(2|1)-dimensional Euclidean field theory

[[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
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###Context###
#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Supergeometry
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[[!include supergeometry - contents]]
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This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]

See there for background and context.

This entry here is about the definition of $(2|1)$-dimensional [[super-cobordism]] categories where cobordisms are [[Euclidean supermanifold]]s, and about $the (2|1)$-dimensional [[FQFT]]s given by functors on these.

=--

Previous:

 * [[Axiomatic field theories and their motivation from topology]]

 * [[(1,1)-dimensional Euclidean field theories and K-theory]]

 * [[(2,1)-dimensional Euclidean field theories and tmf]]

 * [[bordism categories following Stolz-Teichner]]

#Contents#

* tic
{:toc}

#Idea#

Previously we had defined smooth categories of [[Riemannian cobordism]]s. Now we pass from [[Riemannian manifold]]s to [[Euclidean supermanifold]]s and define the corresponding smooth [[cobordism category]]. Then we define $(d|\delta)$-dimensional Euclidean field theories to be smooth representations of these categories.

As described at  [[(2,1)-dimensional Euclidean field theories and tmf]], the idea is that $(2|1)$-dimensional Euclidean field theories are a geometric model for [[tmf]] [[cohomology theory]]. While there is no complete proof of this so far, in the next and final session

* [[modular forms from partition functions]]

it will be shows that the claim is true at least for the [[cohomology ring]] over the point: the [[partition function]] of a $(2|1)$-dimensional EFT is a modular form. Hence $(2|1)$-dimensional EFTs do yield the correct [[cohomology ring]] of [[tmf]] over the point.

#Details#

Let [[SDiff]] be the [[category]] of [[supermanifold]]s.

We will define a [[stack]]/[[fibered category]] on $SDiff$ called $E Bord_{2|1}$ whose morphisms are smooth families of (2|1)-dimensional [[super-cobordism]]s, and a [[stack]]/[[fibered category]] $sTV^{fam}$ of topological super vector bundles.

So recall

* [[supergeometry - contents|supergeometry]].

**question**: What is the right notion of Riemannian or Euclidean structure on [[super-cobordism]]s?

**strategy**: From the [[path integral]] perspective we need some structure on $\Sigma$ such that the "space" of maps $Maps(\Sigma,X)$ naturally carries some measure that allows to perform a [[path integral]].

This perspective suggests certain generalizations of the notion of [[Riemannian manifold]] to [[supermanifold]]s which may be a little different than what one might have thought of naively.

We want to define [[Euclidean supermanifold]]s as a generalization of [[Riemannian manifold]] with _flat_ Riemannian metric.

notice that there is a canonical bijection between

* flat [[Riemannian metric]]s on a $d$-dimensional [[manifold]] $X$

* a maximal [[atlas]] on $X$ consisting of charts such that all transition functions belong the the **Euclidean group** or **Galileo group**

  $$
    Eucl(\mathbb{R}^d)
    :=
    \mathbb{R}^d \rtimes O(\mathbb{R}^d)
  $$

  (rigid translations and rotations)

In analogy to that we define:

Similarly a **Euclidean structure** on a $(d|\delta)$-dimensional [[supermanifold]] is defined using the Euclidean [[super Lie group]] given by the [[semidirect product]]

$$
  Eucl(\mathbb{R}^{d|\delta})
  :=
  \mathbb{R}^{d|\delta}
  \rtimes
  Spin(\mathbb{R}^d)
$$

where the [[Spin]] group acts on the translations in $\mathbb{R}^{d|\delta}$ in a way to be specified.

first recall the notion of

* [[complex supermanifold]]

**goal** replace the standard Euclidean group $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$ by the [[super Euclidean group]]
$(X,G)$ where $X$ is a suitable [[supermanifold]]
and $G$ a suitable [[super Lie group]].

This leads to the notion of

* [[Euclidean supermanifold]].

The morphisms of the category $E Bord_{(2|1)}$ will be [[cobordism]]s that are [[Euclidean supermanifold]]s.

**goal** define the [[fibered category]]

$$
  \array{
    E Bord_{d|\delta}^{sfam}
    \\
    \downarrow
    \\
    cSDiff
  }
$$

where $cSDiff$ is the category of [[complex supermanifold]]s.

The objects of this fibered category are

$$
  \array{
    Y^{\pm} &\to& Y &\leftarrow& Y^c
    \\
    & \searrow & \downarrow & \swarrow
    \\
    && S
  }
$$

where $Y \to S$ is a family of [[Euclidean supermanifold]]s of dimension $(d|\delta)$.

For the non-super, non-family version of **Euclidean bordism** we require that the core $Y^c$ is totally geodesic in $Y$.

now for the superversion we require that there exist charts (in the open atlas) of $Y \to S$ covering all of $Y^c$ such that

$$
  \array{
    && S
    \\
    & \nearrow && \nwarrow
    \\
    Y \supset_{open} U &&\stackrel{\phi}{\to}&&
    V \subset S \times \mathbb{R}^{d|\delta}_{cs}
    \\
    \downarrow^{\supset} &&&& \downarrow^{\supset}
    \\
    Y^c \supset U \cap Y^c
    &&\stackrel{\simeq}{\to}&&
    V \cap S \times \mathbb{R}^{d-1|\delta}
    \subset S \times \mathbb{R}^{d-1|\delta}_{cs}
  }
$$

next, a **Euclidean superbordism** from $Y_0 \to S$ to $Y_1 \to S$ is a diagram

$$
  \array{
    Y_1
     &\stackrel{i_1}{\to}&
    \Sigma
     &\stackrel{i_0}{\leftarrow}& Y_1
    \\
     & \searrow & \downarrow & \swarrow
    \\
    && S
  }
$$

where $i_0, i_1$ are morphisms (of families of $(X,G)$-spaces) satisfying the (+)-condition and the (c)-condition described at [[bordism categories following Stolz-Teichner]].

Now a morphism in $E Bord^{sfam}_{d|\delta}$ from $Y_0 \to S_0$ to $Y_1 \to S_1$ is a bordism fitting into a diagram

$$
  \array{
      \Sigma &\stackrel{i_1}{\leftarrow}&
      f^* Y_1 &\to& Y_1
     \\
     \uparrow^{i_0} &\searrow& \downarrow && \downarrow
     \\
     Y_1 &\to & S_0 &\stackrel{f}{\to}& S1
  }
$$

and we identify bordisms $\Sigma, \Sigma'$ if they are isometric -- namely isomorphic in the category of [[Euclidean supermanifold]]s --  "rel boundary".

**definition** A **$(d|\delta)$-dimensional Euclidean field theory** is a [[symmetric monoidal functor]]

$$
  E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam},
  TV^{sfam})
$$

of [[symmetric monoidal category|symmetric monoidal]] [[fibered category|fibered cateories]] (i. [[symmetric monoidal functor]] as well as [[cartesian functor]] ) over the category $cSDiff$ of [[complex supermanifold]]s.

**Definition** (roughly) $TV^{sfam}$ is the category of families of topological vector spaces parameterized by [[complex supermanifold]]s.

Recall that the ordinary category $TV$ is the category of complete [[Hausdorff space|Hausdorff]], locally convex [[topological vector space]].

define the [[projective tensor product]] of two such $V, W \in TV$. This is a certain completion of the algebraic tensor product $V \otimes_{alg} W$ with respect to the projective topology on $V \otimes_{alg} W$.

This will be the coarsest [[topology]] (the one with the least open sets) making the following maps $f'$

$$
  \array{
     V \otimes_{alg} W &\to& Z
     \\
     & \nearrow_{f'}
     \\
     V \times W
   }
$$

continuous.

**Remark**

$$
  \array{
    C^\infty(M \times N)
    &\leftarrow&
    C^\infty(M) \otimes_{alg} C^\infty(N)
    \\
    & {}_{\simeq}\nwarrow & \downarrow^{\subset}
    \\
    && C^\infty(M) \otimes C^\infty(N)
  }
$$

**Definition** the objects of $TV^{sfam}$ are pairs $(S,V)$ for $S$ a [[supermanifold]] and $V$ is a [[sheaf]] of locally complex $\mathbb{Z}_2$-graded [[topological vector space]] with the structure of a sheaf of modules of the [[structure sheaf]] $O_S$.

**goal** define the [[partition function]] of of a $(2|1)$-dimensional Euclidean field theory.

**definition** Let $E$ be an EFT as above.

We may think of $\mathbb{R}_+ \times h$ (positive axis times upper half plane) as moduli space of Euclidean tori, where for $(\ell, \tau) \in \mathbb{R}_+ \times h$ we get a torus (regarded as a [[cobordism]]) denoted $T_{\ell,\tau}$. This is the torus given by the lattice spanned by $(1,0)$ and $\ell Re(\tau) + Im(\tau)$ in the upper half plane. Then for the ordinary EFT we would define

$$
  Z_E : \mathbb{R}_+ \times h \to \mathbb {C}
$$

$$
  (\ell,\tau) \mapsto E(T_{\ell,\tau})
$$

For the superversion we put

$$
  Z_{E} := Z_{E_{red}}
$$

where

$$
  \array{
    && E Bord_{2|1}^{sfam}
    \\
    & {}^{\rho}\nearrow
    &&
    \searrow^{E}
    \\
    E Bord_{2, Spin}^{fam}
    &\stackrel{E_{red}}{\to}&
    TV^{fam}
    &
     \hookrightarrow
    &
    TV^{sfam}
  }
$$

# Examples #

## explicit description of $E Bord_{1}^{fam}$ ##

See [[bordism categories following Stolz-Teichner]].

