reciprocal algebra

[[!redirects reciprocal algebras]]

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## Idea

An algebra with a reciprocal that behaves like $\frac{1}{x}$ in the rational and real numbers.

## Definition

A **reciprocal algebra** is a possibly [[non-associative algebra]] $A$, typically over a [[field]] $k$, which has the property that for non-zero element $b$, there exists an element $c$, called the __[[reciprocal]]__ of $b$, such that for every element $a$

* $a (c b) = a$
* $(c b) a = a$
* $a (b c) = a$
* $(b c) a = a$

## Examples

* Every [[Cayley-Dickson algebra]] is a reciprocal algebra. Examples include the [[real numbers]] $\mathbb{R}$, the [[complex numbers]] $\mathbb{C}$, the [[quaternions]] $\mathbb{H}$, and the [[octonions]] $\mathbb{O}$.

* The ring of rational [[polynomials]] $\mathbb{R}(X)$ is a real reciprocal algebra.

* There exists a reciprocal algebra with nonzero zero divisors.

### Counter-examples

* There exists a [[division algebra]] with identity that does not have two-sided inverses for every nonzero element.

## Related concepts

* [[division algebra]]
* [[reciprocal ring]]


reciprocal function

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## Definition ##

### In Heyting fields

Given a [[Heyting field]] $F$, let us define the [[type]] of all [[terms]] in $F$ [[apartness|apart]] from 0:

$$F_{#0} \coloneqq \{a \in F \vert a # 0\}$$

The **reciprocal** or **reciprocal function** is a [[partial function]] $f:F_{#0} \to F$ such that for all $a \in F_{#0}$ we have $a \cdot f(a) = 1$ and $f(a) \cdot a = 1$

### In dense sequentially Cauchy complete ordered integral domains

Let $R$ be a [[ordered integral domain]], and for all elements $a \in R$ and $b \in R$ let $(a, b)$ be the [[open interval|open subinterval]] containing all elements greater than $a$ and less than $b$. Then the sequences

$$f(p)(x) \coloneqq \sum_{n=0}^{p} x^n$$
$$g(p)(x) \coloneqq \sum_{n=0}^{p} (-1)^n x^n$$

indexed by natural number $p \in \mathbb{N}$ are [[Cauchy sequences]] for all elements $x \in (-1, 1)$, and if $R$ is [[sequentially Cauchy complete]], it has a limit for elements $x \in (-1, 1)$ as

$$f_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} x^n$$
$$g_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} (-1)^n x^n$$

If $R$ is also [[dense]] with given element $a \in (0, 1)$, then there are sequences of sequences

$$f'(i)(p)(x) \coloneqq a^i \sum_{n=0}^{p} (-a^i)^n (x-f_\infty(-a^i+1))^n$$
$$g'(i)(p)(x) \coloneqq a^i \sum_{n=0}^{p} (-a^i)^n (x+g_\infty(a^i-1))^n$$

indexed by natural numbers $i \in \mathbb{N}$ and $p \in \mathbb{N}$, the first which is Cauchy for elements $x \in (-2 f_\infty(-a^i+1), 0)$ and the second which is Cauchy for $x \in (0, 2 g_\infty(a^i-1))$. Since $R$ is sequentially Cauchy complete, both have limits as

$$f_\infty'(i)(x) \coloneqq \lim_{p \to \infty} a^i \sum_{n=0}^{p} (-a^i)^n (x-f_\infty(-a^i+1))^n$$
$$g_\infty'(i)(x) \coloneqq \lim_{p \to \infty} a^i \sum_{n=0}^{p} (-a^i)^n (x+g_\infty(a^i-1))^n$$

which themselves are Cauchy, and thus have limits
$$f' '(x) \coloneqq \lim_{i \to \infty} f_\infty'(i)(x)$$
$$g' '(x) \coloneqq \lim_{i \to \infty} g_\infty'(i)(x)$$

Since both $f_\infty(-a^i+1)$ and $g_\infty(a^i-1)$ go to infinity as $i$ goes to infinity, the domain of $f' '$ is $(-\infty, 0)$ and the domain of $g' '$ is $(0, \infty)$.

The __reciprocal__ $\frac{1}{(-)}:(-\infty, 0)\union (0, \infty) \to R$ is a piecewise defined [[partial function]] defined as

$$
\frac{1}{x} \coloneqq
\begin{cases}
f' '(x) & x \in (-\infty, 0) \\
g' '(x) & x \in (0, \infty)
\end{cases}
$$

Thus, every dense sequentially Cauchy complete ordered integral domain is an [[ordered field]].

### In sequentially Cauchy complete ordered integral rational algebras

Let $R$ be a [[ordered integral domain]] which is a [[rational algebra|$\mathbb{Q}$-algebra]], and for all elements $a \in R$ and $b \in R$ let $(a, b)$ be the [[open interval|open subinterval]] containing all elements greater than $a$ and less than $b$. Then the sequence

$$g(p)(x) \coloneqq \sum_{n=0}^{p} \frac{(-1)^n x^n}{n + 1}$$

indexed by natural number $p \in \mathbb{N}$ is a [[Cauchy sequence]] for all elements $x \in (-1, 1)$, and if $R$ is [[sequentially Cauchy complete]], it has a limit for elements $x \in (-1, 1)$ as

$$g_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} \frac{(-1)^n x^n}{n + 1}$$

There is a sequence of sequences

$$g'(i)(p)(x) \coloneqq i \sum_{n=0}^{p} \frac{(-i)^n (x+g_\infty(i-1))^n}{n + 1}$$

indexed by natural numbers $i \in \mathbb{N}$ and $p \in \mathbb{N}$, which is Cauchy for $x \in (0, 2 g_\infty(i-1))$.

Since $R$ is sequentially Cauchy complete, the function has a limit as

$$g_\infty'(i)(x) \coloneqq \lim_{p \to \infty} i \sum_{n=0}^{p} \frac{(-i)^n (x+g_\infty(i-1))^n}{n + 1}$$

which itself is Cauchy, and thus has a limit
$$\ln(x) \coloneqq \lim_{i \to \infty} g_\infty'(i)(x)$$
called the [[natural logarithm]]. Since $g_\infty(a^i-1)$ goes to infinity as $i$ goes to infinity, the domain of $\ln(x)$ is $(0, \infty)$.

The [[exponential function]] is defined as as

$$\exp(x) \coloneqq \lim_{n \to \infty} \sum_{i = 0}^{n} \frac{x^i}{i!}$$

and the reciprocal function is defined as

$$
\frac{1}{x} \coloneqq
\begin{cases}
-\exp(- \ln(-x)) & x \in (-\infty, 0) \\
\exp(- \ln(x)) & x \in (0, \infty)
\end{cases}
$$

## See also ##

* [[Heyting field]]
* [[rational function]]
* [[reciprocal algebra]]
* [[real reciprocal function]]
* [[division]]

[[!redirects reciprocal functions]]

[[!redirects reciprocal]]
[[!redirects reciprocals]]


reciprocity law

#Contents#
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## Examples

* [[Hilbert reciprocity law]], [[Artin reciprocity law]], [[Weil reciprocity law]]

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## References

* Wikipedia, _[Reciprocity law](http://en.wikipedia.org/wiki/Reciprocity_law)_

[[!redirects reciprocity laws]]


recognition of relative loop spaces

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#Contents#
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## Idea

The recognition principle for finite [[relative loop spaces]] states that, for $1 \leq N \lt \infty$, a pair of [[pointed topological space]]s is of the [[homotopy type]] of a relative $N$-loop space pair of some $(N-2,N-1)$-connected inclusion map if and only if it is a grouplike algebra over the [[Swiss cheese operad]] $\mathcal{SC}_N$, in the sense of an [[algebra over an operad]].

The recognition principle for infinite [[relative loop spaces]] states that a pair of pointed topological spaces is of the homotopy type of a relative $\infty$-loop space pair of some 1-shifted morphism of connective spectra if and only if it is a grouplike algebra over $\mathcal{SC}_\infty\coloneqq \colim_{N\to\infty}\mathcal{SC}_N$.

These are relative versions of [[May recognition theorem]] ([May 72](#May72), [May 74](#May74)) of loop spaces.

The recognition principle for relative $N$-loop spaces was proved in ([Vieira 2020](#VVieira2020)) for $3\leq N\leq \infty$. The proof there works in the cases $N=1,2$ if we assume the spaces connected. The general case for $N=1$ was proved in ([Hoefel, Livernet, Stasheff 2016](#HLS16)). The unconnected case for $N=2$ remains open.

