate notion of a "category of states", one can explore functorial properties of the GNS construction as in Arthur Parzygnat's work (see [Parzygnat2016](#Parzygnat2016) and [Parzygnat2018](#Parzygnat2018)).

The construction here is from a slightly different perspective, using a mildly richer category of states based on the natural poset structure on the collection of (normal) states on a fixed $W^\ast$ or $C^\ast$-algebra.
The posetal relation is intimately related to the notion of a Radon-Nikodym derivative.

## The category of (normal) states over a fixed (W$^\ast$) C$^\ast$-algebra.

Begin by recalling a well-known posetal relation on the set of states $\text{State(E)}$ on a $C^\ast$-algebra $E$:
\begin{definition}
For any $f, g \in \text{State}(E)$, define the relation $\leq$ by $f \leq g$ if $f - g$ is a state, where $-$ denotes pointwise subtraction; equivalently: $f \leq g$ if $f(e^\ast e) \leq g(e^\ast e)$ for all $e \in E$
\end{definition}

It is a straightforward exercise to show this relation is posetal.

The story for normal states in the $W^\ast$-algebra world is nearly identical: let $\text{NState(A)}$ denote the set of normal states on a $W^\ast$-algebra $A$ (a subset of $\text{State}(A)$); for any $f, g \in \text{NState(A)}$ we can define a relation $\leq_{\text{norm}}$ by: $f \leq_{\text{norm}} g$ if $f - g$ is a normal state.
Because the space of linear functionals splits as an orthogonal sum into normal and singular parts, it follows immediately that $f \leq_{\text{norm}} g$ for normal linear functionals $f$ and $g$ if and only if $f \leq g$.
As a result we can simply write $\leq$ when discussing states on $C^\ast$-algebras or normal states on $W^\ast$-algebras without any issues if we forget down from the category of $W^\ast$-algebras to the category of $C^\ast$-algebras.

\begin{remark}
We can easily define weaker versions of the relation $\leq$ that may also be of interest: for $f$ and $g$ positive linear functionals on a $C^\ast$ algebra $E$, we can define relations $\ll$ and $\lesssim$ in the following manner:

* $f \ll g$ if there is an inclusion of $\ker^{L}(g) \subseteq \ker^{L}(f)$, where, for any positive linear functional $h$ on $E$, the "left kernel" or "vanishing ideal" is defined by:
\[
	\ker^{L}(h) \coloneqq \{e \in E: h(e^\ast e ) = 0 \}.
\]

* $f \lesssim g$ if there exists a $C \geq 0$ such that $f(e^\ast e) \leq C g(e^\ast e)$ for every $e \in E$.

We have the following immediate properties:

* $f \leq g \Rightarrow f \lesssim g \Rightarrow f \ll g$.

* $\leq$ is a partial order as it is antisymmetric: $f \leq g$ and $g \leq f$ if and only if $f = g$.

However, the weaker preorders $\ll$ and $\lesssim$ do not satisfy antisymmetry: e.g. $2 f \lesssim f$ and $f \lesssim 2f$.

Moreover, while $\lesssim$ is only a mild weakening of $\leq$, the weakest preorder $\ll$ is vastly different.
In particular, we will show there is a unique Radon-Nikodym derivative associated to every pair of functionals related by the preorders $\leq$ and $\lesssim$; the result can be weakened for $\ll$, but the Radon-Nikodym derivative will be an unbounded (densely defined) operator.
See [Gudder](#Gudder).
\end{remark}

\begin{definition}
Let $E$ be a $C^\ast$-algebra.
The category $\mathbf{State}(E)$ is the opposite of the thin category determined by the preorder $\leq$, i.e. it is the category whose objects are all states (positive linear functionals) on $E$, and whose morphism sets are given by
\[
  \mathbf{State}(E)(\omega, \nu) =
  \left\{
    \begin{array}{ll}
       \{\star\}, & \nu \leq \omega\\
       \emptyset, & \text{otherwise}
    \end{array}
\right.
\]
where $\{\star\}$ denotes the one point set.
Similarly if $A$ is a $W^\ast$-algebra, let $\mathbf{NState}(A)$ be defined as the full subcategory of $\mathbf{State}(A)$ whose objects are *normal* states.
\end{definition}

\begin{definition}
For $E$ a $C^\ast$-algebra, the category $\mathbf{Rep}^{\bullet}(E)$ is the category with:

* objects given by pointed $\ast$-representations of $E$: triples $(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H})$ of a Hilbert space $\mathcal{H}$ a vector $v \in \mathcal{H}$ and a $\ast$-representation $\pi$;

* morphisms given by contractive linear point-preserving intertwiners: a morphism from $(\mathcal{H}_{1}, v_{1}, \pi_1)$ to $(\mathcal{H}_2, v_2, \pi_2)$ is a linear map $f: \mathcal{H}_{1} \to \mathcal{H}_{2}$ such that $\|f\| \leq 1$ with respect to the operator norm, $f v_{1} = v_{2}$, and
\[
  f \circ \pi_{1}(e) =  \pi_{2}(e) \circ f
\]
for all $e \in E$.

$\mathbf{Rep}^{\odot}(E)$ is the full subcategory of $\mathbf{Rep}^{\bullet}(E)$ whose objects are *cyclic* representations: $(\mathcal{H}, v, \pi: E \to \mathrm{B} \mathcal{H})$ such that the subspace $E v = \{e v : e \in E \}$ is dense in $\mathcal{H}$.
\end{definition}

We also define normal versions of the above.
\begin{definition}
Similarly, for $A$ a $W^\ast$-algebra, the category $\mathbf{NRep}^{\bullet}(A)$ is the category whose objects are pointed normal $\ast$-representations of $A$, and morphisms are contractive linear intertwiners (equivalently the full subcategory of $\mathbf{Rep}^{\bullet}(A)$ generated by objects whose underlying representation is normal).
$\mathbf{NRep}^{\odot}(A)$ is the full subcategory of $\mathbf{NRep}^{\bullet}(A)$ generated by cyclic representations.
\end{definition}

+-- {: .query}
Maybe add a bit here about how NRep and Rep interact.
=--

The following are a few straightforward observations about the subcategories $\mathbf{(N)Rep}^{\odot}(\ldots)$ generated by cyclic objects.

\begin{remark}
The cyclic subcategories are thin: there is at most one morphism between two objects, every morphism is uniquely determined by where distinguished point is sent.
\end{remark}

\begin{remark}
Just as the full subcategory of cyclic pointed $Q$-modules (for $Q$ an $R$-algebra) is reflective inside of the category of pointed $A$-modules (c.f.\ Remark \ref{rem:mod_reflective}),
the subcategory $\mathbf{Rep}^{\odot}(E)$ (resp. $\mathbf{NRep^{\odot}}(A)$) is a [[reflective subcategory|reflective subcategories]] of $\mathbf{Rep}^{\bullet}(A)$ (resp. $\mathbf{NRep^{\bullet}}(A)$).
The right adjoint of the inclusion functor is given on objects by assigning to any pointed representation the completion of the orbit of the distinguished point;
it is defined on morphisms by restriction.
\end{remark}