The category $E Bord_1^{fam}$ is generated from

*  the _family of right elbows__

   $$
   \array{
     1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt)
     & \ni&  R  & :=  \mathbb{R}_+ \times \mathbb{R}
     \\
     && \downarrow
     \\
     && S := \mathbb{R}_+
    }
   $$

* the point-family of the left elbox
  $$
    \array{
      L_0
      \\
      \downarrow
      \\
      S := pt
    }
  $$

* the family of intervals in $E Bord^{fam}_{\mathbb{R}^+}(pt,pt)$

  $$
    \array{
      I
      \\
      \downarrow
      \\
      \mathbb{R}_{\geq 0}
    }
  $$

Because:

$
  E \in Fun^{\otimes}_{Diff}(E Bord^{fam}, TV^{fam})
$

is determined by

$$
  E(pt) =: V \in TV
$$

$$
  E(L_0) =: \lambda : V \otimes V \to \mathbb{R}
$$

$$
  E(R) =:  \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)
$$

$$
  E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))
$$

forms a _smooth_ semigroup under composition generated by

$$
  H \in End(V)
$$

(the [[Hamiltonian operator]])

$$
  \array{
    V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&&
    End(V)
    \\
    & {}_{\rho}\nwarrow && \nearrow_{e^{- t H}}
    \\
    && \mathbb{R}_+
  }
$$

so due to smoothness the data collapses to the infinitesimal data

$$
  (V, \lambda, H)
$$

**example -- ordinary quantum mechanics** Let $M$ be a [[Riemannian manifold]]. Then  set

* $H:= \Delta$ the corresponding [[Laplace operator]];

* $V := C^\infty(M) \subset L^2(M)$;

* $\lambda$ is the restriction of the $L^2(M)$ inner product to $V$

where $e^{-t H}$ is "trace class" in the non-standard sense described above in that it makes the above diagram commute.

So everything as known from standard [[quantum mechanics]] textbooks, except that we don't use the full [[Hilbert space]] of states, but just the [[Frechet space]] of smooth functions.

## explicit description of $E Bord_{2}$ ##

The category $E Bord_{2}_{oriented}^{fam}$ has the following generators:

objects are generated from

* the **circle** $K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell$  of length $\ell \gt 0$ (with collars!! that's why it looks like a cylinder of circumference $\ell$)

  notice that we may think of $\ell $ as parameteriing translation by $\ell$ in $\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)$

  and the circle with $(+)$/$(-)$-collars reversed

morphisms are generated from

* **cylinders** $C_{\ell,\tau}$ which as a manifold is $\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell$ where $\tau$ parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by $\ell \cdot Re(\tau)$ upwards and rotated by $\ell \cdots Im(\tau)$.

* **right elbows** which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.

* the **thin left elbow** $L_\ell$, similar to the above, with arbitrary $\ell$ but $\tau = 0$

* the **torus** $T_\tau$ obtained from the cylinder by gluing incoming and outgoing

**notice** the **pair of pants** is not a morphism in the category at all! since, recall, we require all bordisms to be _flat_ and all boundaries to be _geodesics_ . There is no way to put such a flat metric on the trinion.

satisfying the relations

$$
  L_\ell \circ R_{\ell, \tau}
  =
  T_{\ell, \tau}
$$

as for the non-family version, but now also with the new relations

$$
  T_{\ell', \tau'} = T_{\ell, \tau}
$$

whenever $\ell' = \ell \cdot|c \tau + d|$ and $\tau' = \frac{a \tau + b }{c \tau + d}$

for $\left(\array{a & b \\ c & d }\right) \in SL_2(\mathbb{Z})$.

Notice that $SL_2(\mathbb{Z})$ is generated by

* translation $(\ell, \tau) \mapsto (\ell, \tau + 1)$

* S-matrix $(\ell, \tau) \mapsto (\ell \cdot |\tau|, - \frac{1}{\tau})$

and now there is **one more relation**

$$
  T_{\ell, \tau}
  =
  T_{\ell |\tau|, - \frac{1}{\tau}}
$$

as usual write $q := e^{2 \pi  i \tau}$ which is on the pointed unit disk since $\tau$ is half plane since $\tau$

## explicit description of $E Bord_{2}^{fam}$ ##

thwe category $E Bord_{2, oriented}^{fam}$ is generated from

objects:

* $\array{K \\ \downarrow \\ S = \mathbb{R}_+}$

* morphisms

  $$
    \array{
      L \\ \downarrow \\ \mathbb{R}_+
    }
  $$

  $$
    \array{
      R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z}
    }
  $$

  $$
    \array{
      C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z}
    }
  $$

subject to the relations

... as before (homework 3, problem 4).. and the furhter one:

for

$$
  \array{
    T  && \alpha^* T
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}_+ \times h
    &\stackrel{\alpha}{\leftarrow}&
    \mathbb{R}_+ \times h
    \\
    \\
    (\ell\cdot |\tau|, -\frac{1}{\tau})
    &\stackrel{}{\lt\leftarrow}&
    (\ell, \tau)
  }
$$

the relation is

$$
  \alpha^* T  = T
  \,.
$$

## References

* [[Stephan Stolz]], [[Peter Teichner]], *[[What is an elliptic object?]]* in: _Topology, geometry and quantum field theory_, London Math. Soc. LNS **308**, Cambridge Univ. Press (2004) 247-343  [[pdf](https://math.berkeley.edu/~teichner/Papers/Oxford.pdf), [[Stolz-Teichner_EllipticObject.pdf:file]]]

* {#Stolz} [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf))

* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_ , in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Symposia in Pure Mathematics (2011) [[arXiv:1108.0189](http://arxiv.org/abs/1108.0189)]

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?* [[arXiv:2303.09138](https://arxiv.org/abs/2303.09138)]

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23)]

[[!redirects (2,1)-dimensional Euclidean field theories]]


(bo, ff) factorization system

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### Context
#### Factorization systems
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[[!include factorization systems - contents]]
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#### Category theory
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[[!include category theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Definition

There is an [[orthogonal factorization system]] on the category [[StrCat]], whose left class is the class of [[bo functor|bijective-on-objects functors]], or "bo functors" and whose right class is the class of [[full and faithful functor|full and faithful functors]], or "ff functors".

This means that each functor $f$ decomposes as a composition of the form $j e$, where $e$ is bijective on objects and $j$ fully faithful; and if

$$\array{
A &\overset{u}{\longrightarrow}& C
\\
\mathllap{{}^{e}}\big\downarrow &&\big\downarrow \mathrlap{{}^{j}}
\\
B &\underset{v}\longrightarrow& D
}$$

is a commutative diagram with $e$ bijective on objects and $j$ fully faithful, then there is a unique functor $h \colon B\to C$ such that $h e = u$ and $j h = v$. The object through which $f$ factors is called the [[full image]] of $f$.

In fact, this can be generalized to a square commuting up to invertible [[natural transformation]], in which case one still concludes that $h e = u$ but that $j h \cong v$, with the isomorphism composing with $e$ to give the original isomorphism.  This means that this is an [[enhanced factorization system]].

## Properties

This factorization system can be constructed using [[generalized kernels]].

For [[essentially surjective functors]], one can relax both the commuting and the uniqueness to obtain a [[factorization system in a 2-category]].

[[!redirects (bo,ff) factorization system]]
[[!redirects bo-ff factorization system]]
[[!redirects (bo,ff) factorization]]
[[!redirects bo-ff factorization]]
[[!redirects (bo,ff) factorisation system]]
[[!redirects bo-ff factorisation system]]
[[!redirects (bo,ff) factorisation]]
[[!redirects bo-ff factorisation]]


(classical) axiom of multiple choice

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### Context
#### Foundations
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[[!include foundations - contents]]
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# The axiom of multiple choice
* table of contents
{: toc}

This article is about a [[classical mathematics|classical]] [[set theory]] [[axiom]].  Some literature instead uses this name for an unrelated weakening of [[axiom of choice|AC]].  For that notion, see _[[axiom of multiple choice]]_.

## Idea

The **axiom of multiple choice** (AMC) weakens the [[axiom of choice]] by allowing [[choice functions]] to choose [[finite sets]], rather than particular [[elements]].

##Statement

Recall that one common statement of the [[axiom of choice]] is:

:For every [[set]] $S$ of [[inhabited set|non-empty sets]], there is a [[function]] $f$ defined on $S$ such that $\forall X\in S$, $f(X)\in X$.

Such an $f$ is called a __choice function__ for $S$.

The axiom of multiple choice weakens the axiom of choice by allowing choice functions to pick out finite [[subsets]], rather than finite sets.  It says:

:For every set $S$ of non-empty sets, there is a function $f$ defined on $S$ such that $\forall X\in S$, $f(X)\subseteq X$ and $f(X)$ is finite and non-empty.

## Relationships to other axioms

The axiom of multiple choice is [[logical equivalence|equivalent]] to the axiom of choice modulo [[ZF]] set theory.  However, it is strictly weaker in [[ZFA]] and other similar set theories.  AMC holds in any [[permutation model]] with finite supports where each atom has a finite orbit.  For a detailed proof, see [Jech's "The Axiom of Choice"](https://www.gwern.net/docs/math/1973-jech-theaxiomofchoice.pdf), chapter 9.

The [[axiom of multiple choice|constructive axiom by the same name]] is not historically related, and the two axioms are independent.  Any permutation model will satisfy [[SVC]], which [Rathjen](#Rathjen) proves implies the constructive axiom, but this AMC can fail in a permutation model.

## References

* Jech, _The Axiom of Choice_ (1973), ISBN : 0444104844 (New York)

* A. Lévy. Axioms of multiple choice. Fundamenta mathematicae, vol. 50 no. 5 (1962), pp. 475–483

The constructive axiom by the same name is discussed in:

* {#Rathjen} Rathjen, "Choice principles in constructive and classical set theories"

category: foundational axiom


(co)isotropic subspaces - table

|   type of [[subspace]] $W$ of [[inner product space]]  | condition on [[orthogonal]] space $W^\perp$  | |
|--|--|--|
|  [[isotropic subspace]] | $W \subset W^\perp $ | |
|  [[coisotropic subspace]] | $W^\perp \subset W$ |  |
|  [[Lagrangian subspace]] |  $W = W^\perp$ | (for [[symplectic form]]) |
|  [[symplectic vector space|symplectic space]] | $W \cap W^\perp = \{0\}$ | (for [[symplectic form]])|


(d,d) Klein space

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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=--

> not to be confused with [[Klein geometry]]

#Contents#
* table of contents
{:toc}

## Idea

The [[Euclidean space|flat space]] $\mathbb{R}^{d,d}$ of $2d$ dimensions equipped with a [[pseudo-Riemannian metric]] of $(d,d)$ signature.