Notoriously the loop space functors that appear in these recognition theorems are not right adjoint, thus they cannot be proved as an equivalence of homotopy categories induced by a [[Quillen equivalence]]. May's original strategy is to define what essentially constitutes adjunction units up to resolutions induced by the [[two-sided bar construction]]. The strategy of ([Vieira 2020](#VVieira2020)) is to define a generalization of [[Quillen adjunction]] that allow for the units and counits to exist only up to functorially determined resolutions, providing a natural model theoretical axiomatization of the essential elements of May’s original proof. For the full proof, homotopical versions of the concepts of [[idempotent monad]] and [[idempotent adjunction]] require similar generalizations. This is because we are interested in the homotopy subcategories of grouplike algebras and $(N-2,N-1)$-connected spaces/connective spectra.

Bellow the proof of the relative recognition of $N$-loop spaces for $3\leq N\leq \infty$ using the machinery of idempotent quasiadjunctions in ([Vieira 2020](#VVieira2020)) is sketched.

In ([Vieira 2023](#VVieira2020)) idempotent quasiadjunctions were used to prove a recognition theorem of $\infty$-loop pairs of commutative [[algebra spectra]] over commutative [[ring spectra]]. There it is shown that the multiplicative structure induced by the [[linear isometries operad]] is compatible with the model theoretical machinery presented here.

## Idempotent weak Quillen quasiadjunctions

### Weak Quillen quasiadjunction

To construct the unit and counit natural transformations of an adjuction between homotopy categories of model categories it suffices to construct a unit natural span and counit natural cospan at the model categories level, plus some additional compatibility conditions.

\begin{definition}\label{QsiAdjQuilFr} ([Vieira 2020, Definition 2.1.1](#VVieira2020)) Let $\mathcal{T}$ and $\mathcal{A}$ be model categories. A weak Quillen quasiadjunction, or just quasiadjunction, between $\mathcal{T}$ and $\mathcal{A}$, denoted by
$$
(S \dashv_{\ \mathscr{C},\mathscr{F}}\Lambda):\mathcal{T}\rightleftharpoons \mathcal{A},
$$
is a quadruple of functors
$$
\mathscr{C}:\mathcal{T} \to \mathcal{T}
,\quad
\mathscr{F}:\mathcal{A} \to \mathcal{A}
,\quad
S:\mathcal{T} \to \mathcal{A}
,\quad
\Lambda:\mathcal{A} \to \mathcal{T}
$$
with $S$ the left quasiadjoint and $\Lambda$ the right quasiadjoint, equipped with a natural span in $\mathcal{T}$ and a natural cospan in $\mathcal{A}$
$$
\eta':\mathscr{C}\xRightarrow{\sim} Id_{\mathcal{T}}
,\quad
\eta:\mathscr{C}\Rightarrow \Lambda S
,\qquad
\epsilon:S \Lambda \Rightarrow \mathscr{F},
\quad
\epsilon':Id_{\mathcal{A}}\xRightarrow{\sim}\mathscr{F}
$$

such that

 (i) $S$ is left derivable;

 (ii) $\Lambda$ is right derivable;

 (iii) $\mathscr{C}$ and $\mathscr{F}$ preserve cofibrant and fibrant objects;

 (iv) $\eta'$ and $\epsilon'$ are natural weak equivalences;

 (v) If $X\in\mathcal{T}$ is cofibrant then $\epsilon_{SX}S\eta_X\simeq \epsilon'_{SX}S\eta_X'$;

 (vi) If $Y\in\mathcal{A}$ is fibrant then $\Lambda\epsilon_Y \eta_{\Lambda Y}\simeq \Lambda\epsilon'_Y\eta'_{\Lambda Y}$.
\end{definition}

If $\mathscr{C}$, $\mathscr{F}$, $\eta'$ and $\epsilon'$ are all identities and the homotopy equivalences in (v) and (vi) are equalities we recover the notion of a weak Quillen adjunction. If we further require $S$ to preserve cofibrations and $\Lambda$ to preserve fibrations we get a Quillen adjunction. Though the above information is weaker than that of a Quillen adjunction it suffices to construct an adjunction at the homotopy categories level.

\begin{theorem} ([Vieira 2020, Theorem 2.1.2](#VVieira2020))
    A quasiadjunction induces an adjunction
$$
    (\mathbb{L}S \dashv\mathbb{R}\Lambda):\mathcal{H}o \mathcal{T}\rightleftharpoons \mathcal{H}o \mathcal{A}
$$
between the homotopy categories.
\end{theorem}

### Idempotent quasi(co)monads

The following generalization of Quillen idempotent monads (see [[Bousfield-Friedlander theorem]]) was also introduced following the same principle of only requiring the existence of a unit natural span, and they also induce [[Bousfield localization]]s.

\begin{definition}([Vieira 2020, Definition 2.3.1](#VVieira2020))
    Let $\mathcal{T}$ be a right proper model category. A Quillen idempotent quasimonad on $\mathcal{T}$, or simply an idempotent quasimonad, is a pair of endofunctors $Q,\overline{\mathscr{C}}:\mathcal{T}\rightarrow\mathcal{T}$ equipped with a natural span
$$
  \eta':\overline{\mathscr{C}}\xRightarrow{\sim}Id_{\mathcal{T}}
,\qquad
\eta:\overline{\mathscr{C}}\Rightarrow Q
$$
such that:

(i) $\eta'$ is a natural weak equivalence;

(ii) $Q$ preserves weak equivalences;

(iii) $Q\eta$ and $\eta_{Q}$ are natural weak equivalences;

(iv) If $f\in\mathcal{T}(X,B)$, $p\in F(E,B)$ and $\eta_E,\eta_B,Qf\in W$ then $Q(f^\ast p)\in W$;

(v) If $\iota\in C(\overline{\mathscr{C}} X,K)$ then $\iota_\ast \eta'\in W$.
\end{definition}

\begin{theorem}([Vieira 2020, Theorem 2.3.5 / Proposition 2.3.6](#VVieira2020))
  An idempotent quasimonad induces a left Bousfield localization $\mathcal{T}_Q=(\mathcal{T};W_Q,C_Q,F_Q)$ with $W_Q=Q^{-1}W$, $C_Q=C$ and $F_Q\subset F$ the subset composed of the fibrations $(p:E\twoheadrightarrow B)\in F$ such that
\begin{center}
\begin{xymatrix@=0.5cm}    E\ar@{->>}[d]_p\ar[r]^(0.25){i_E}&E\sqcup_{\overline{C} E}QE\ar[d]^{(p,Qp)}\\
    B\ar[r]_(0.25){i_B}&B\sqcup_{\overline{C} B}QB
\end{xymatrix}
\end{center}
is a homotopy pullback.

The resulting homotopy category is the reflective subcategory
$$\mathcal{H}o \mathcal{T}_Q:=\{X\in \mathcal{H}o \mathcal{T}\mid (i_X:X\rightarrow X\sqcup_{\overline{\mathscr{C}} X}QX)\in W\}
$$
of $Q$-fibrant objects.

\end{theorem}

The above definition can be dualized. The resulting idempotent quasicomonads induce right Bousfield localizations and associated coreflective homotopy subcategories.

### Idempotent quasiadjunctions

A quasiadjunction $(S\dashv_{\ \mathscr{C},\mathscr{F}}\Lambda):\mathcal{T}\rightleftharpoons \mathcal{A}$ induces the following natural span on $\mathcal{T}$ and natural cospan on $\mathcal{A}$
$$
  cof \eta'_{Cof}:\mathscr{C} Cof\xRightarrow{\sim} Id_{\mathcal{T}},
\quad
(\Lambda fib_S \eta)_{Cof}:\mathscr{C} Cof\Rightarrow \Lambda Fib S Cof
$$
$$
  (\epsilon S cof_{\Lambda})_{Fib}: S Cof \Lambda Fib\Rightarrow \mathscr{F} Fib,
\quad
\epsilon'_{Fib}fib:Id_{\mathcal{A}}\xRightarrow{\sim} \mathscr{F} Fib
$$
where $cof : Cof \xRightarrow{\sim} Id$ and $fib : Id \xRightarrow{\sim} Fib$ are the functorial (co)fibrant resolutions of the model structures.

\begin{definition}([Vieira 2020, Definition 2.3.7](#VVieira2020))
    An idempotent quasiadjunction is a quasiadjunction such that the induced span and cospan are respectively an idempotent quasimonad and an idempotent quasicomonad.
\end{definition}

\begin{theorem}([Vieira 2020, Theorem 2.3.8](#VVieira2020))
An idempotent quasiadjunction induces an equivalence between the associated (co)reflective homotopy subcategories.
$$
(\mathbb{L}S\dashv\mathbb{R}\Lambda):
\mathcal{H}o \mathcal{T}_{\Lambda Fib S Cof} \leftrightharpoons \mathcal{H}o \mathcal{A}_{S Cof \Lambda Fib}.
$$
\end{theorem}

## Finite relative recognition theorem

In this section fix some $1\leq N\lt \infty$. Let $\mathcal{SC}_N$ be the cubical Swiss cheese relative operad, a colored operad over the colors $\{c, o\}$, respectively referred to as the closed and open colors. It induces a monad $SC_N$ on the category $Top_*^{\{c, o\}}$ of pairs of pointed topological spaces (see [[algebra over an operad]]).  We denote pairs of pointed spaces as $X=(X_{c},X_{o})$. Denote by $\mathcal{SC}_N[Top]$ the category of algebras over $SC_N$, which we equip with the [[mixed model structure]] transferred from the one on $Top_*^{\{c, o\}}$.