The observations in the above remark are shadows of adjunctions ($E$ a $C^{\ast}$-algebra and $A$ a $W^{\ast}$-algebra)
\begin{center}
  \begin{tikzcd}
    \mathbf{State}(E)
     \arrow[r, shift right=6pt, "\widehat{\mathsf{L^2}}"', "\bot"]
    &
    \mathbf{Rep}^{\bullet}(E)
     \arrow[l, shift right=6pt, "\mathsf{Fnl}"']
  \end{tikzcd}
  \begin{tikzcd}
    \mathbf{NState}(A)
     \arrow[r, shift right=6pt, "\widehat{\mathsf{L^2}}"', "\bot"]
    &
    \mathbf{NRep}^{\bullet}(A)
     \arrow[l, shift right=6pt, "\mathsf{Fnl}"']
  \end{tikzcd}
\end{center}
which restricts to an equivalence of categories:
\begin{center}
  \begin{tikzcd}
    \mathbf{State}(E)
     \arrow[r, shift right=6pt, "\mathsf{L^2}"']
    &
    \mathbf{Rep}^{\odot}(E)
     \arrow[l, shift right=6pt, "\mathsf{Fnl}"']
  \end{tikzcd}
  \begin{tikzcd}
    \mathbf{NState}(A)
     \arrow[r, shift right=6pt, "\mathsf{L^2}"']
    &
    \mathbf{NRep}^{\odot}(A)
     \arrow[l, shift right=6pt, "\mathsf{Fnl}"']
  \end{tikzcd}
\end{center}
The analogous statements hold for normal states/pointed representations.
Our next step is to unravel the functors in the above diagrams, both which should be clear to those familiar with the GNS construction

\begin{definition}
The functor:
\[
	\mathsf{Fnl} \colon \mathbf{Rep}^{\bullet}(E) \longrightarrow \mathbf{State}(E)
\]
acts on objects by:
\[
	\mathsf{Fnl} \colon (\mathcal{H}, v, \pi: E \rightarrow \operatorname{B} \mathcal{H}) \mapsto (e \mapsto \langle v, \pi(e) v \rangle^{\mathcal{H}} ),
\]
and takes a morphism $f \in \mathbf{Rep}^{\bullet}(E)[(\mathcal{H}_{1}, v_{1}, \pi_{1}), (\mathcal{H}_{2}, v_{2}, \pi_{2})]$ to the unique morphism in $\mathbf{State}(E)$ defined by the relation:
\[
     \langle v_{2}, \pi_{2}( e^{*} e ) v_{2} \rangle = \langle f v_{1}, \pi_{2}( e^{*} e ) f v_{1} \rangle =  \langle v_{1}, f^{*} f \pi_{1}( e^{*} e ) v_{1} \rangle
       \leq
       \langle v_{1}, \pi_{1}( e^{*} e) v_{1} \rangle,
\]
for all $e \in E$.
\end{definition}

Before defining the functor $\mathsf{L^2}$, we note that to any positive linear functional $\omega \colon E \to \mathbb{C}$, there is an associated left ideal $\ker^{L} \omega$.
Thus, repeating the observations of [Section](#baby), there is an associated pointed (algebraic) left $E$-module $E/\ker^{L} \omega$

\begin{definition}
The functor $\mathsf{L^2} \colon \mathbf{State}(E) \rightarrow \mathbf{Rep}^{\bullet}(E)$ acts on objects by taking a positive linear functional $\omega$ to its corresponding GNS representation, and acts on a morphism $\nu \to \omega$ corresponding the relation $\omega \leq \nu$ by assigning the quotient map $\overline{q}(\omega, \nu) \colon \mathsf{L^{2}} \nu \to \mathsf{L^{2}} \omega$
\end{definition}

## References

* {#Mainiero2019} [[Tom Mainiero]], *Homological Tools for the Quantum Mechanic* (2019) [[arXiv:1901.02011](https://arxiv.org/abs/1901.02011), [10.10.11.6/handle/1/1339](http://10.10.11.6/handle/1/1339)]

* {#Parzygnat2016} [[Arthur Parzygnat]], _From observables and states to Hilbert space and back: a 2-categorical adjunction_, ([arXiv:1609.08975](https://arxiv.org/abs/1609.08975))

* {#Parzygnat2018} [[Arthur Parzygnat]], _Stinespring's construction as an adjunction_ ([arXiv:1807.02533](https://arxiv.org/abs/1807.02533))