## Related concepts

* [[Klein geometry]]

* [[pseudo-Riemannian manifold]]

* [[smooth Lorentzian space]]

## References:

* [[John Barrett]], [[Gary W. Gibbons]], M. J. Perry, [[Christopher N. Pope]], P. J. Ruback, *Kleinian Geometry and the $N=2$ Superstring*, Int. J. Mod. Phys. **A9** (1994) 1457-1494  [[doi:10.1142/S0217751X94000650](https://doi.org/10.1142/S0217751X94000650), [arXiv:hep-th/9302073](https://arxiv.org/abs/hep-th/9302073)]


(dense,closed)-factorization

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[topology|point set topology]], every subspace $X$ of a space $A$ has a unique largest subspace in which $X$ is [[dense subspace|dense]], namely simply the closure $\overline{X}$. Using maps, this amounts to say that $X\hookrightarrow A$ factors as $X\hookrightarrow\overline{X}$ followed by $\overline{X}\hookrightarrow A$.

The **(dense,closed)-factorization** generalizes this idea from topology to [[topos theory]]. It can be viewed as a way to associate to every [[subtopos]] $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ a closure $\overline{Sh_{j}(\mathcal{E})}$.

## Statement

A [[geometric embedding]] of [[elementary toposes]]

$$
  Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}
$$

factors as

$$
  Sh_j(\mathcal{E})
    \hookrightarrow
  Sh_{c(ext(j))}(\mathcal{E})
    \hookrightarrow
  \mathcal{E}
$$

where $ext(j)$ (the "exterior" of $j$) denotes the $j$-closure of $\emptyset \rightarrowtail 1$ and

$$
  \bar j \coloneqq c(ext(j))
$$

the [[closed subtopos|closed topology]] that corresponds to the [[subterminal object]] $ext(j)$.

Here the first inclusion exhibits a [[dense subtopos]] and the second a [[closed subtopos]].

## Remark

$Sh_{c(ext(j))}(\mathcal{E})$ can be viewed as the 'best approximation' of $Sh_j(\mathcal{E})$ by a closed subtopos and therefore might be called the _closure_ $Cl(Sh_j(\mathcal{E}))$ of $Sh_j(\mathcal{E})$.[^sga4]

[^sga4]: ([SGA4](#SGA4), p.461) uses the term '_l'adhérence_' for it.

Its complement, the [[open subtopos]] $Ext(Sh_j(\mathcal{E}))$ corresponding to the [[subterminal object]] $ext(j)$ deserves in turn to be called the _exterior_ of $Sh_j(\mathcal{E})$.

## The (dominant,closed)-factorization

The (dense,closed)-factorization is a special case for inclusions of a slightly more general factorization which attaches to a general [[geometric morphism]] the closure of its image.

Recall that an inclusion is [[dense subtopos|dense]] precisely if it is a [[dominant geometric morphism]], hence the following is pertinent for the (dense,closed)-factorization as well.

+-- {: .num_prop #dense-closed}
###### Proposition
Let $i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E}$ be dominant and a [[closed subtopos|closed inclusion]] at the same time. Then $i$ is an isomorphism.
=--

**Proof**: Recall that $X\in\mathcal{E}$ are in the [[closed subtopos]] precisely when they satisfy $X\times U\cong U$ with $U$ the [[subterminal object]] associated to $i$. But $i$ is dominant, or what comes to the same for inclusions: dense, hence $\emptyset$ is in $Sh_{c(U)}(\mathcal{E})$ and therefore $\emptyset\times U\cong U$ . From this follows $U\cong\emptyset$, which in turn implies that all $X\in\mathcal{E}$ are in $Sh_{c(U)}(\mathcal{E})$ . $\qed$

+-- {: .num_prop #dominant-closed}
###### Proposition
Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a [[dominant geometric morphism]] $d$ followed by a [[closed subtopos|closed inclusion]] $c$.
=--

**Proof**: Let $i\circ d_1$ be the [[(geometric surjection, embedding) factorization system|surjection-inclusion factorization]] of $f$. Since $d_1$ is surjective, it is dominant (cf. [this proposition](dominant+geometric+morphism#dominant_surjection)). Then we use the (dense,closed)-factorization to factor $i$ into $c\circ d_2$. Since both $d_i$ are dominant, so is $d:=d_2\circ d_1$ and $c\circ d$ yields the demanded factorization of $f$. $\qed$

## Related entries

* [[dense]]

* [[dense subtopos]]

* [[dominant geometric morphism]]

* [[(geometric surjection, embedding) factorization system]]

## References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.I_ , Oxford UP 2002. (around Lemma A 4.5.19, p. 219)

* {#Caramello09} [[Olivia Caramello]], _Lattices of theories_ , [arXiv:0905.0299](http://arxiv.org/abs/0905.0299) (2009). (section 8)


(epi, mono) factorization system

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#### Factorization systems
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#### Category theory
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#Contents#
* table of contents
{:toc}

## Definition

An **epi-mono factorization system** is an [[orthogonal
factorization system]] in which the left class is the class of [[epimorphisms]] and the right class is the class of [[monomorphisms]].  Such a factorization system exists on any (elementary) [[topos]], and indeed on any [[pretopos]]. It provides the factorization through the [[image]] of any morphism.

## Properties

Note that any category which admits an epi-mono factorization system is necessarily [[balanced category|balanced]].  This excludes many commonly occurring categories.  More common are ([[strong epimorphism|strong epi]], mono) and (epi, strong mono) factorization systems; the former exists in any [[regular category]] and the latter in any [[quasitopos]], as well as in other categories such as [[Top]].

The epi-mono factorization system in a topos is the special case of the [[n-connected/n-truncated factorization system]] in an [[(∞,1)-topos]] for the case that $(n = -1)$ and restricted to [[0-truncated]] [[object]]s.

## Related concepts

* **(epi, mono) factorization system**

* [[(eso+full, faithful) factorization system]]

* [[n-connected/n-truncated factorization system]].

## References

For instance:

* {#Borceux94} [[Francis Borceux]], vol 1, section 4.4. of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

[[!redirects (epi, mono) factorization system]]

[[!redirects (epi,mono)-factorization]]
[[!redirects (epi,mono) factorization]]
[[!redirects (epi, mono)-factorization]]
[[!redirects (epi, mono) factorization]]

[[!redirects (epi, mono)-factorization system]]
[[!redirects (epi, mono)-factorization systems]]

[[!redirects (epi,mono) factorization system]]
[[!redirects (epi,mono) factorization systems]]
[[!redirects (epi,mono)-factorization system]]
[[!redirects (epi,mono)-factorization systems]]

[[!redirects epi/mono factorization system]]
[[!redirects epi/mono factorization systems]]
[[!redirects epi-mono factorization system]]
[[!redirects epi-mono factorization systems]]

[[!redirects (effective epi, mono) factorization system]]
[[!redirects (effective epi, mono) factorization systems]]
[[!redirects (effective epi,mono) factorization system]]
[[!redirects (effective epi,mono) factorization systems]]

[[!redirects epi-mono factorization]]
[[!redirects epi-mono factorizations]]


(eso and full, faithful) factorization system

[[!redirects (eso+full, faithful) factorization system]]

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#### Category theory
+-- {: .hide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In the [[2-topos]] [[Cat]], the [[pair]] of [[classes]] of [[morphisms]]

* left class: [[essentially surjective and full functors]]

* right class: [[faithful functors]]

forms a [[factorization system in a 2-category]].  This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].

In fact, an analogous factorization system exists in any [[2-exact 2-category]] and any [[(2,1)-exact (2,1)-category]], including any [[Grothendieck 2-topos]] or [[(2,1)-topos]]; see [[michaelshulman:full morphism|here]].

## Properties

* When restricted to [[Grpd]], this is the special case of the [[n-connected/n-truncated factorization system]] in the [[(∞,1)-topos]] [[∞Grpd]] for the case that $(n = 0)$ and restricted to [[1-truncated]] [[objects]].

* For $f : X \to Y$ a functor between groupoids, its factorization is through a groupoid $im_2 f$ which is, up to equivalence, given as follows;

  * [[objects]] are those of $X$;

  * a [[morphism]] $[\phi] : x_1 \to x_2$ is an [[equivalence class]] of morphisms in $X$ where $[\phi] = [\phi']$ if $f(\phi) = f(\phi')$.

More on this is at _[infinity-image --  Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.

## Related concepts

* [[epi-mono factorization system]]

* [[(eso, fully faithful) factorization system]]
* **(eso+full, faithful)-factorization system**

* [[n-connected/n-truncated factorization system]].

[[!redirects essentially surjective and full/faithful factorization system]]


(eso, fully faithful) factorization system

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#### Category theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In the [[2-topos]] [[Cat]], the pair of classes of morphisms ([[essentially surjective functor|essentially surjective]] functors, [[fully faithful functors]]) form a [[factorization system in a 2-category]].  This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].

e.g. ([Dupont-Vitale 03, 7.9, example 2](#DupontVitale03))

In fact, an analogous factorization system exists in any regular 2-category and any (2,1)-exact (2,1)-category, including any Grothendieck 2-topos or (2,1)-topos; see [[michaelshulman:regular morphism|here]].

## Properties

* The factorization of a functor in this factorization system is the construction of its [[full image]].

* More on this for the case of functors between [[groupoids]] is at _[infinity-image --  Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.

## Related concepts

* [[(epi, mono) factorization system]]

* **(eso, fully faithful)-factorization system**
* [[(eso+full, faithful) factorization system]]

* [[ternary factorization system]]

## References

* {#DupontVitale03} M. Dupont, [[Enrico Vitale]], _Proper factorization systems in 2-categories_, Journal of pure and applied algebra, 179 (2003), pp.65-86

[[!redirects essentially surjective/fully faithful factorization system]]
[[!redirects essentially surjective-fully faithful factorization system]]
[[!redirects eso/ff factorization system]]
[[!redirects eso/fully faithful factorization system]]
[[!redirects eso-ff factorization system]]
[[!redirects eso-fully faithful factorization system]]
[[!redirects (eso, ff) factorization system]]


(g-2) anomaly

#Contents#
* table of contents
{:toc}

## Idea

For the moment see at _[[anomalous magnetic moment]]_ _[this section](https://ncatlab.org/nlab/show/anomalous+magnetic+moment#Anomalies)_

## Related concepts

* [[flavour anomaly]]

* [[Cabibbo anomaly]]

* [[V_cb puzzle]]

[[!redirects (g-2)-anomaly]]


(geometric surjection, embedding) factorization system

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#Contents#
* table of contents
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## Idea

Every [[geometric morphism]] between [[toposes]] factors into a [[geometric surjection]] followed by a [[geometric embedding]]. This exhibits an [[image]] construction in the [[topos theory|topos-theoretic]] sense, and gives rise to a [[factorization system in a 2-category]] for [[Topos]].