Denote by $Top_*^\rightarrow$ the category of pointed maps, equipped with the mixed [[projective model structure]]. We denote maps as $Y=(Y:Y_0\rightarrow Y_1)$.

\begin{definition}
The relative $N$-loop pair functor is
$$
  \Omega^N_2:Top_*^{\rightarrow}\rightarrow Top_*^{\{c,o\}},\qquad \Omega^N_2(Y)\coloneqq(Y_1^{\mathbb{S}^N},(Y_0\times_{Y_1} Y_1^I)^{\mathbb{S}^{N-1}}).
$$

The relative $N$-suspension functor is
$$
\Sigma^N_\to : Top_*^{\{c,o\}}\to Top_*^\to, \qquad
\Sigma^N_\to(X)\coloneqq
(X_{o} \wedge \mathbb{S}^{N-1} \to ((X_o \wedge I) \vee(X_c \wedge \mathbb{S}^1)) \wedge \mathbb{S}^{N-1} ).
$$
\end{definition}

\begin{remark}
  See *[[relative loop space]]* for details on why we use this definition for relative loop spaces. Note that the above functor outputs both a relative loop space and a loop space of the total space, since the structure of interest here includes the action of $Y_1^{\mathbb{S}^N}$ on $(Y_0\times_{Y_1} Y_1^I)^{\mathbb{S}^{N-1}}$.
\end{remark}

\begin{theorem} ([Vieira 2020, Proposition 2.2.3](#VVieira2020))
  We have a weak Quillen adjunction
$$
(\Sigma^N_\to \dashv \Omega^N_2):Top_*^{\{c,o\}}\leftrightharpoons Top_*^\to.
$$
\end{theorem}

This adjunction transfers a new model structure on $Top_*^\to$, with weak equivalences the commutative squares $f\in Top_*^\to(X,Y)$ such that $f_{0,*}:\pi_q X_0\to \pi_q Y_0$ are isomorphisms for all $q\geq N$ and $(f_0,f_1^I)_*:\pi_q(X_0\times_{X_1}X_1^I)\to \pi_q(Y_0\times_{Y_1}Y_1^I)$ are isomorphisms for all $q\geq N-1$. All objects of $Top_*^\to$ are fibrant in this model structure. We will say a pointed map $Y\in Top_*^\to$ is $(N-2,N-1)$-connected if $Y_0$ is $(N-2)$-connected and $Y_1$ is $(N-1)$-connected. The cofibrant objects are the $Y\in Top_*^\to$ that are homotopy equivalent to $(N-2,N-1)$-connected inclusions of relative CW-pairs. We denote the category of pointed maps equipped with this model structure as $Top^\to_{N-2,N-1}$.

The images of $\Omega^N_2$ are naturally algebras over $\mathcal{SC}_N$, so we have an induced functor $\Omega^N_2:Top_*^\to\rightarrow \mathcal{SC}_N[Top]$. This functor is not a right adjoint, but it does have a left weak Quillen quasiadjoint induced by the [[two-sided bar construction]].

\begin{definition}
  The relative $N$-delooping functor is
$$
B^N_\to:\mathcal{SC}_N[Top]\to Top_*^\to,\qquad B^N_\to(X)\coloneqq B(\Sigma^N_\to,SC_N,X).
$$
\end{definition}

\begin{definition}
  The resolution of $\mathcal{SC}_N$-spaces functor is
$$
\overline{B}_2:\mathcal{SC}_N[Top]\to\mathcal{SC}_N[Top],\qquad \overline{B}_2(X)=B(SC_N,SC_N,X).
$$
\end{definition}

\begin{theorem} ([Vieira 2020, Theorem 4.3.5, Theorem 4.3.8](#VVieira2020))
  We have an idempotent weak Quillen quasiadjunction
$$
  (B^N_\to\dashv_{\ \overline{B}_2,Id_{Top_*^\to}}\Omega^N_2):\mathcal{SC}_N[Top]\leftrightharpoons Top^\to.
$$
If $N\geq 3$ then this quasiadjunction induces an equivalence
$$
(\mathbb{L}B^N_\to\dashv\mathbb{R}\Omega^N_2):\mathcal{H}o\mathcal{SC}_N[Top]_{grp}\leftrightharpoons \mathcal{H}o Top^\to_{N-2,N-1}
$$
between the homotopy subcategory of grouplike Swiss cheese algebras and the homotopy category of $(N-2,N-1)$-connected pointed maps.
\end{theorem}

\begin{remark}
In the reference [Vieira 2020](#VVieira2020) a cofibrant resolution of $\mathcal{SC}_N$ is used. The fact that $\mathcal{SC}_N$ is a $\Sigma$-cofibrant operad (see [[model structure on operads]]) when we consider the mixed model structure of collections means this assumption is not necessary.
\end{remark}

\begin{remark}
The reason the proof of the above theorem doesn't extend to the cases $N=1,2$ is due to the fact that the constructed unit of the quasiadjunction is induced by a natural map $\alpha:SC_N\Rightarrow \Omega^N_2\Sigma^N_\to$ which is a natural pair of [[group completion]]s if and only if $N\geq 3$.
\end{remark}

## Infinite relative recognition theorem

  Let $Sp$ be the category of [[sequential spectra]], and $Sp^{\nearrow}$ the category composed of pairs of spectra $Y_0,Y_1\in Sp$ equipped with a 1-shifted spectra map $Y:Y_0\to Y_1[1]$. Morphisms are pairs of spectra maps that commute with the shifted spectra maps in the appropriate sense. It admits a strict mixed model structure with weak equivalences the level-wise weak homotopy equivalences and with fibrations the level-wise Hurewicz fibrations.

\begin{definition}
The [[spectrification]] functor is
$$
\widetilde{\Omega}:Sp\to Sp,\qquad \widetilde{\Omega}(Y)_\bullet\coloneqq \colim_{q\to\infty} \widetilde Y_{\bullet+q}^{\mathbb{S}^q}
$$
where $\widetilde{Y}$ is a certain inclusion prespectrum constructed from $Y$, equipped with a quotient map $Y\to \widetilde{Y}$ ([LMS 86, Appendix 1](#LewisMaySteinberger86)). If $Y$ is already an inclusion spectrum then $\widetilde Y=Y$.

This functor induces a spectrification functor
$$
\widetilde{\Omega}_{\nearrow}:Sp^{\nearrow}\to Sp^{\nearrow}
,\qquad \widetilde{\Omega}_{\nearrow}(Y)\coloneqq(\widetilde{\Omega}(Y):\widetilde{\Omega}(Y_0)\to \widetilde{\Omega}(Y_1)[1]).
$$
\end{definition}

We have a natural [[stable weak homotopy equivalence]] $\epsilon':Id_{Sp^{\nearrow}}\Rightarrow\widetilde{\Omega}_{\nearrow}$ which gives a Quillen idempotent monad structure on $\widetilde{\Omega}_{\nearrow}$. The stable mixed model structure on $Sp^{\nearrow}$ is the one induced by this idempotent monad, which is the left Bousfield localization of the strict mixed model structure on the pairs of stable weak homotopy equivalences. The fibrant objects in this model structure are the [[Omega-spectra]].

\begin{definition}
  The relative base pair of spaces functor is
$$
\Lambda^\infty_2:Sp^{\nearrow}\to Top_*^{\{c,o\}},
\qquad \Lambda^\infty_2(Y)\coloneqq (Y_{10},Y_{00}\times_{Y_{11}}Y_{11}^I).
$$
The relative $\infty$-loop pair of spaces functor is
$$\Omega^\infty_2:=\Lambda^\infty_2\widetilde{\Omega}_{\nearrow}:Sp^{\nearrow}\to Top_*^{\{c,o\}}.
$$
\end{definition}

The images of $\Omega^\infty_2$ are naturally algebras over $\mathcal{SC}_\infty$, so we have an induced functor $\Omega^\infty_2:Sp^{\nearrow}\rightarrow \mathcal{SC}_\infty[Top]$.