* {#MO247626}  _MathOverflow: In which sense the GNS construction is a functor?_ ([link](https://mathoverflow.net/questions/247626/in-which-sense-the-gns-construction-is-a-functor))

* {#Gudder} [[Stanley Gudder]], _A Radon-Nikodym Theorem for $\ast$-Algebras_ ([link](https://projecteuclid.org/download/pdf_1/euclid.pjm/1102785959))

[[!redirects Functorial Aspects of the GNS Representation]]
[[!redirects Functorial Aspects of the GNS Representation]]
[[!redirects functorial aspects of the gns representation]]


functorial factorization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A __functorial factorization__ is a [[mathematical structure|structure]] on a [[category]] that factors any [[morphism]] into a [[composition|composite]] of two morphisms, in a way that depends [[functor|functorially]] on [[commutative squares]].

Functorial factorizations play a prominent role in [[model category]] theory.  On the one hand, many constructions there do rely on the factorizations into (acyclic) [[cofibrations]] and (acyclic) [[fibrations]] to be functorial, while, on the other hand, via the [[small object argument]] many examples of model categories do in fact carry a functorial factorization.  (As a result, some authors include functorial factorization in the axioms of a model category right away.)

Functorial factorizations also play an important role as an ingredient in [[algebraic weak factorization systems]].

## Definition

### Concrete explicit

+-- {: .un_defn}
###### Definition
A __functorial factorization__ on a category $\mathcal{C}$ is a way of assigning to any arrow $f$ in $\mathcal{C}$ a pair of composable arrows $f_L, f_R$ such that $f = f_R \circ f_L$, together with for any [[commuting square]]

$$
  \array{
    & \overset{h}{\longrightarrow}
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    & \overset{k}{\longrightarrow}
  }
$$

a morphism $E(h,k)$ completing their factorizations $f = f_R \circ f_L$ and $g = g_R \circ g_L$ to a further [[commuting diagram]]

$$
  \array{
    & \overset{h}{\longrightarrow}
    \\
    {}^{\mathllap{f_L}}\downarrow && \downarrow^{\mathrlap{g_L}}
    \\
    & \overset{E(h,k)}{\longrightarrow}
    \\
    {}^{\mathllap{f_R}}\downarrow && \downarrow^{\mathrlap{g_R}}
    \\
    & \overset{k}{\longrightarrow}
  }
  \,,
$$

in a way that depends functorially on the given commutative square, i.e. $E(h_1\circ h_2,k_1\circ k_2) = E(h_1,k_1) \circ E(h_2,k_2)$.
=--

### Abstract version

Write $\Delta[1] = \{0 \to 1\}$ and $\Delta[2] = \{0 \to 1 \to 2\}$ for the [[ordinal numbers]], regarded as [[posets]] and hence as [[categories]]. The [[arrow category]] $Arr(\mathcal{C})$ is equivalently the [[functor category]] $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image.
By precomposition, this induces [[functors]] $d_i  \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here

* $d_1$ sends a pair of composable morphisms to their [[composition]];

* $d_2$ sends a pair of composable morphisms to the first morphism;

* $d_0$ sends a pair of composable morphisms to the second morphism.

+-- {: .num_defn #FunctorialFactorization}
###### Definition

For $\mathcal{C}$ a [[category]], a **functorial factorization** of the morphisms in $\mathcal{C}$ is a [[functor]]

$$
  fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}
$$

which is a [[section]] of the [[composition]] functor $d_1 \;\colon\; \mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.