## Statement

+-- {: .num_prop}
###### Proposition

There is a [[factorization system on a 2-category|factorization system]] on the [[2-category]] [[Topos]] whose left class is the
[[surjective geometric morphism]]s and whose right class is the [[geometric embeddings]].

Moreover, the factorization of a given geometric morphism $f : \mathcal{E} \to \mathcal{F}$ is, up to [[equivalence of categories|equivalence]], through the canonical surjection onto the [[topos of coalgebras]] $f^* f_* CoAlg(\mathcal{E})$ of the [[comonad]]  $f^* f_* : \mathcal{E} \to \mathcal{E}$:

$$
  \array{
     \mathcal{E} &&\stackrel{f}{\to}&& \mathcal{F}
     \\
     & {}_{\mathllap{F}}\searrow && \nearrow
     \\
     && f^* f_* CoAlg(\mathcal{E})
  }
  \,E.
$$

=--

This appears for instance as ([MacLaneMoerdijk, VII 4., theorem 6](#MacLaneMoerdijk)).

We use the following lemma

+-- {: .num_lemma #ConditionForSheafFactorization}
###### Lemma

Let $j$ be a [[Lawvere-Tierney topology]] on a [[topos]] $\mathcal{E}$ and write $i : Sh_j(\mathcal{E}) \to \mathcal{E}$ for the corresponding [[geometric embedding]].

Then a [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ factors through $i$ precisely if

* the [[direct image]] $f_*$ takes values in $j$-sheaves;

or, equivalently

* the [[inverse image]] $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s.

=--

This appears as ([MacLaneMoerdijk, VII 4. prop. 2](#MacLaneMoerdijk)).

+-- {: .proof}
###### Proof of the lemma

We first show the first statement, that for $f$ to factor it is sufficient for $f_*$ to take values in $j$-sheaves: in that case, set

$$
  p_* := i^* f_*: \mathcal{F} \to Sh_j(\mathcal{E})
  \,.
$$

Since by assumption the [[unit of an adjunction|unit]] map $x \to i_* i^* x$ is an [[isomorphism]] on the image of $f_*$ this indeed serves to factor $f_*$:

$$
  i_* p_* \simeq i_* i^* f_* \simeq f_*
  \,.
$$

The [[left adjoint]] to $p_*$ is then

$$
  p^* \simeq f^* i_*
  \,,
$$

because

$$
  \begin{aligned}
    \mathcal{F}(g^* E, F)
     & \simeq \mathcal{F}(f^* i_* E, F)
    \\
    & \simeq \mathcal{E}(i_* E, f_* F)
    \\
    & \simeq \mathcal{E}(i_* E, i_* i^* f_* F)
    \\
    & \simeq Sh_j\mathcal{E} (E, i^* f_* F)
    \\
    & \simeq Sh_j(E, p_* F)
  \end{aligned}
  \,,
$$

where in the middle steps we used that $f_* F$ is a $j$-sheaf, by assumption, and that $i_*$ is full and faithful.

It is clear that $p^*$ is left exact, and so $(p^* \dashv p_*)$ is indeed a factorizing geometric morphism.

We now show that $f_*$ taking values in sheaves is equivalent to $f^*$ mapping dense monos to isos.

Let $u : U \hookrightarrow X$ be a $j$-[[dense monomorphism]] and $A \in \mathcal{E}$ any object. Consider the induced naturality square

$$
  \array{
    \mathcal{E}(X, f_* A) &\stackrel{\simeq}{\to}& \mathcal{F}(f^* X, A)
    \\
    {}^{\mathllap{\mathcal{E}(u, f_* A)}}\downarrow
     &&
     \downarrow^{\mathrlap{\mathcal{F}(f^* u, A)}}
    \\
    \mathcal{E}(U, f_* A)
      &\stackrel{\simeq}{\to}&
    \mathcal{F}(f^* U, A)
  }
$$

of the adjunction [[natural isomorphism]]. If now $f_* A$ is a $j$-sheaf and $u$ a [[dense monomorphism]], then by definition the left vertical morphism is also an isomorphism and so is the right one. By the [[Yoneda lemma]] this being an iso for all $A$ is equivalent to $f^* u$ being an iso. And conversely.

=--

+-- {: .proof}
###### Proof of the proposition

Let $f : \mathcal{F} \to \mathcal{E}$ be any [[geometric morphism]].

We first discuss the existence of the factorization, then its uniqueness.

To construct the factorization, we shall give a [[Lawvere-Tierney topology]] on $\mathcal{E}$ and factor $f$ through the [[geometric embedding]] of the corresponding [[sheaf topos]].

Take the closure operator $\overline{(-)} : Sub(-)_{\mathcal{E}} \to Sub(-)_{\mathcal{E}}$ to be given by sending $U \hookrightarrow X$ to the [[pullback]]

$$
  \array{
    \overline{U} &\to& f_* f^* U
    \\
    \downarrow && \downarrow
    \\
    X &\to& f_* f^* X
  }
  \,,
$$

where the bottom morphism is the $(f^* \dashv f_*)$-[[unit of an adjunction|unit]].  One checks that this is indeed a closure operator by the fact that $f^*$ preserves both pullbacks and pushouts.

Notice that this implies that for two [[subobject]]s $U_1, U_2 \hookrightarrow X$ we have

\[
  \label{ASubobjectRelation}
  (U_1 \subset \overline{U_2})
   \;\;\;
   \Leftrightarrow
   \;\;\;
  (f^* U_1 \subset f^* U_2)
\]

Write $j$ for the corresponding [[Lawvere-Tierney topology]] and

$$
  i : Sh_j(\mathcal{E}) \to \mathcal{E}
$$

for the corresponding [[geometric embedding]].

By lemma \ref{ConditionForSheafFactorization} we get a factorization through $I$ if $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s. But if $U \hookrightarrow X$ is dense so that $X \subset \overline{U}$ then, by (eq:ASubobjectRelation), $f^* X \subset f^* U$ and hence $f^* X = f^* U$.

Write

$$
  f : \mathcal{F} \stackrel{p}{\to} Sh_j(\mathcal{E}) \stackrel{i}{\to}
  \mathcal{E}
$$

for the factorization thus established. It remains to show that $p$ here is a [[geometric surjection]]. By one of the equivalent characterizations discussed there, this is the case if $p^*$ induces an injective homomorphism of subobject lattices.

So suppose that for subobjects $U_1, U_2 \subset X$ we have $p^* U_1 \simeq p^* U_2$. Observe that then also $f^* i_* U_1 \simeq f^* i_* U_2$, because

$$
  \begin{aligned}
    f^* i_* U_1 & \simeq p^* i^* i_* U_1
    \\
    & \simeq p^* U_1
    \\
    & \simeq p^* U_2
    \\
    & \simeq p^* i^* i_* U_2
    \\
    & \simeq f^* i:* U_2
  \end{aligned}
$$

by the fact that $i_*$ is [[full and faithful functor|full and faithful]].  With (eq:ASubobjectRelation) it follows that also

$$
  i_* U_1 \simeq \overline{i_* U_2}
$$

and hence

$$
  \cdots \simeq i_* U_2
$$

by the very fact that $i_*$ includes $j$-sheaves in general, hence $j$-closed subobjects in particular. Finally since $i_*$ if a [[full and faithful functor]] this means that

$$
  U_1 \simeq U_2
  \,.
$$

So $p^*$ is indeed injective on subobjects and so $p$ is a [[geometric surjection]].

This establishes the existence of a surjection/embedding factorization. Next we discss that this is unique.

So consider two factorizations

$$
  \array{
    && \mathcal{A}
    \\
    & {}^{\mathllap{p_1}}\nearrow &\Downarrow^\simeq& \searrow^{\mathrlap{i_1}}
    \\
    \mathcal{F}
    &&\stackrel{f}{\to}&&
    \mathcal{E}
    \\
    & {}_{\mathllap{p_2}}\searrow &\downarrow^{\simeq}& \nearrow_{\mathrlap{i_2}}
    \\
    && \mathcal{B}
  }
$$

into a geometric surjection followed by a geometric embedding.

We will now argue that $i_1$ factors -- essentially uniquely -- through $i_2$ in a way that makes

$$
  \array{
    && \mathcal{A}
    \\
    & {}^{\mathllap{p_1}}\nearrow && \searrow^{\mathrlap{i_1}}
    \\
    \mathcal{F}
    &&\downarrow^g&&
    \mathcal{E}
    \\
    & {}_{\mathllap{p_2}}\searrow && \nearrow_{\mathrlap{i_2}}
    \\
    && \mathcal{B}
  }
$$

commute up to natural isomorphisms. By the same argument for the upside-down situation we find an essentially unique middle vertical morphism $h : \mathcal{B} \to \mathcal{A}$ the other way round. Then essential uniqueness of these factorizations implies that $g \circ h \simeq Id$ and $h \circ g \simeq Id$. This means that the original two factorizations are equivalent.

To find $g$ and $h$, use again that every [[geometric embedding]] (by the discussion there) is, up to equivalence, an inclusion of $j$-sheaves for some $j$. Find such a $j$ the bottom morphism and then use again lemma \ref{ConditionForSheafFactorization} that $i_1$ factors through $i_2$ -- essentially uniquely -- precisely if $i_1^*$ sends [[dense monomorphism]]s to isomorphisms.

To see that it does, let $IU \to X$ be a dense mono and consider the naturality square

$$
  \array{
    p_2^* i_2^* U &\stackrel{\simeq}{\to}& p_1^* i_1^* U
    \\
    \downarrow && \downarrow
    \\
    p_2^* i_2^* X &\stackrel{\simeq}{\to}& p_1^* i_1^* X
  }
  \,.
$$

Since $i_2^*(U \to X)$ is an iso by definition, the left vertical morphism is, and thus so is the right vertical morphism. But since $p_1$ is a [[geometric surjection]] we have (by the discussion there) that $p_1^*$ is [[conservative functor|conservative]], and hence also $i_1^* U \to i_1^* X$ is an isomorphism.