\begin{definition}
The relative $\infty$-delooping functor is
$$
B^\infty_\nearrow:\mathcal{SC}_\infty[Top]\to Sp_\nearrow,
\qquad
B^\infty_\nearrow(X)_\bullet\coloneqq B(\Sigma^{\bullet+1}_\to,SC_{\bullet+1},X).
$$
\end{definition}

\begin{definition}
  The resolution of $\mathcal{SC}_\infty$-spaces functor is
$$
\overline{B}_2:\mathcal{SC}_\infty[Top]\to\mathcal{SC}_\infty[Top],\qquad \overline{B}_2(X)=B(SC_\infty,SC_\infty,X).
$$
\end{definition}

\begin{theorem}([Vieira 2020, Theorem 4.3.5, Theorem 4.3.8](#VVieira2020))
  We have an idempotent weak Quillen quasiadjunction
$$
  (B^\infty_\nearrow\dashv_{\ \overline{B}_2,\widetilde{\Omega}_\nearrow}\Omega^\infty_2):\mathcal{SC}_\infty[Top]\leftrightharpoons Sp^\nearrow
$$
which induces an equivalence
$$
(\mathbb{L}B^\infty_\nearrow\dashv\mathbb{R}\Omega^\infty_2):\mathcal{H}o\mathcal{SC}_\infty[Top]_{grp}\leftrightharpoons \mathcal{H}o Sp^\nearrow_{con}
$$
between the homotopy category of grouplike Swiss cheese algebras and the homotopy category of shifted spectra maps between connective spectra.
\end{theorem}

## Related concepts

* [[relative loop space]]

* [[May recognition theorem]]

* [[Quillen equivalence]]

## References

* {#HLS16} [[Eduardo Hoefel]], [[Muriel Livernet]], [[Jim Stasheff]], _$A_\infty$-actions and recognition of relative loop spaces_, Topology and its Applications **206** (2016) 126-147 [[arXiv:1312.7155](https://arxiv.org/abs/1312.7155), [doi:10.1016/j.topol.2016.03.023](https://doi.org/10.1016/j.topol.2016.03.023)]

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics  **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)]

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Lecture Notes in Mathematics, Springer 1972 ([doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491), [pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))

* {#May74} [[Peter May]], _$E_\infty$-Spaces, group completions, and permutative categories_, New Developments in Topology, London Math. Soc. Lecture Note Series 11 (1974) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/13.pdf))

* {#VVieira2020} [[Renato Vasconcellos Vieira]], _Relative recognition principle_, Algebr. Geom. Topol. 20(3): 1431-1486 (2020) [[arXiv:1802.01530](https://arxiv.org/abs/1802.01530), [doi:10.2140/agt.2020.20.1431](https://doi.org/10.2140/agt.2020.20.1431)]

* {#VVieira2023} [[Renato Vasconcellos Vieira]], _Recognition of connective commutative algebra spectra
through an idempotent quasiadjunction_, Algebr. Geom. Topol. 23(1):[arXiv:2101.03052](https://arxiv.org/abs/2101.03052), 295-338 (2023) & [doi:10.2140/agt.2023.23.295](https://doi.org/10.2140/agt.2023.23.295)]


recollement

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The __recollement__ situation is a [[diagram]] of six [[additive functors]]

$$
  \mathcal{A}'
    \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
  \mathcal{A}\stackrel{\overset{j_!}{\longleftarrow}}{\stackrel{\overset{j^*}{\longrightarrow}}{\underset{j_*}{\longleftarrow}}}\mathcal{A}''
$$

among three [[abelian category|abelian]] or [[triangulated categories]] satisfying a strong list of [[exact functor|exactness]] and [[adjoint functor|adjointness]] axioms.

The paradigmatic situation is about the categories of [[abelian sheaves]] $\mathcal{A}' = Sh(C)$, $\mathcal{A} = Sh(X)$, $\mathcal{A}'' = Sh(U)$, where $U\subset X$ is an [[open subset]] of a [[topological space]], $C = X\backslash U$ the [[complement]], and the functors among the [[category of sheaves|sheaf categories]] are induced by the [[open embedding]] $j \colon U\hookrightarrow X$ and [[closed embedding]] $i \colon C\hookrightarrow X$. As suggested by this example, recollement may in fact be regarded as the additive or triangulated version of [[Artin gluing]].

## Definition

A modern treatment for the recollement of abelian categories is in ([Franjou-Pirashvili 04](#FranjouPirashvili04)), where the following axioms are listed:

(i) $j_!\dashv j^*\dashv j_*$

(ii) the [[unit of an adjunction|unit]] $Id_{\mathcal{A}''}\to j^* j_!$ and the [[counit of an adjunction|counit]] $j^* j_*\to Id_{\mathcal{A}''}$ are [[isomorphism|iso]] ([hence](adjoint%20functor#FullyFaithfulAndInvertibleAdjoints) $j_\ast$ and $j_!$ are [[fully faithful functor|fully faithful]])

(iii) $i^*\dashv i_*\dashv i^!$

(iv) the unit $Id_{\mathcal{A}'}\to i^! i_*$ and the counit $i^* i_*\to Id_{\mathcal{A}'}$ are iso

(v) the functor $i_*:\mathcal{A}'\to Ker(j^*)$ is an equivalence of categories.

In fact (i) and (ii) for $j^*:\mathcal{A}\to\mathcal{A}''$ enable one to define $\mathcal{A}'$ as the full subcategory of $\mathcal{A}$ whose objects $a$ satisfy $j^* a = 0$ such that one satisfies the recollement situation.

A standard treatment for the sequence of triangulated functors
$$
  \mathcal{D}'
   \overset{i_*}{\to}
  \mathcal{D}\overset{j^*}{\to}\mathcal{D}''
$$

is in ([Beilinson-Bernstein-Deligne 82](#BeilinsonBernsteinDeligne82)) where in 1.4.3 the following axioms are listed

(a) $i_* = i_!$ admits a triangulated left adjoint $i^*$ and triangulated right adjoint $i^!$

(b) $j^* = j^!$ admits a triangulated left adjoint $j_*$ and triangulated right adjoint $j_!$

(c) $j^* i_* = 0$ (hence by adjointness, also $i^*j_! = 0$ and $i^! j_*=0$)

(d) given $d\in Ob{\mathcal{D}}$, there exist (necessarily unique) distinguished triangles

$$
i_! i^! d \to d\to j_* j^* d\to (i_! i^! d) [1]
$$
$$
j_! j^! d \to d\to i_* i^* d\to (j_! j^! d) [1]
$$

(e) $i_*, j_*, j_!$ are full embeddings.

Again in good situations, less data is needed to provide the recollement.

## Examples

* The [[forgetful functor]] from [[global equivariant stable homotopy theory]] to plain [[stable homotopy theory]] exhibits a recollement,

See at _[global equivariant stable homotopy theory -- Relation to plain stable homotopy theory](global+equivariant+stable+homotopy+theory#RelationToPlainStableHomotopyTheory)_.

## Related concepts

* [[adjoint modality]]

## References

* {#FranjouPirashvili04} V. Franjou, [[T. Pirashvili]], _Comparison of abelian categories recollements_, Doc. Math. 9 (2004), 41–56, [MR2005c:18008](http://www.ams.org/mathscinet-getitem?mr=2054979), [pdf](http://www.math.uni-bielefeld.de/documenta/vol-09/03.pdf)

* {#BeilinsonBernsteinDeligne82} A.A. Beilinson, J. Bernstein, [[Pierre Deligne]], _Faisceaux pervers. Analysis and topology on singular spaces_, I (Luminy, 1981), 5–171, Aste'risque 100, Soc. Math. France, Paris 1982.

In references

* E. Cline, B. Parshall, L. Scott, _Finite dimensional algebras and highest weight categories_, J. Reine Angew. Math, 1988, 391: 85—99, [MR90d:18005](http://www.ams.org/mathscinet-getitem?mr=961165), [goettingen](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002205971)
* E. Cline, B. Parshall, L. Scott, _Algebraic stratification in representative categories_, J. of Algebra 117, 1988, 504—521.

one studies the following kind of sources of recollement situations for triangulated categories: $k$ is a commutative [[field]], $A$ a finite dimensional unital associative $k$-algebra, $e$ an idempotent, and $D^b(A)$ the bounded derived category of right $A$-modules. Suppose $eA(1-e) = 0$ and the global dimension of $A$ is finite. Then there is a recollement of triangulated categories involving $D^b(eAe)$, $D^b(A)$ and $D^b((1-e)A(1-e))$.