=--

## Examples

\begin{example}\label{InCombinatorialModelCategories}
  In a [[combinatorial model category]] there exists functorial factorizations into

1. [[acyclic cofibrations]] followed by [[fibrations]],

1. [[cofibrations]] followed by [[acyclic fibrations]],

hence in particular there exists functoral [[fibrant replacement]] and functorial [[cofibrant replacement]].
\end{example}
This is due to unpublished results by [[Jeff Smith]], see [Dugger (2001), Prop. 2.3 and §7](#Dugger01) and [Barwick (2010), Prop. 2.5](#Barwick10).

\begin{example}
All [[orthogonal factorization systems]] are automatically functorial.
\end{example}

\begin{example}
Not all [[weak factorization systems]] are functorial, although most are.  This includes those produced by the [[small object argument]], with due care, and also all [[algebraic weak factorization systems]].
\end{example}

## Properties
 {#Properties}

### Enriched functorial factorizations

Sufficient conditions in [[enriched category theory]] (in particular [[enriched model category]]-theory) for functorial factorization to exist as an [[enriched functor]] is discussed in [Hirschhorn 02, Theorem 4.3.8](#Hirschhorn02), [Shulman 06, Prop. 24.2](#Shulman06)

### Equivalence to pointed endofunctors

+-- {: .un_prop}
###### Proposition
The following are equivalent:

1. A functorial factorization on $\mathcal{C}$.
2. A [[pointed endofunctor]] $R$ of $\mathcal{C}^{\Delta[1]}$ such that $cod : \mathcal{C}^{\Delta[1]} \to \mathcal{C}$ is a strict morphism of pointed endofunctors from $R$ to $Id_{\mathcal{C}}$, i.e. $cod \circ R = cod$ and $cod$ maps the point $Id_{\mathcal{C}^{\Delta[1]}}\to R$ to the identity map of $Id_{\mathcal{C}}$.  (This is called a **pointed endofunctor over $cod$**.)
3. Dually, a copointed endofunctor $L$ of $\mathcal{C}^{\Delta[1]}$ under $dom$.
=--
+-- {: .proof}
###### Proof
A endofunctor $R$ of $\mathcal{C}^{\Delta[1]}$ maps every morphism $f$ in $\mathcal{C}$ to a morphism $f_R$, in a functorial way.  A point of such an endofunctor  assigns to each $f$ a pair of morphisms $f_L, f_M$ such that $f_R \circ f_L = f_M\circ f$, naturally with respect to commutative squares.  To say $cod \circ R = cod$ means that $f_R$ has the same codomain as $f$, and to say that $cod$ respects the points says that $f_M$ is an identity.  Thus, $f = f_R \circ f_L$ is a factorization.  For functoriality, the functoriality of $R$ gives the commutative square $k \circ f = f_R \circ E(h,k)$ and the functoriality of $E(-,-)$, while the naturality of $(-)_L$ gives the commutative square $E(h,k) \circ f = f_L \circ h$.  The converse and dual are straightforward.
=--

An [[algebraic weak factorization system]] is a functorial factorization together with compatible enhancements of these endofunctors to a monad and comonad.  This can often be detected with the help of a [[composition law for factorizations]].

## Related concepts

* [[fibrant replacement]]

## References

* {#Hirschhorn02} [[Philip Hirschhorn]], _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([AMS](http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

Existence of functorial factorization in [[combinatorial model categories]] (following unpublished results by [[Jeff Smith]]):

* {#Dugger01} [[Daniel Dugger]], Prop. 2.3 and §7 in: *Combinatorial model categories have presentations*, Advances in Mathematics **164** 1  (2001) 177-201 [[arXiv:math/0007068](https://arxiv.org/abs/math/0007068), [doi:10.1006/aima.2001.2015](https://doi.org/10.1006/aima.2001.2015)]

* {#Barwick10} [[Clark Barwick]], Prop. 2.5 in: *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320 [[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

Further discussion for [[enriched functors]]:

* {#Shulman06} [[Michael Shulman]], _Homotopy limits and colimits and enriched homotopy theory_ ([arXiv:math/0610194](https://arxiv.org/abs/math/0610194))

[[!redirects functorial factorizations]]

[[!redirects functorial factorization system]]
[[!redirects functorial factorization systems]]