Hence $i_1$ factors via some $g$ through $i_2$ and the proof is completed by the above argument.

=--

## Examples

* For $f : X \to Y$ a [[continuous function]] between [[topological space]]s and $X \to im(f) \to Y$ its ordinary [[image]] factorization through an [[embedding]], the corresponding composite of geometric morphisms of [[sheaf topos]]es

  $$
    Sh(X) \to Sh(im(f)) \to Sh(Y)
  $$

  is a geometric surjection/geometric embedding factorization.

* For $\mathcal{E}$ any topos, $f : X \to Y$ any morphism in $\mathcal{E}$, and $X \to im(f) \to Y$ its [[image]] factorization, the corresponding composite of [[base change geometric morphisms]]

  $$
    \mathcal{E}/X  \to \mathcal{E}/im(f) \to \mathcal{E}/Y
  $$

  is a geometric surjection/embedding factorization.

* For $f : C \to D$ any [[functor]] between [[categories]], write $C \to im(f) \to D$ for its [[essential image]] factorization. Then the induced composite geometric morphism of presheaf toposes

  $$
    [C^{op}, Set] \stackrel{}{\to} [im(f)^{op}, Set] \to
    [D^{op}, Set]
  $$

  is a geometric surjection/embedding factorization.

See ([MacLaneMoerdijk, p. 377](#MacLaneMoerdijk)).

## Properties

### As idempotent approximation

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ induces via the adjunction $f^\ast\vdash f_\ast$ a [[monad]] on $\mathcal{E}$. Due to a general result by S. Fakir this induces an associated [[idempotent monad]] on $\mathcal{E}$ and this idempotent approximation coincides with the monad induced by $i^\ast\vdash i_\ast$ given by the inclusion $i$ from the factorization $f=i\circ q$.

For references and further details on the idempotent approximation see at [[idempotent monad]].

### A logical description

Let $T$ be a [[geometric theory]] over a signature $\Sigma$ and $f:\mathcal{E}\to Set[T]$ a geometric morphism to its [[classifying topos]]. Then by the general properties of a classifying topos, $f$ corresponds to a certain $T$-model $M$ in $\mathcal{E}$.

Notice that every geometric morphism $f$ between [[Grothendieck toposes]] is of this form for some geometric theory $T$ and hence corresponds to some model $M$ ! This model permits to attach a geometric theory to $f$ as well:

The **theory of M** $Th(M)$ consists of all geometric sequents $\sigma$ over $\Sigma$ such that $M\models \sigma$.

Then the following holds ([Caramello 2009](#Caramello09), p.57):

+-- {: .num_prop}
###### Proposition

The topos occurring in the middle of the surjection-embedding factorization of $f$ is precisely the classifying topos for $Th(M)$: $\mathcal{E}\twoheadrightarrow Set[Th(M)]\hookrightarrow Set[T]$.

=--

## Related entries

* [[open subtopos]]

* [[(dense,closed)-factorization]]

## References

* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_ , Academic Press 1977 (Dover reprint 2014). (section 4.1, pp.103-107)

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002. (section A4.2, pp.172ff)

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (section VII.4)
 {#MacLaneMoerdijk}

* [[Olivia Caramello|O. Caramello]], _Lattices of theories_ , arXiv:0905.0299v1 (2009). ([pdf](http://arxiv.org/pdf/0905.0299v1)) {#Caramello09}

[[!redirects geometric surjection/embedding factorization]]
[[!redirects geometric surjection/inclusion factorization]]
[[!redirects geometric surjection/embedding factorization system]]
[[!redirects geometric surjection/inclusion factorization system]]
[[!redirects (geometric surjection, inclusion) factorization system]]


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#### Factorization systems
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#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[hyperconnected geometric morphism|Hyperconnected geometric morphisms]] are the left class of a 2-categorical [[orthogonal factorization system]] on the [[2-category]] [[Topos]] of [[toposes]]; the right class is the class of [[localic geometric morphisms]].

## References

* [[Peter Johnstone]], _Factorization theorems for geometric morphisms_ Cahiers, 22, no1 (1981) ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_1_3_0))
{#Johnstone}

[[!redirects (hyperconnected, localic) factorization system]]
[[!redirects (hyperconnected,localic) factorization system]]


(infinity,0)-category

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#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Following the terminology of *[[(n,r)-category|(n,r)-categories]]*, an __$(\infty,0)$-category__ is an [[∞-category]] in which every $j$-morphism (for $j \gt 0$) is an [[equivalence]].

So in an $(\infty,0)$-category _every_ morphism is an [[equivalence]]. Such [[∞-categories]] are usually called _[[∞-groupoid]]s_.

This is directly analogous to how a [[0-category]] is equivalent to a [[set]], a [[(1,0)-category]] is equivalent to a [[groupoid]], and so on.  (In general, an [[(n,0)-category]] is equivalent to an [[n-groupoid]].)

The term "$(\infty,0)$-category" is rarely used, but does for instance serve the purpose of amplifying the generalization from [[Kan complex]]es, which are one model for [[∞-groupoid]]s, to [[quasi-category|quasi-categories]], which are a model for [[(∞,1)-categories]].

## References

On [[model categories]] [[presentable (infinity,1)-category|presenting]] $(\infty,0)$-categories, namely models for [[infinity-groupoids|$\infty$-groupoids]] (such as the [[model structure on simplicial groupoids]]) akin to corresponding models for [[(infinity,1)-categories|$(\infty,1)$-categories]] (such as the [[model structure on simplicial categories]]):

* [[Julia E. Bergner]], *Adding inverses to diagrams II: Invertible homotopy theories are spaces*, Homology, Homotopy Appl. **10** 2 (2008) 175-193 [[doi:10.4310/HHA.2008.v10.n2.a9](https://dx.doi.org/10.4310/HHA.2008.v10.n2.a9), [doi:0710.2254](https://arxiv.org/abs/0710.2254), erratum:[doi:10.4310/HHA.2012.v14.n1.a15](https://dx.doi.org/10.4310/HHA.2012.v14.n1.a15)]

[[!redirects (infinity,0)-categories]]

[[!redirects (∞,0)-category]]
[[!redirects (∞,0)-categories]]


(infinity,1)-bimodule

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#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

## Definition

### $(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners

Write $BMod^\otimes$ for the [[(∞,1)-category of operators]] of the [[(∞,1)-operad]] [[operad for bimodules]]. Write

$$
  \iota_{\pm} \colon Assoc \to BMod
$$

for the two canonical inclusions of the [[associative operad]] (as discussed at _[operad for bimodules - relation to the associative operad](#RelationToTheAssociativeOperad)_).

+-- {: .num_defn #NotationForWeaklyBiEnrichedInfinityCategory}
###### Definition (Notation)

For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]], write

$$
  \mathcal{C}^\otimes_{\pm}
  \coloneqq
  \mathcal{C}^\otimes \underset{BMod^\otimes}{\times}^\pm Assoc^\otimes
$$

for the two [[fiber products]] of $p$ with the inclusions $\iota_\pm$. The canonical [[projection]] maps

$$
  \mathcal{C}^\otimes_{\pm}
  \to
  Assoc^\otimes
$$

exhibit these as two [[planar (∞,1)-operads]].

Finally write

$$
  \mathcal{C}
    \coloneqq
  \mathcal{C}^\otimes \underset{BMod^\otimes}{\times} \{\mathfrak{n}\}
$$

for the [[(∞,1)-category]] over the object labeled $\mathfrak{n}$.

=--

([Lurie, notation 4.3.1.11](#Lurie)).

+-- {: .num_remark }
###### Remark

This exhibits $\mathcal{C}$ as equipped with [[weak tensoring]] over $\mathcal{C}_-$ and reverse weak tensoring over $\mathcal{C}_+$.

=--

The most familiar special case of these definitions to keep in mind is the following.

+-- {: .num_remark #MonoidalCategoryAsBitensoredOverItself}
###### Remark

For $\mathcal{C}^\otimes \to Assoc^\otimes$ a [[coCartesian fibration of (∞,1)-operads]], hence exhibiting $\mathcal{C}^\otimes$ as a [[monoidal (∞,1)-category]], [[pullback]] along the canonical map $\phi \colon BMod^\otimes \to Assoc^\otimes$ gives a fibration

$$
  \phi^* \mathcal{C}^\otimes \to BMod^\otimes
$$

as in def. \ref{NotationForWeaklyBiEnrichedInfinityCategory} above. In the terminology there this exhibts $\mathcal{C}$ as weakly enriched (weakly [[tensoring|tensored]]) over itself from the left and from the right.

This is the special case for which bimodules are traditionally considered.

=--

([Lurie, example 4.3.1.15](#Lurie))

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

For $\mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (∞,1)-operads]] we say that the corresponding [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]]

$$
  BMod(\mathcal{C}) \coloneqq Alg_{/BMod}(\mathcal{C})
$$

is the **$(\infty,1)$-category of $(\infty,1)$-bimodules** in $\mathcal{C}$.

Composition with the two inclusions $\iota_{1,2}\colon Assoc BMod$ of the [[associative operad]] yields a [[fibration]] in the [[model structure for quasi-categories]] $BMod(\mathcal{C}) \to Alg(\mathcal{C}_-)\times Alg(\mathcal{C}_+)$. Then for $A_- \in Alg_{\mathcal{C}_-}$ and $A_+ \in Alg_{\mathcal{C}_+}$ two algebras the [[fiber product]]

$$
  {}_A BMod_{B}(\mathcal{C})
  \coloneqq
  \{A\}
   \underset{Alg(\mathcal{C}_-)}{\times}
  BMod(\mathcal{C})
   \underset{Alg(\mathcal{C}_-)}{\times}
  \{B\}
$$

we call the **$(\infty,1)$-category of $A$-$B$-bimodules**.