* S. Koenig, _Tilting complexes, perpendicular categories and recollements of derived module categories of rings._, [MR92k:18009](http://www.ams.org/mathscinet-getitem?mr=1124785), doi
* Qinghua Chen,Yanan Lin, _Recollements of extension algebras_, Science in China 46, 4, 2003 [pdf](http://www.springerlink.com/content/h185372128726855/fulltext.pdf)

Another source of examples is due MacPherson and Vilonen

* Kari Vilonen, _Perverse sheaves and finite dimensional algebras_, Trans.
A.M.S. 341 (1994), 665–676, [MR94d:16012](http://www.ams.org/mathscinet-getitem?mr=1135104), [doi](http://dx.doi.org/10.2307/2154577)

* Michael Artin, Grothendieck Topologies. Harvard University, 1962.
* Yuri Berest, Oleg Chalykh, Farkhod Eshmatov, _Recollement of deformed preprojective algebras and the Calogero-Moser correspondence_, Mosc. Math. J. 8 (2008), no. 1, 21–37, 183, [arxiv/0706.3006](http://arxiv.org/abs/0706.3006), [MR2009h:16030](http://www.ams.org/mathscinet-getitem?mr=2422265)
* Roy Joshua, [pdf](http://www.math.osu.edu/~joshua.1/GV.pdf)
* Yang Han, _Recollements and Hochschild theory_, [arxiv/1101.5697](http://arxiv.org/abs/1101.5697)

For treatment in the setting of $\infty$-categories:

* [[Jacob Lurie]], section A.8 of _[[Higher Algebra]]_,

* [[Clark Barwick]], [[Saul Glasman]], _A note on stable recollements_ ([arXiv:1607.02064](https://arxiv.org/abs/1607.02064))


reconstruction of covering spaces from monodromy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]], the _[[fundamental theorem of covering spaces]]_ asserts that for $X$ a [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space]], then the [[functor]] given by sending a [[covering space]] over $X$ to its [[monodromy]] [[permutation representation|permutation]] [[groupoid representation]] of the [[fundamental groupoid]] $\Pi_1(X)$ of $X$

$$
  Fib
    \;\colon\;
  Cov(X) \longrightarrow Set^{\Pi_1(X)}
$$

is an [[equivalence of categories]] between the [[category of covering spaces]] and that of [[permutation representation|permutation]] [[groupoid representations]] of $\Pi_1(X)$. This means that there exists a functor the other way around

$$
  Rec
    \;\colon\;
  Set^{\Pi_1(X)}
    \longrightarrow
  Cov(X)
$$

which is an [[inverse functor]] up to [[natural isomorphism]]. This functor "reconstructs" a covering space $Rec(\rho)$ from a [[permutation representation]] $\rho$ of the [[fundamental groupoid]], such that the [[monodromy]] of $Rec(\rho)$ is [[natural isomorphism|naturally isomorphic]] to the original representation $Fib(Rec(\rho)) \simeq \rho$.

## Details
 {#Details}

We give several equivalent discussions:

1. An [Elementary description](#ElementaryDescription) using just basic [[point-set topology]];

1. A [Description via Coends](#InTermsOfACoend) using [[category theory|category theoretic]] tools.

### Elementary description
 {#ElementaryDescription}

The following is a description of the reconstruction in terms of elementary [[point-set topology]].

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

Let

1. $(X,\tau)$ be a [[locally path-connected topological space|locally path-connected]] [[semi-locally simply connected topological space|semi-locally simply connected]] [[topological space]],

1. $\rho \in Set^{\Pi_1(X)}$ a [[permutation representation]] of its [[fundamental groupoid]].

Consider the [[disjoint union]] [[set]] of all the sets appearing in this representation

$$
  E(\rho) \;\coloneqq\;  \underset{x \in X}{\sqcup} \rho(x)
$$

For an [[open subset]] $U \subset X$ which is [[path-connected topological space|path-connected]] and for which every element of the [[fundamental group]] $\pi_1(U,x)$ becomes trivial under $\pi_1(U,x) \to \pi_1(X,x)$, and for $\hat x \in \rho(x)$ with $x \in U$ consider the subset

$$
  V_{U,\hat x}
    \coloneqq
  \left\{
    \rho(\gamma)(\hat x)
    \;\vert\;
    x' \in U
    \,,\phantom{A}
    \gamma \,\text{path from}\, x \,\text{to}\, x'
  \right\}
  \;\subset\;
  E(\rho)
  \,.
$$

The collection of these defines a [[base for a topology]] (prop. \ref{ElementaryReconstructedCoveringSpaceWellDefined} below). Write $\tau_{\rho}$ for the corresponding topology. Then

$$
  (E(\rho), \tau_{\rho})
$$

is a [[topological space]]. It canonically comes with the function

$$
  \array{
    E(\rho) &\overset{p}{\longrightarrow}& X
    \\
    \hat x \in \rho(x) &\mapsto& x
  }
  \,.
$$

Finally, for

$$
  f \;\colon\; \rho_1 \longrightarrow \rho_2
$$

a [[homomorphism]] of permutation representations, there is the evident induced function

$$
  \array{
    E(\rho_1)
     &\overset{Rec(f)}{\longrightarrow}&
    E(\rho_2)
    \\
    (\hat x \in \rho_1(x)) &\mapsto& (f_x(\hat x) \in \rho_2(x))
  }
  \,.
$$

=--

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

The construction $\rho \mapsto E(\rho)$ in def. \ref{ElementaryReconstructionCoveringSpace}
is well defined and yields a [[covering space]] of $X$.

Moreover, the construction $f \mapsto Rec(f)$ yields a homomorphism of covering spaces.

=--

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

First to see that we indeed have a [[topological space|topology]], we need to check  (by [this prop.](topological+base#Recognition)) that every point is contained in some base element, and that every point in the intersection of two base elements has a base neighbourhood that is still contained in that intersection.

So let $x \in X$ be a point. By the assumption that $X$ is [[semi-locally simplc connected space|semi-locally simply connected]] there exists an [[open neighbourhood]] $U_x \subset X$ such that every loop in $U_x$ on $x$ is contractible in $X$. By the assumption that $X$ is [[locally path-connected topological space]], this contains an open neighbourhood $U'_x \subset U_x$ which is [[path-connected topological space|path connected]]. As every subset of $U_x$, it still has the property that every loop in $U'_x$ based on $x$ is contractible as a loop in $X$. Now let $\hat x \in E$ be any point over $x$, then it is contained in the base open $V_{U'_x,x}$.

The argument for the base open neighbourhoods contained in intersections is similar.

Then we need to see that $p \colon E(\rho) \to X$ is a [[continuous function]]. Since taking pre-images preserves unions ([this prop.](interactions+of+images+and+pre-images+with+unions+and+intersections#PreImagePreservesUnionsAndIntersections)), and since by semi-local simply connectedness every neighbourhood contains an open $U \subset X$ that labels a base open, it is sufficient to see that $p^{-1}(U)$ is a base open. But by the very assumption on $U$, there is a unique morphism in $\Pi_1(X)$ from any point $x \in U$ to any other point in $U$, so that $\rho$ applied to these paths establishes a bijection of sets

$$
  p^{-1}(U)
    \;\simeq\;
  \underset{\hat x \in \rho(x)}{\sqcup} V_{U,\hat x}
    \;\simeq\;
  U \times \rho(x)
  \,,
$$

thus exhibiting $p^{-1}(U)$ as a union of base opens.

Finally we need to see that this continuous function $p$ is a covering projection, hence that every point $x \in X$ has a neighbourhood $U$ such that $p^{-1}(U) \simeq U \times \rho(x)$. But this is again the case for those $U$ all whose loops are contractible in $X$, by the above identification via $\rho$, and these exist around every point by semi-local simply-connetedness of $X$.

This shows that $p \colon E(\rho) \to X$ is a covering space. It remains to see that $Rec(f) \colon E(\rho_1) \to E(\rho_2)$ is a homomorphism of covering spaces. Now by construction it is immediate that this is a function over $X$, in that this [[commuting diagram|diagram commutes]]:

$$
  \array{
    E(\rho_1) && \overset{Rec(f)}{\longrightarrow}&& E(\rho_2)
    \\
    & \searrow && \swarrow
    \\
    && X
  }
  \,.
$$

So it only remains to see that $Rec(f)$ is a [[continuous function]]. So consider $V_{U, y_2 \in \rho_2(x)}$ a base open of $E(\rho_2)$. By [[natural transformation|naturality]] of $f$ its pre-image under $Rec(f)$ is

$$
  Rec(f)^{-1}(V_{U, y_2 \in \rho_2(x)})
  =
  \underset{y_1 \in f^{-1}(y_2)}{\sqcup} V_{U,y_1}
$$

and hence a union of base opens.

=--

### In terms of a coend
 {#InTermsOfACoend}

The following is a description of the reconstruction functor in terms of tolls from [[category theory]].

Given a space $B$, let $|B|$ be $B$ retopologized with the discrete topology, and consider the [[pullback]] in [[Top]] of the [[path space]] $B^I$ to the [[product space]] ${\vert B \vert} \times B$:

$$\array{
Path(B) & \to & B^I & \\
\downarrow & (pb) & \downarrow & \langle ev_0, ev_1 \rangle\\
|B| \times B & \underset{id \times id}{\to} & B \times B &
}$$

Let $\overline{Path}(B)$ be the [[quotient space]] of $Path(B)$ by the [[equivalence relation]] "[[homotopy]] relative to the [[boundary]]". We can think of $\overline{Path}(B)$ as a sum of spaces

$$\sum_{b \in B} \tilde{B}_b,$$

fibered in the obvious way over $|B|$ (the set of all basepoints $b$), where $\tilde{B}_b$ is the space of paths in $B$ which begin at $b$, modulo homotopy-rel-boundary. The space $\tilde{B}_b$ can be thought of the universal covering space over the connected component of a point $b \in B$, considered as a space based at $b$.