=--

([Lurie, def. 4.3.1.12](#Lurie))

+-- {: .num_example}
###### Example

For the special case of remark \ref{MonoidalCategoryAsBitensoredOverItself} where the bitensored structure on $\mathcal{C}$ is induced from a monoidal structure $\mathcal{C}^\otimes \to Asoc^\otimes$, we have by the [[universal property]] of the [[pullback]] that

$$
  BMod(\mathcal{C})
  \simeq
  {Alg_{BMod}}_{/Assoc}(\mathcal{C})
  \simeq
  \left\{
    \array{
       && \mathcal{C}
       \\
       &{}^{\mathllap{(A,B,N)}}\nearrow& \downarrow
       \\
       BMod^\otimes &\to& Assoc^\otimes
    }
  \right\}
$$

=--

+-- {: .num_remark}
###### Remark

Let $\mathcal{C}$ be a [[1-category]], for simplicity.
Then a [[morphism]]

$$
  (A_1,B_1,N_1) \to $(A_2,B_2,N_2)$
$$

in $BMod(\mathcal{C})$ is a pair $\phi_1 \colon A_1 \to A_1$, $\rho \colon B_1 \to B_2$ of algebra homomorphisms and a morphism $\kappa \colon N_1 \to N_2$ which is "linear in both $A$ and $B$" or "is an [[intertwiner]]" with respect to $\phi$ and $\rho$ in that for all $a \in A$, $b \in B$ and $n \in N$ we have

$$
  \kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n)
  \,.
$$

It is natural to depict this by the square diagram

$$
  \array{
    A_1 &\stackrel{N_1}{\to}& B_1
    \\
    {}^{\mathllap{\phi}}\downarrow
    &
    \Downarrow^{\kappa}
    &
    \downarrow^{\mathrlap{\rho}}
    \\
    A_2 &\underset{N_2}{\to}& B_2
  }
  \,.
$$

This notation is naturally suggestive of the existence of the further [[horizontal composition]] by [[tensor product of (∞,1)-modules]], which we come to [below](#TensorProductOfBimodules).

On the other hand, a morphism $N_1 \to N_2$ in ${}_A BMod(\mathcal{C})_B$ is given by the special case of the above for $\phi = id$ and $\rho = id$.

=--

### Tensor products of $(\infty,1)$-Bimodules
 {#TensorProductOfBimodules}

+-- {: .num_defn #NotationForTensS}
###### Definition (Notation)

Write $Tens^\otimes$ for the [[generalized (∞,1)-operad]]
discussed at _[[tensor product of ∞-modules]]_.

For $S \to \Delta^{op}$ an [[(∞,1)-functor]] (given as a map of simplicial sets from a [[quasi-category]] $S$ to the [[nerve]] of the [[simplex category]]), write

$$
  Tens^\otimes_{S}
   \coloneqq
  Tens^\otimes \underset{\Delta^{op}}{\times} S
$$

for the [[fiber product]] in [[sSet]].

Moreover, for $\mathcal{C}^\otimes \to Tens^\otimes_S$ a [[fibration]] in the [[model structure for quasi-categories]] which exhibits $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], write

$$
  Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)
$$

for the full [[sub-(∞,1)-category]] on those [[(∞,1)-functors]] which send inert morphisms to inert morphisms.

=--

([Lurie, notation 4.3.4.15](#Lurie))

### The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules

We discuss the generalization of the notion of bimodules to [[homotopy theory]], hence the generalization from [[category theory]] to [[(∞,1)-category theory]]. ([Lurie, section 4.3](#Lurie)).

Let $\mathcal{C}$ be [[monoidal (∞,1)-category]] such that

1. it admits [[geometric realization]] of [[simplicial objects in an (∞,1)-category]] (hence a [[left adjoint|left]] [[adjoint (∞,1)-functor]] ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a [[presentable (∞,1)-category]];

1. the [[tensor product]] $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.

Then there is an
[[(∞,2)-category]] $Mod(\mathcal{C})$ which given as an [[(∞,1)-category object]] internal to [[(∞,1)Cat]] has

* $(\infty,1)$-category of objects

  $$
    Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})
  $$

  the [[A-∞ algebras]] and [[∞-algebra]] [[homomorphisms]] in $\mathcal{C}$;

* $(\infty,1)$-category of morphisms

  $$
    Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})
  $$

  the $\infty$-bimodules and bimodule homomorphisms ([[intertwiners]]) in $\mathcal{C}$

This is ([Lurie, def. 4.3.6.10, remark 4.3.6.11](#Lurie)).

Morover, the [[horizontal composition]] of bimodules in this [[(∞,2)-category]] is indeed the relative
[[tensor product of ∞-modules]]

$$
  \circ_{A,B,C}
  =
  (-) \otimes_B (-)
  \;\colon\;
  {}_A Mod_{B} \times {}_{B}Mod_C \to {}_A Mod_C
  \,.
$$

This is ([Lurie, lemma 4.3.6.9 (3)](#Lurie)).

Here are some steps in the construction:

The **idea** of the following constructions is this:
we start with a [[generalized (∞,1)-operad]]
$Tens^\otimes \to FinSet_* \times \Delta^{op}$ which is such that
the [[(∞,1)-algebras over an (∞,1)-operad]] over its fiber over $[k] \in \Delta^{op}$ is equivalently the collection of $(k+1)$-tuples of
[[A-∞ algebras]] in $\mathcal{C}$ together with a string of $k$ $\infty$-bimodules between them. Then we turn that into a [[simplicial object in an (∞,1)-category|simplicial object]] in [[(∞,1)Cat]]

$$
  Mod(\mathcal{C}) \in ((\infty,1)Cat)^{\Delta^{op}}
  \,.
$$

This turns out to be an [[internal (∞,1)-category]] object in [[(∞,1)Cat]], hence an [[(∞,2)-category]] whose object of objects is the category $Alg(\mathcal{C})$ of [[A-∞ algebras]] and [[homomorphisms]] in $\mathcal{C}$ between them, and whose object of morphisms is the category $BMod(\mathcal{C})$ of $\infty$-bimodules and [[intertwiners]].

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

Define $Mod(\mathcal{C}) \to \Delta^{op}$ as the map of [[simplicial sets]] with the [[universal property]] that for every other map of simplicial set $K \to \Delta^{op}$ there is a canonical bijection

$$
  Hom_{sSet/\Delta^{op}}(K, Mod(\mathcal{C}))
  \simeq
  Alg_{Tens_K / Assoc}( \mathcal{C} )
  \,,
$$

where

* on the left we have the hom-simplicial set in the [[slice category]]

* on the right we have the [[(∞,1)-category]] of [[(∞,1)-algebras over an (∞,1)-operad]] given by lifts $\mathcal{A}$ in

  $$
    \array{
       && \mathcal{C}^\otimes
       \\
       &{}^{\mathcal{A}}\nearrow& \downarrow
       \\
      Tens_K &\to& Assoc
    }
    \,.
  $$

=--

This is ([Lurie, cor. 4.3.6.2](#Lurie)) specified to the case of ([Lurie, lemma 4.3.6.9](#Lurie)). Also ([Lurie, def. 4.3.4.19](#Lurie))

## References

The general theory in terms of [[higher algebra]] of [[(∞,1)-operads]] is discussed in section 4.3 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Specifically the homotopy theory of [[A-infinity bimodules]] is discussed in

* Volodymyr Lyubashenko, Oleksandr Manzyuk, _A-infinity-bimodules and Serre A-infinity-functors_ ([arXiv:math/0701165](http://arxiv.org/abs/math/0701165))

and section 5.4.1 of

* [[Boris Tsygan]], _Noncommutative calculus and operads_ in
Guillermo Cortinas (ed.) _Topics in Noncommutative geometry_, Clay Mathematics Proceedings volume 16

The generalization to [[(infinity,n)-modules]] is in

* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))

[[!redirects (infinity,1)-bimodules]]

[[!redirects ∞-bimodule]]
[[!redirects ∞-bimodules]]
[[!redirects (∞,1)-bimodule]]
[[!redirects (∞,1)-bimodules]]

[[!redirects (∞,1)-category of (∞,1)-bimodules]]
[[!redirects (infinity,1)-category of (∞,1)-bimodules]]

[[!redirects (∞,1)-categories of (∞,1)-bimodules]]
[[!redirects (infinity,1)-categories of (∞,1)-bimodules]]

[[!redirects infinity-bimodule]]
[[!redirects infinity-bimodules]]


(infinity,1)-categorical hom-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Where an ordinary [[category]] has a [[hom-set]], an [[(∞,1)-category]] has an [[∞-groupoid]] of morphisms between any two objects, a _hom-space_.

There are several ways to _present_ an [[(∞,1)-category]] $\mathbf{C}$ by an ordinary [[category]] $C$ equipped with some extra structure: for instance $C$ may be a [[category with weak equivalences]] or a [[model category]] or even a [[simplicial model category]]. In all of these presentations, given two objects $X, Y \in C$, there is a way to construct a [[simplicial set]] $\mathbb{R}\mathbf{C}(X,Y)$ that presents the hom-[[∞-groupoid]] $\mathbf{C}(X,Y)$. This simplicial set -- or rather its [[homotopy type]] -- is called the
_derived hom space_ or  _homotopy function complex_ and denoted $\mathbf{R}Hom(X,Y)$ or similarly.

## Presentations

There are many ways to present an [[(∞,1)-category]] by [[category theory|category theoretic data]], and for each of these there are corresponding tools for explicitly computing the derived hom spaces.

The most basic data is that of a [[category with weak equivalences]]. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called _[[simplicial localization]]_. This we discuss below in _[For a category with weak equivalences](#ForACategoryWithWeakEquivalences)_.

Particularly useful extra structure on a [[category with weak equivalences]] that helps with computing the derived hom spaces is the structure of a _[[model category]]_. Using this one can construct simplicial resolutions of objects -- called _framings_ -- that generalize [[cylinder objects]] and [[path objects]], and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in _[For a model category](#Framings)_.

Still a bit more helpful structure on top of a bare model category is that of a [[simplicial model category]]. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the [[sSet]]-[[hom objects]]. This we discuss below in _[For a simplicial model category](#EnrichedHomsCofToFib)_.

### For a category with weak equivalences
 {#ForACategoryWithWeakEquivalences}

Let $(C,W \subset Mor(C))$ be a [[category with weak equivalences]].

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

Fix $n \in \mathbb{N}$. For $X,Y \in Obj(C)$, define a category $wMor_C^n(X,Y)$

* whose objects are [[zig-zag]]s of morphisms in $C$ of length $n$

  $$
    X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y
  $$

  such that each morphism going to the left, $X_{2k}\leftarrow X_{2k +1}$, is a [[weak equivalence]], an element in $W$;

* morphisms between such objects $(X,X_i,Y) \to (X',X'_i,Y')$ are collections of weak equivalences $(X_i \to X'_i)$ for all $0 \lt i \lt n $ such that all triangles and squares commute.