We have a [[span]]

$$\array{
& & \overline{Path}(B) & & \\
& \swarrow & & \searrow & \\
|B| & & & & B
}$$

with an obvious (contravariant) composition action $comp$ of the fundamental groupoid $\Pi_1(B)$, itself regarded as a span

$$\array{
& & \Pi_1(B) & & \\
& \swarrow & & \searrow & \\
|B| & & & &  |B|
}$$

with a [[monad]] structure in the [[bicategory]] of [[span]]s. The [[action]] gives a map

$$comp: \Pi_1(B) \times_{|B|} \overline{Path}(B) \to \overline{Path}(B),$$

of spans from $|B|$ to $B$.

Now suppose given an object $F$ of $Set^{\Pi_1(B)}$, i.e., a covariant action of the fundamental groupoid, that is to say a span $F: 1 \to |B|$ equipped with an action $\alpha$ of the monad $\Pi_1(B): |B| \to |B|$ in $Span(Top)$. The data of a right-handed action $comp$ on $\overline{Path}(B)$ and the left-handed action $\alpha$ on $F$ gives rise to a two-sided [[bar construction]]

$$B(\overline{Path}(B), \Pi_1(B), F),$$

which here is a [[simplicial object]] in the category of spans from $1$ to $B$, whose two face maps from degree 1 to degree 0 take the form:

$$\array{
  & F \times_{|B|} \Pi_1(B) \times_{|B|} \overline{Path}(B) &
  \\
  &
  {}^{\mathllap{F \times_{|B|} comp}}\downarrow \downarrow^{\mathrlap{\alpha \times_{|B|} \overline{Path}(B)}}
  &
  \\
  &
  F \times_{|B|} \overline{Path}(B) &
}$$

The [[coequalizer]] of this pair provides a canonical augmentation of the two-sided bar construction, and may be called the tensor product

$$\overline{Path}(B) \otimes_{\Pi_1(B)} F$$

(the seemingly opposite placement of the two tensor factors, as compared against the span constructions above, is simply an artifact of the discrepancy between [[diagrammatic order]] of composition, and the traditional order in which right actions are covariant and left actions contravariant).

As a span from $1$ to $B$, that is as a bundle over $B$, this tensor product is indeed a covering space over $B$, assuming that $B$ is [[locally connected topological space|locally connected]] and [[semi-locally simply connected topological space|semi-locally simply connected]]. Finally, the functor

$$\overline{Path}(B) \otimes_{\Pi_1(B)} -: Set^{\Pi_1(B)} \to Cov/B$$

is under these conditions quasi-inverse to the fiber functor

$$Fiber: Cov/B \to Set^{\Pi_1(B)}$$

and thus establishes the [[equivalence of categories]] known as the [[fundamental theorem of covering spaces]].

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

An abstract way of considering the functor $Fiber$ is that it is obtained by homming:

$$Fiber(p: E \to B)(b) = (Cov/B)(\tilde{B}_b, p)$$

and this forces its [[left adjoint]] to be given by the tensor product construction described above.

=--

## Examples

### The universal covering space

As a special case, consider the [[permutation representation]] $\Pi_1(B) \to Set$ given by the discrete fibration

$$cod: \Pi_1(B) \to |B|$$

+-- {: .query}
[[David Roberts]]: shouldn't such a discrete fibration then give rise to a functor $|B| \to Set$? If you mean $Mor(\Pi_1(B))$, then this could probably be described as the total tangent groupoid, which is the action groupoid for the action of $\Pi_1(B)$ on itself.

[[Todd Trimble]]: I didn't make myself clear then. Recall that if $C$ is an internal category in a category $E$ (with $E = Set$ in this discussion), then one defines $E^C$ by taking its objects to be internal discrete fibrations, defined as arrows $F \to C_0$ equipped with the data of an action by the internal category $C$, considered as a monoid in spans from $C_0$ to $C_0$. (This is a standard usage of the term "discrete fibration"; see Johnstone's Topos Theory for instance.) Looking over this again, I guess I really should have had $F = Mor(\Pi_1(B))$, and $|B|$ here means the underlying set of $B$. But hopefully my meaning is now clear.

[[David Roberts]]: Yes, I see now.
=--

(as a span from $1$ to $|B|$) equipped with the obvious (covariant) action of the monad $\Pi_1(B)$ (as a span from $|B|$ to $|B|$). This is essentially the "[[regular representation]]" of the fundamental groupoid. The tensor product of the previous section,

$$\overline{Path}(B) \otimes_{\Pi_1(B)} cod,$$

is a way of realizing the [[universal covering space]] over $B$.

Here is a way of thinking of this construction which links it to the description of universal bundles by Roberts and Schreiber, which is based on considering tangent spaces of the fundamental groupoid. If the fundamental groupoid $G = \Pi_1(B)$ is connected, its universal bundle (as a fibration of groupoids) may be realized as the "tangent groupoid at $b$" or slice

$$T_b(G) := (b/G) \to  G$$

 for a chosen basepoint $b \in B$. Note that this slice groupoid is the pullback

$$\array{
(b/G) & \to & G^I & \\
\downarrow & & \downarrow & ev_0\\
\{b\}  & \to & G &
}$$

with $I$ the groupoid $(0 \overset{\sim}{\to} 1)$. This is then a groupoid over $G$ by the restriction of $ev_1$.

Since the set of arrows of $G$ is obtained as a quotient of the set of paths in $B$, it inherits naturally a topology (a quotient of the compact-open topology on $B^I$) which, together with the given topology on $G_0 = B$, makes $G$ a topological groupoid. Then we recover the universal covering space $B^{(1)}_b$ (I prefer this notation for the 1-connected cover, rather than the usual $\tilde{B}$, because it generalises to $B^{(n)}$ for $n$-connected covers - DR) over $B$ by pulling back along the functor $B \to G$, where we consider $B$ as a topological groupoid with only identity arrows. The assumptions on the topology of $B$ mean that $G$ is a [[locally trivial groupoid]] with discrete hom-spaces, which implies that $B^{(1)}_b$ is a locally trivial bundle with discrete fibres.  Local path-connectedness implies that it is locally trivial, and the local condition on $\pi_1$ holds if and only if the fibres are discrete - this last result is due to Daniel Bliss.

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

Another way to consider the topological conditions on $B$ is to realise that $\Pi_1(B)$, with its inherited topology, is equivalent to a topologically discrete groupoid (in some appropriate localisation of the 2-category of topological groupoids) if and only if $B$ is locally path-connected and semi-locally simply-connected. Otherwise one has to consider the pro-homotopy 1-type of $B$, as in the theory of algebraic fundamental groups (recall that varieties with appropriate topologies - e.g Zariski - are topologically badly behaved).
=--

+--{: .query}
[[David Roberts]]: Is there a prodiscrete completion of a topological groupoid? Maybe we need to assume it is locally trivial, so it is weakly equivalent (in the said localised 2-category) to a groupoid enriched in $Top$, considered as being internal to $Top$. We could then talk about quotients by wide subgroupoids being topologically discrete. Or even quotients being discrete and having finite Leinster cardinality?? Hmm...
=--

In this analysis, the universal covering space $E_b$ of (path-connected) $B$ is retrieved as the quotient of the space of paths which start at the basepoint $b$, modulo homotopy-rel-boundary; the projection to $B$ takes a class of a path $\phi$ to its terminal point $\phi(1)$. This last description is what one would find in any textbook on algebraic topology dealing with covering spaces. This covering space is, strictly speaking, universal among *connected* covering spaces

More generally, if $S \subset |B|$ is a set of basepoints (Thanks, Ronnie Brown! - DR), we can form the pullback

$$\array{
(S/G) & \to & G^I & \\
\downarrow & & \downarrow & ev_0\\
S  & \to & G &
}$$

which is again a groupoid over $G$ by restriction of $ev_1$. Then pullback of $(S/G) \to G$ along the inclusion $B \to G$ is a covering space which is the sum
$$
B^{(1)}\langle S\rangle = \sum_{b\in S} B^{(1)}_b
$$
of connected, 1-connected covering spaces based at the points in $S$. Thus for not-necessarily-connected $B$, taking $S$ such that it intersects each component of $B$ once we can get a universal covering space of $B$ (universal among covering spaces $E \to B$ that induce isomophisms $\Pi_0(E) \to \Pi_0(B)$).

This construction is functorlal (for general $S\subset |B|$), since a map $(B,S) \to (B',S')$ of pairs (remember we are giving $S,S'$ the discrete topology, not the subspace topology) induces a functor of (topological) groupoids $\Pi_1(B) \to \Pi_1(B')$, which by universality of the pulbacks in the above construction gives a map
$$
B^{(1)}\langle S\rangle \to B'^{(1)}\langle S'\rangle
$$
covering the given map $B \to B'$.

The dependence on basepoints is of course spurious; we can make this explicit by considering the colimit obtained by pasting together the universal covering spaces $B^{(1)}_b$ along isomorphisms induced by paths $b \to c$. But this is in effect how our tensor product construction of the universal covering space works: $\overline{Path}(B)$ is precisely the sum

$$\sum_{c \in |B|} B^{(1)}_c$$

which can be viewed as a topological span from $|B|$ to $B$. The fundamental groupoid acts contravariantly on this sum, and the tensor product

$$\overline{Path}(B) \otimes_{\Pi_1(B)} (cod: \Pi_1(B) \to |B|)$$

is the same thing as the coequalizer of the pair of arrows

$$\sum_{[\phi]: b \to c} \sum_c B^{(1)}_c \overset{\to}{\to} \sum_c B^{(1)}_c$$

in $Top/B$, where one arrow is projection and the other is given by the action of pulling back along classes of paths; this coequalizer is a precise description of the pasting colimit alluded to above. It should be noted that this coequaliser is isomorphic to the covering space $B^{(1)}\langle S\rangle$ when $S$ has one point in each component of $B$, but the description as the tensor product is a priori functorial without reference to a set of basepoints.

+-- {: .query}
[[David Roberts]]: I think, though, due to the lifting theorems for covering spaces, that given a map $f:B \to B'$ and basepoint sets $S \subset |B|$, $S' \subset |B'|$ that are not necessarily preserved by $f$, there should be a unique lift of $B^{(1)}\langle S\rangle \to B'$ to $B'^{(1)}\langle S'\rangle$ anyway. This would also make this construction independent, up isomorphism, of the choice of basepoints and probably also functorial.

[[David Roberts]]: It won't be functorial - the lift referred to isn't unique. The up-to-isomorphism is a non-canonical isomorphism.
=--

(David or Urs: please feel free to sprinkle your own sugar over this, by adapting or even copying what David wrote below based on your paper.)

([[David Roberts]]: unless someone feels the discussion below is essential, it can be deleted.)

+-- {: .query}
[[David Roberts]]: My personal favourite way of doing this is to topologise the fundamental groupoid, then form the following strict pullback of topological groupoids
$$
\array{
\widetilde{B} & \to & T_b\Pi_1(B) \\
\downarrow && \downarrow\\
B & \to &\Pi_1(B)
}
$$
where $b\in B$ is a chosen basepoint and $T_b\Pi(B)$ is the tangent groupoid at the object $b$. This links the ideas that the tangent groupoid is the contractible cover of a groupoid, that the fundamental groupoid is the 1-type of a space and the Whitehead construction of connected covers (pull back the path-fibration along the inclusion of a space into the appropriate Postnikov section).

The topology on the fundamental groupoid can either be constructed with the assumption that $B$ is locally path-connected and semi-locally simply-connected, or be given the quotient topology from the free path space $B^I$. With this inherited topology, the fundamental groupoid is equivalent (in the bicategory of topological groupoids and [[anafunctor]]s) to the same groupoid considered with the discrete topology if and only if $B$ satisfies the usual conditions for the universal covering space to exist. Thus even when $\Pi_1(B)$ is topologised, it still represents a 1-type for nice $B$. One thing which interests me, even though I have no idea about how to approach it, is how for general $B$ the topologised fundamental groupoid can be considered as a pro-homotopy type, that is, the limit of discrete groupoids, taken in the appropriate (bi)category of topological groupoids.

I would like see several expositions of the construction of the universal covering space, since they illustrate different ideas. They seem tautologously related, but things show a bit more of the differences when one passes to bigroupoids.

The universal covering space is

* the source-fibre (at a basepoint) of the topologised fundamental groupoid

* the pullback of the tangent groupoid as described above

* The pullback of the map $(s,t):Mor(\Pi_1(B)) \to Obj(B)\times Obj(B)$ along the inclusion $\{b\}\times B \to B\times B$

[[Todd Trimble|Todd]] I'll get back to writing more of what I had planned soon. I haven't had a chance to digest what you're writing yet, but I prefer to proceed without having to choose basepoints. I'd like to get you and Urs to have a look though when I get back to this within a few days.

[[David Roberts|David]]: Of course - hence the theorem about functors from the fundamental groupoid and not the fundamental group. This is where the full tangent groupoid comes in: it is the pullback
$$
\array{
TG & \to & G^I & \\
\downarrow && \downarrow& dom\\
Obj(G) & \to & G &
}
$$
or equivalently the slice $Obj(G)\downarrow id_G$ for an internal groupoid $G$ (internal in $Top$, but extensions to other categories work too). The tangent groupoid at a point $g$ is just the subgroupoid of this gotten by pulling back $TG \to Obj(G)$ along the inclusion $\{g\} \to Obj(G)$. I hadn't thought about applying this construction to my personal universal covering space recipe, so maybe we need to take the discrete topology on $Obj(G)$. That's what your pullback square above seems to indicate. Urs' and my paper [arXiv:0708.1741] has stuff on tangent groupoids for anyone who interested in pitching in.
=--

[[!redirects reconstructing covering spaces from monodromy]]
[[!redirects reconstructing a covering space from monodromy]]


Reconstruction of Groups

[[!redirects Reconstruction of groups]]

This entry is to try understanding the article

* [[A. L. Rosenberg]], _Reconstruction of groups_, Selecta Math. N.S. 9:1 (2003) [doi](http://dx.doi.org/10.1007/s00029-003-0322-x)

aimed at a reasonably general reconstruction theorem for [[locally compact topological groups]] from a suitable category of continuous infinite-dimensional representations, generalizing the [[Pontrjagin duality]] theorem for locally compact abelian groups and the [[Tannaka-Krein theorem]] for compact groups. More precisely,  it asks

> what are conditions on a monoidal subcategory of the category of continuous representations of a locally compact group sufficient to reconstruct the group?

The first generalization of that type is the Tatsuuma reconstruction theorem

* N. Tatsuuma, _A duality theorem for locally compact groups_, J. Math., Kyoto Univ. 6 (1967), 187–293.

which asserts that for a monoidal subcategory $C$ of the category of [[unitary representation]]s of a locally compact group $G$ to be sufficient for the reconstruction from the [[fiber functor]] $C\to Hilb$, it suffices that $C$ contains  the regular representation of $G$ on $L^2(G,\mu)$ where $\mu$ is the left Haar measure on $G$.

Table of contents

$1.$ $\ast$-categories, $\ast$-functors

Here monoidal categories, monoidal functors and their morphisms are reviewed and (antilinear) $\ast$-involutions on $\mathbb{C}$-linear monoidal categories. (Small)  monoidal $\ast$-categories form a category $Cat^\ast$. For any (monoidal) $\ast$-category, and any group $G$, the category $Rep^G(C)$ of representation of $G$ in objects in $C$ has a structure of a (monoidal) $\ast$-category $Rep^G(C)^{\tilde\ast}$ and the forgetful functor $Rep^G(C)\to C$ is a strict monoidal functor of $\ast$-categories.

$2.$ Duality construction

$2.1$ The case of abstract groups

Here a basic pair of adjoint functors is defined $\mathcal{R}_{C\tilde{}^\ast}\dashv Aut(C\tilde{}^\ast)$ where $Aut(C\tilde{}^\ast): Cat^\ast/C\tilde{}^\ast\to Group$ gives the group of automorphisms of a monoidal $\ast$-functor with codomain $C\tilde{}^\ast$ and its left adjoint is $\mathcal{R}_{C\tilde{}^\ast} : G\mapsto Rep^G(C)\tilde{}^\ast$.

$2.2$ The case of topological groups

Here one supposes that $C$ is enriched over the category of topological (complex) vector spaces, so that $Aut(V)$ is a topological group for all $V\in Ob C$. The construction of an adjoint pair of functors above works with the category $Group$ replaced by $TopGroup$. However, in the representation theory one assumes only that the representations are continuous in the [[weak operator topology]]; and the group of invertibles in $End V$ in the enriched sense is not always a topological group in that topology. Instead one works with the category of groups with a topology in which only left and right translations are required to be continuous.

$2.3$ Reflexivity

Reflexive monoidal categories are defined: they have an involution $\sigma$ and functorial isomorphisms $\sigma(X)\otimes X\to Id$ for all objects $X$. For example, rigid symmetric monodial categories and rigid braided monoidal categories are in that class; the category of Hilbert spaces and if a category is reflexive than the category of representations of a group is reflexive.