=--

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

Write $N(wMor_C^n(X,Y))$ for the [[nerve]] of this category, a [[simplicial set]].

The _[[hammock localization]]_ $L_W^H C$ of $C$ is the [[simplicially enriched category]] with objects those of $C$ and [[hom-objects]] given by the [[colimit]] over the length of these hammock hom-categories

$$
  L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y))
  \,.
$$

The [[Kan fibrant replacement]] of this simplicial set is the derived hom-space between $X$ and $Y$ of the $(\infty,1)$-category modeled by $(C,W)$.

=--

### For a model category
 {#Framings}

The derived hom spaces of a model category $C$ may always be computed
in terms of simplicial resolutions with respect to the [[Reedy model structure]]
$[\Delta^{op}, C]_{Reedy}$. These resolutions are often called
_framings_ ([Hovey](#Hovey)). These constructions are originally due to
([Dwyer-Hirschhorn-Kan](#DHK)).

Let $C$ be any [[model category]].

+-- {: .num_prop }
###### Observation

There is an [[adjoint triple]]

$$
  (const \dashv ev_0 \dashv (-)^{\times^\bullet})
  :
  C
  \stackrel{\overset{const}{\longrightarrow}}{\stackrel{\overset{ev_0}{\longleftarrow}}{\underset{(-)^{\times^\bullet}}{\longrightarrow}}}
  \,,
  [\Delta^{op}, C]
  \,,
$$

where

1. $const X : [n] \mapsto X$;

1. $ev_0 X_\bullet = X_0$;

1. $X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}$.

=--

+-- {: .num_remark #CoDiscreteIsReedyFibrant}
###### Remark

For $X \in C$ fibrant, $X^{\times^\bullet}$ is fibrant in
the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.

=--

+-- {: .proof}
###### Proof

The matching morphisms are in fact [[isomorphisms]].

=--

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

Let $C$ be a model category.

1. For $X \in C$ any object, a _simplicial frame_ on $X$ is a factorization of $const X \to X^{\times^\bullet}$ into a weak equivalence followed by a fibration in the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.

1. A _right framing_ in $C$ is a functor $(-)_\bullet :  C \to [\Delta^{op}, C]$
   with a [[natural isomorphism]] $(X)_0 \simeq X$ such that $X_\bullet$ is a simplicial
   frame on $X$.

Dually for _cosimplicial frames_.

=--

This appears as ([Hovey, def. 5.2.7](#Hovey)).

+-- {: .num_remark}
###### Remark

By remark \ref{CoDiscreteIsReedyFibrant} a simplicial frame $X_\bullet$ in the above is in particular fibrant in $[\Delta^{op}, C]_{Reedy}$.

=--

+-- {: .num_prop #SimplicialFunctionComplexes}
###### Proposition

For $X \in C$ cofibrant and $A \in C$ fibrant, there
are weak equivalences in $sSet_{Quillen}$

$$
  Hom_C(X^\bullet, A)
  \stackrel{\simeq}{\to}
  diag Hom_C(X^\bullet, A_\bullet)
  \stackrel{\simeq}{\leftarrow}
  Hom_C(X, A_\bullet)
  \,,
$$

(where in the middle we have the diagonal of the [[bisimplicial set]] $Hom(X^\bullet, A_\bullet)$).

=--

This appears as ([Hovey, prop. 5.4.7](#Hovey)).

Either of these simplicial sets is a model for the derived hom-space $\mathbb{R}Hom(X,A)$.

+-- {: .num_remark }
###### Remark

By developing these constructions further, one obtains
a canonical  [[simplicial model category]]-resolution of
(left proper and combinatorial) model categories $C$, such that
the simplicial resolutions given by framings are just the
cofibrant$\to$fibrant $sSet$-hom objects as discussed
[below](#EnrichedHomsCofToFib).

This is discussed at _[Simplicial Quillen equivalent models](http://ncatlab.org/nlab/show/simplicial+model+category#SimpEquivMods)_.

=--

+-- {: .num_prop }
###### Proposition

Let $C$ be a model category, let $\mathrm{c}_\mathrm{w} C$ be the full subcategory of $[\Delta, C]$ spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let $\mathrm{s}_\mathrm{w} C$ be the full subcategory of $[\Delta^{op}, C]$ spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.

1. $const : C \to \mathrm{c}_\mathrm{w} C$ is the right half of an adjoint homotopical equivalence of [[homotopical category|homotopical categories]], and $const : C \to \mathrm{s}_\mathrm{w} C$ is the left half of an adjoint homotopical equivalence of homotopical categories.
2. The functor $\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet$ admits a right [[derived functor]].
3. The induced functor $(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet$ is the derived hom-space functor.

=--

### For a simplicial model category
 {#EnrichedHomsCofToFib}

We describe here in more detail properties of [[derived hom-functors]] (see there for more) in a [[simplicial model category]].

The crucial axiom used for this is the axiom of an [[enriched model category]] $C$ which says that

* the [[copower|tensor operation]]

  $$
    \cdot : C \times SSet \to C
  $$

  is a [[Quillen bifunctor]];

* or equivalently that for $X \to Y$ a cofibration
  and $A \to B$ a fibration the induced morphism

  $$
    C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)
  $$

  is a fibration, which is acyclic if either
  $X \to Y$ or $A \to B$ is.

First of all the first statement directly implies that for $\emptyset \in C$ the [[initial object]] and $A \in C$ any object, the simplicial set $C(\emptyset,A) = {*}$ is the terminal simplicial set (see also [this Prop.](powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial)): because for any simplicial set $S$

$$
  \begin{aligned}
    SSet(S,C(\emptyset, A))
    & =  Hom_C(\emptyset \cdot S, A)
    \\
    & = Hom_C(colim_{\emptyset} \cdot S, A)
    \\
    & = Hom_C(\emptyset, A)
    \\
    &= {*}
  \end{aligned}
  \,,
$$

where we use that the [[copower|tensor]] [[Quillen bifunctor]] is required to respect [[colimit]]s and that the empty colimit is the [[initial object]]. (All equality signs here denote [[isomorphisms]], to distinguish them from weak equivalences.)

Similarly one has for all $X$ that $C(X,{*}) = {*}$.

Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.

+-- {: .num_lemma }
###### Lemma

In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $A \in C$ fibrant, the [[simplicial set]] $C(X,A)$ is a [[Kan complex]].

=--

+-- {: .proof}
###### Proof

We apply the [[enriched model category]] axiom to the cofibration $\emptyset \to X$ and the fibration $A \to {*}$ to obtain a fibration

$$
  C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*})
  \,.
$$

The right hand is the [[pullback]] of the terminal simplicial set ${*} = \Delta^0$ to itself, hence is itself the point. So we have a fibration $ C(X,A) \to {*}$
and $C(X,A)$ is a fibrant object in the standard [[model structure on simplicial sets]], hence a [[Kan complex]].
.
=--

+-- {: .num_lemma }
###### Lemma

In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $f : A \to B$ a fibration, the morphism of [[simplicial set]]s $C(X,f) : C(X,A) \to C(X,B)$ is a [[Kan fibration]] that is a [[weak homotopy equivalence]] if $f$ is acyclic.

Dually, for $i : X \to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A) : C(X,A) \to C(Y,A)$ is a cofibration of simplicial sets.

=--

+-- {: .proof}
###### Proof

This is as before. Explicity, consider the first case, the second one is the formal dual of that:

We enter the enrichment axiom with the morphisms $\emptyset \to X$ and $A \to B$ and find for the required [[pullback]] that

$$
  C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B)
  =
  {*} \times_{*} C(X,B)
  =
  C(X,B)
$$

and hence that $C(X,A) \to C(X,B)$ is, indeed, a fibration, which is acyclic if $A \to B$ is.

=--

+-- {: .num_proposition }
###### Proposition

Let $C$ be a [[simplicial model category]].

Then for $X$ a cofibant object and

$$
   f : A \stackrel{\simeq}{\to}  B
$$

a weak equivalence between fibrant objects, the [[enriched functor|enriched]] [[hom-object|hom-functor]]

$$
  C(X,f) : C(X,A) \to C(X,B)
$$

is a [[weak homotopy equivalence]] of [[Kan complex]]es.

Similarly, for $A$ a fibrant object and $j : X \stackrel{\simeq}{\to} Y$ a weak equivalence between cofibrant objects, the morphism

$$
  C(j,A) : C(X,A) \to C(Y,A)
$$

is a [[weak homotopy equivalence]] of [[Kan complex]]es.

=--

+-- {: .proof}
###### Proof

The second case is formally dual to the first, so we restrict attention to the first one.

By the above, the axioms of an [[enriched model category]] ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.

This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the _factorization lemma_ that is described in great detail at [[category of fibrant objects]]. By this lemma, for every morphism $f : A \to B$ between fibrant objects there is a commutative diagram

$$
  \array{
    && \mathbf{E}_f B
    \\
    & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}}
    \\
    A &&\stackrel{\simeq}{\to}&& B
  }
$$

Since $f$ is assumed a weak equivalence it follows by [[category with weak equivalences|2-out-of-3]] that $\mathbf{E}_f B$ is also a weak equivalence.

Therefore by the above properties of simpliciall enriched categories we obtain a [[span]] of acyclic fibrations of [[Kan complex]]es

$$
  C(X,A) \stackrel{\simeq}{\leftarrow}
  C(X, \mathbf{E}_f B)
  \stackrel{\simeq}{\to}
  C(X,B)
  \,.
$$

By the [[Whitehead theorem]] every weak equivalence of Kan complexes is a [[homotopy equivalence]], hence there is a weak equivalence

$$
  C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)
$$

that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.

=--

### Comparison

Let $C$ be a [[model category]]. We discuss how its
simplicial function complexes from prop. \ref{SimplicialFunctionComplexes}
are related to the simplicial localization from def. \ref{ZigZagCategories} and def. \ref{HammockLocalization}.

Suppose now that $Q : C \to C$ is a [[cofibrant replacement functor]] and $R : C \to C$ a [[fibrant replacement functor]], $\Gamma^\bullet : C \to (cC)_c$ a [[cosimplicial resolution functor]] and $\Lambda_\bullet : C \to (sC)_f$ a [[simplicial resolution functor]] in the [[model category]] $C$.

+-- {: .num_theorem #DKTheorem}
###### Theorem
**(Dwyer--Kan)**

There are natural weak equivalences between the following equivalent realizations of this [[SSet]] [[hom-object]]:

$$
  \array{
    Mor_C(\Gamma^\bullet X, R Y)
    &\stackrel{\simeq}{\to}&
    diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y)
    &\stackrel{\simeq}{\leftarrow}&
    Mor_C(Q X, \Lambda_\bullet Y)
    \\
    &&
    \uparrow^\simeq
    \\
    &&
    hocolim_{p,q \in \Delta^{op} \times \Delta^{op}}
    Mor_C(\Gamma^p X, \Lambda_q Y)
    \\
    &&\downarrow^\simeq
    \\
    &&N wMor_C^3(X,Y)
    \\
    &&\downarrow^\simeq
    \\
    &&Mor_{L^H C}(X,Y)
  }
  \,.
$$

=--

The top row weak equivalences are those of prop. \ref{SimplicialFunctionComplexes}

### In a category of fibrant objects
 {#InACategoryOfFibrantObjects}

There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a [[category of fibrant objects]].
This is described in ([Cisinksi 10](#Cisinski10)) and ([Nikolaus-Schreiber-Stevenson 12, section 3.6.2](#NSS12)).

## Properties

### Hom-spaces of equivalences
 {#SpacesOfEquivalences}

+-- {: .num_theorem #DKTheorem}
###### Theorem

For $C$ a [[simplicial model category]] and $X$ an object, the [[delooping]] of the [[automorphism ∞-group]]

$$
  Aut_W(X) \subset \mathbb{R}Hom(X,X)
$$

has the [[homotopy type]] of the component on $X$ of the [[nerve]] $N(C_W)$ of the [[subcategory]] of weak equivalences:

$$
  \mathbf{B} Aut_W(X) \simeq N(C_W)_X
  \,.
$$

The equivalence is given by a finite sequence of [[zig-zags]] and is natural with respect to [[sSet]]-[[enriched functors]] of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.

=--

This is [Dwyer-Kan 84, 2.3, 2.4](#DK84).

+-- {: .num_cor }
###### Corollary

For $C$ a model category, the simplicial set $N(C_W)$ is a model for the
[[core]] of the [[(∞,1)-category]] determined by $C$.

=--

+-- {: .proof}
###### Proof

That core, like every [[∞-groupoid]] is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.

=--

## Related concepts

* [[hom-object]]

* [[hom-set]], [[hom-functor]]

* [[hom-category]]

* [[hom-space]], [[cocycle space]]

* [[simplicial mapping complex]]

[[!include homotopy-homology-cohomology]]

## References

For some original references by [[William Dwyer]] and [[Dan Kan]] see  [[simplicial localization]]. For instance

* {#DK84} [[William Dwyer]], [[Dan Kan]], _A classification theorem for diagrams of simplicial sets_, Topology 23 (1984), 139-155.

On the derived [[function complexes]] in a [[projective model structure on simplicial presheaves]]:

* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [doi:10.1016/1385-7258(83)90051-3, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]

Discussion in terms of [[quasi-categories]]:

* [[Jacob Lurie]], Section 1.2.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

* [[Dan Dugger]], [[David Spivak]], *Mapping spaces in quasi-categories*, Algebraic & Geometric Topology **11** (2011) 263–325 [[arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263)]

The theory of _framings_ is due to

* {#DHK} [[William Dwyer]], [[Philip Hirschhorn]], [[Dan Kan]], _Model categories and general abstract homotopy theory_, (1997) ([pdf](http://www.mimuw.edu.pl/~jacho/literatura/ModelCategory/DHK_ModelCateogories1.pdf))

and in parallel section 5 of

* {#Hovey} [[Mark Hovey]], _Model categories_ ([ps](http://math.unice.fr/~brunov/SecretPassage/Hovey-Model%20Categories.ps))

and in sections 16, 17 of

* [[Philip Hirschhorn]], _Model categories and their localization_ .

A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of

* [[Clark Barwick]], _On (enriched) left Bousfield localization of model categories_ ([arXiv](http://arxiv.org/abs/0708.2067))

Discussion of derived hom spaces for [[categories of fibrant objects]] is in

* {#Cisinski10} [[Denis-Charles Cisinski]], _Invariance de la K-théorie par equivalences dérivées_, J. K-theory, 6 (2010), 505–546.

and section 3.6.2 of

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_ ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

[[!redirects (infinity,1)-categorical hom-spaces]]

[[!redirects (∞,1)-categorical hom-space]]
[[!redirects (∞,1)-categorical hom-spaces]]

[[!redirects (infinity,1)-categorical hom space]]
[[!redirects (∞,1)-categorical hom space]]
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[[!redirects (∞,1)-categorial hom space]]
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[[!redirects quasi-categorical hom-spaces]]

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[[!redirects hom-spaces]]

[[!redirects hom-∞-groupoid]]
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[[!redirects hom ∞-groupoids]]
[[!redirects hom infinity-groupoids]]
[[!redirects hom (∞,0)-categories]]
[[!redirects hom (infinity,0)-categories]]

[[!redirects homotopy function complex]]
[[!redirects homotopy function complexes]]

[[!redirects (∞,1)-hom (∞,1)-functor)]]

[[!redirects (∞,1)-categorical hom]]

[[!redirects (∞,1)-categorical hom-spaces]]

[[!redirects derived mapping space]]
[[!redirects derived mapping spaces]]


(infinity,1)-categorification

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[categorification|Categorification]] is a creative nonmechanical process in which categorical structures are promoted to $n$-categorical structures, for some $n\ge 2$.

(∞,1)-categorification is a special instance of this idea, in which the newly created $n$-morphisms are invertible for all $n\ge 2$.

The phrase [[higher structures]] also refers primarily to (∞,1)-categorification.

Sometimes more than one (∞,1)-categorification is possible,
as is the case for [[abelian groups]],
which can be categorified to [[module spectra|$\mathrm{H}\mathbb{Z}$-module spectra]] (as represented by [[simplicial abelian groups]]) or [[connective spectra]].

In addition to this creative choice of a [[Platonic form]] categorifying a given structure, another creative aspect is a choice of a specifc [[homotopy theory FAQ|model]] for the resulting object,
e.g., [[(∞,1)-categories]] can be modeled by [[relative categories]], [[simplicial categories]], [[quasicategories]], etc.

## Simplicial objects

For [[algebras over an algebraic theory]] $T$,
one can construct an (∞,1)-categorification
by passing to [[simplicial objects]] valued in algebras over $T$,
and equipping them with weak equivalences
induced by the forgetful functor to simplicial sets.

In some cases, the result can be different from the result of the animation
procedure described below, e.g., for the algebraic theory that
defines commutative monoids
we get commutative simplicial monoids, equivalently,
[[E-infinity algebras]] over the [[Eilenberg-MacLane spectrum]]
of the [[integers]],
whereas animation produces connective [[E-infinity ring spectra]].

## Animation

In some cases, there is an automatic (∞,1)-categorification. For example, the [[animation]] $\mathrm{Ani}(C)$ of a cocomplete category $C$ that is generated under colimits by its subcategory $C^{\mathrm{sfp}}$ of compact projective objects, is the (∞,1)-category freely generated by $C^{sfp}$ under sifted colimits. (See Kęstutis Česnavičius and Peter Scholze, Sec. 5.1.4.)

For example, the animation of the 1-category of modules over an ordinary [[ring]] $R$ is the (∞,1)-category of connective [[module spectra]] over the [[Eilenberg-MacLane spectrum|Eilenberg-MacLane ring spectrum]] $\mathrm{H}R$.

## Further Examples

In the table below, structures on the left are always understood up to an isomorphism, whereas on the right we explicitly indicate the notion of a weak equivalence used (except for [[Platonic forms]] such as [[(∞,1)-categories]]).

|categorical structure|(∞,1)-categorical structure|
|---|---|
|[[set]]|[[∞-groupoid]]|
|[[set]]|[[simplicial set]] up to a [[simplicial weak equivalence]]|
|[[groups]]|[[simplicial groups]] up to [[simplicial weak equivalences]]|
|[[algebras]] over [[algebraic theory]] $T$|simplicial algebras over $T$ up to [[simplicial weak equivalences]]|
|[[algebras]] over [[algebraic theory]] $T$|weak simplicial algebras over $T$ up to [[simplicial weak equivalences]]|
|[[abelian group]]|nonnegatively graded [[chain complex]] up to a [[quasi-isomorphism]]|
|[[abelian group]]|[[connective spectrum]] up to a [[weak equivalence]]|
|[[category]]|[[(∞,1)-category]]|
|[[category]]|[[relative category]] up to a [[Barwick-Kan equivalence]]|
|[[category]]|[[simplicial category]] up to a [[Dwyer-Kan equivalence]]|
|[[operad]]|[[(∞,1)-operad]]|
|[[operad]]|[[simplicial operad]] up to a [[weak equivalence]]|
|[[algebras over an operad]] $O$|simplicial algebras over a [[cofibrant resolution]] of $O$ up to a [[weak equivalence]]|
|[[Lie groupoid]]|[[Kan simplicial manifold]]|
|[[presheaf]]|[[simplicial presheaf]]|
|[[sheaf]]|[[(∞,1)-sheaf]]|
|[[sheaf]]|[[simplicial presheaf]] that satisfies the [[homotopy descent]] property|
|[[topos]]|[[(∞,1)-topos]]|
|[[elementary topos]]|[[elementary (∞,1)-topos]]|
|[[2-category]]|[[(∞,2)-category]]|
|[[n-category]]|[[(∞,n)-category]]|

[[!redirects homotopy coherent analog]]
[[!redirects homotopy coherent analogue]]
[[!redirects homotopy coherent analogs]]
[[!redirects homotopy coherent analogues]]
[[!redirects homotopy coherent]]
[[!redirects (∞,1)-categorification]]
[[!redirects (∞,1)-categorifications]]
[[!redirects (infinity,1)-categorifications]]