$2.4$ The reconstruction problem

From now on a full subcategory $C\tilde{}$ of the category of reflexive locally convex topological complex vector spaces is fixed with a real structure; it is equipped with a monoidal product such that the forgetful functor to vector spaces is monoidal with help of maps $F(V)\otimes_{\mathbb{C}} F(W)\to F(V\otimes W)$ which are injective and with dense image in weak topology on the tensor product. Given a topological group $G$, one chooses a full (monodial) $\ast$-subcategory $\mathfrak{C}\tilde{}$ of the $\ast$-category of continuous uniformly bounded representations of $G$ in $C\tilde{}$. The group of automorphisms of the forgetful functor $F_{\mathfrak{C}} : \mathfrak{C}{\tilde{}}\to Vec$ is equipped with weak operator topology and the canonical morphism $G\to Aut^\ast(F_{\mathfrak{C}})$ is continuous; however $Aut^\ast(F_{\mathfrak{C}})$
is not a topological group in general. The reconstruction problem is, under which conditions on $\mathfrak{C}{\tilde{}}$, the canonical isomorphism is an isomorphism of topological groups.

$3.$ The algebra of matrix elements

One considers the set $A(\mathfrak{C}\tilde{})$
of matrix elements of representations belonging to the subcategory $A(\mathfrak{C}\tilde{})$. It appears to be a subalgebra of the Banach $\ast$-algebra of complex valued continuous functions on $G$, and is closed with respect to involution $\sigma: f\to f\circ (-)^{-1}$, $f^\sigma(x) = f(x^{-1})$, closed under complex conjugation (which is an anti-involution) and an $M(G)$-submodule. It is proved that each elements of $Aut^\ast(F_{\mathfrak{C}\tilde{}})$ determine an automorphism of the algebra with involution $A(\mathfrak{C})$.

subset  of the algebra of bounded

$4.$ Conditions of continuity

Here certain lemmas on certain category $B\mathfrak{A}$ of algebras with involutions and anti-involutions is proved.

$5.$ Reconstruction theorems

Among several theorems, the theorem 5.3 states that if $G$ is locally compact then a subcategory $\mathfrak{C}\tilde{}$ then for a $\ast$-category of uniformly bounded continuous representations the canonical map $G\to Aut^\ast(F_{\mathfrak{C}\tilde{}})$
to be an isomorphism of topological groups it is sufficient that the following simultaneously hold:

* it separates the elements of $G$

* the algebra $A(\mathfrak{C})$ of matrix elements has nonzero intersection with $L^p(G,\mu)$ for some $p\gt 0$, where $\mu$ is a left-invariant measure on $G$

Classical corollaries are also shown. For example, for Tannaka-Krein one notes that all matrix elements of a compact group are in $L^1(G,\mu)$, hence only the separation property is needed. For Tatsuuma duality one notes that the matrix elements of the regular representation are convolutions of functions from $L^2(G,\mu)$ and uses stronger theorem 5.1 from the paper. Pontrjagin duality is a known corollary of Tatsuuma's duality theorem.


reconstruction theorem

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#### Category theory
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[[!include category theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents
{:toc}

## Idea

Sometimes for [[categories]] having some fixed [[extra property]] and/or [[extra structure]], one can produce a recipe which gives (up to suitable equivalence) all the examples (and nothing else).

There are several typical classes of such *reconstruction theorems* (which are all to some extent related).

## Tannakian reconstruction theorems

These theorems reconstruct an algebraic symmetry object ([[group]], [[groupoid]], [[gerbe]], [[Hopf algebra]], Hopf algebroid) from the [[monoidal category]] of [[representation]]s of that object (typically rigid and [[symmetric monoidal category|symmetric]] or [[braided monoidal category|braided]]). The correspondence between the symmetry object and the corresponding [[category of representations]] is called **[[Tannaka duality]]**.

Examples include the classical Tannaka theorem and Krein theorem, [[Doplicher-Roberts reconstruction theorem]] in physics, [[Deligne's theorem on tensor categories]], Woronowicz's Tannaka duality for compact matrix pseudogroups, Saavedra-Rivano and Deligne reconstruction theorems for neutral and mixed Tannakian categories, Ulrich's reconstruction theorem, reconstruction theorems of Majid, Nori Tannakian theorem, Grothendieck's version of Galois group in algebraic geometry and so on. The notion of the [[fiber functor]] (due to Grothendieck) is central to these considerations.

## Reconstruction theorems for schemes

These theorems for schemes (or varieties only) reconstruct a [[scheme]] (variety) out of the category of quasicoherent or only coherent sheaves (or a [[derived category]] version of them). In that class one can find the Gabriel--Rosenberg theorem, the [[Bondal-Orlov reconstruction theorem]], the reconstruction theorems of [P. Balmer](http://www.math.ucla.edu/~balmer) and of [G. Garkusha](http://www-maths.swan.ac.uk/staff/gg), and so on. There is also a class of reconstructions where for some derived categories a realization as derived categories of representation of quivers can be reconstructed.

There is a large class of __abelian reconstruction theorems__, for example the [[Gabriel-Popescu theorem]]. In _topos theory_ the __[[Giraud theorem]]__ is also a reconstruction theorem (of a [[site]] out of a [[topos]], though a nonuniqueness of the resulting site is involved, not affecting cohomology, hence, according to Grothendieck, nonessential).

Examples:

- [[Freyd-Mitchell embedding theorem]]
- [[Gabriel-Popescu embedding theorem]]
- [[Quillen-Gabriel embedding theorem]]

## Gabriel-Ulmer duality

[[Gabriel-Ulmer duality]] is an equivalence of 2-categories LFP of locally finitely presentable categories and Lex of finitely complete categories. It is related to syntax-semantics adjunction and to Tannaka type reconstruction for coalgebra-like objects, with which has a common generalization (enriched Tannaka duality of Day).

Gabriel–Ulmer duality has a generalisation to [[sound doctrines]]. For [[sifted colimits]] rather than [[filtered colimits]], this gives [[Lawvere's reconstruction theorem]].

## Heuristics

Typically in the proofs of most reconstruction theorems an implicit use of the  Yoneda lemma is involved. Various embedding theorems of classes of categories (as well as theorems on realization as quotient categories) are closely related, e.g. [[Barr embedding theorem]] and [[Freyd-Mitchell embedding theorem]].

Many of these examples are corollaries of the theory of _lex colimits_.

## References

* [[André Joyal]], [[Ross Street]], [An introduction to Tannaka duality and quantum groups](http://www.maths.mq.edu.au/~street/CT90Como.pdf), in Part II of _Category Theory, Proceedings, Como 1990_, eds. A. Carboni, M. C. Pedicchio and G. Rosolini,  Lec. Notes in Mathematics __1488__, Springer, Berlin, 1991, pp. 411–492 [doi:10.1007/BFb0084207](http://dx.doi.org/10.1007/BFb0084207).

* P. Deligne, [[Catégories Tannakiennes]], [[Grothendieck Festschrift]], vol. II,  Birkhäuser Progress in Math. 87 (1990) pp.111--195.

* Alexander L. Rosenberg, _The existence of fiber functors_, in 'The Gelfand Mathematical Seminars 1996--1999', pp. 145--154. Birkhäuser, Boston, MA, 2000.

* A. L. Rosenberg, [[Reconstruction of groups]], Selecta Math. N.S. 9:1 (2003) [doi](http://dx.doi.org/10.1007/s00029-003-0322-x)

* N. Saavedra Rivano, _Catégories Tannakiennes_, Springer LNM 265, 1972

* Bodo Pareigis, [Quantum groups and noncommutative geometry](http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html), WS 1999, chapter 3, online notes.

* S. Majid, _Foundations of quantum group theory_, chapter 9, Camb. Univ. Press 1995, 2002.

* S. Majid, _Tannaka--Krein theorem for quasiHopf algebras and other results_, Contemp. Math. 134 (1992) 219--232.

* A. L. Rosenberg, _Reconstruction of groups_, Selecta Math. N.S. __9__:1 (2003)[doi:10.1007/s00029-003-0322-x](http://dx.doi.org/10.1007/s00029-003-0322-x) (nlab remark: this paper is on a generalization of Tannaka--Krein and not of the Gabriel--Rosenberg kind of reconstruction)

* A. Rosenberg, _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, [MPIM1996-108](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948) (1996).

* [[A. L. Rosenberg]], _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945), _Underlying spaces of noncommutative schemes_, MPIM2003-111, [dvi](http://www.mpim-bonn.mpg.de/preblob/1947), [ps](http://www.mpim-bonn.mpg.de/preblob/1948)

* [[P. Gabriel]], [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France __90__ (1962), 323--448, [numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0)

* A. I. Bondal, [[D. O. Orlov]], _Reconstruction of a variety from the derived category and groups of autoequivalences_, Compos. Math. 125 (2001), 327–344 [doi:10.1023/A:1002470302976](http://dx.doi.org/10.1023/A:1002470302976)

*  [[K. Szlachányi]], _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_ [arxiv:0907.1578](http://arxiv.org/abs/0907.1578)

*  Phùng Hô Hai, _Tannaka--Krein duality for Hopf algebroids_, [arxiv:math.QA/0206113](http://arxiv.org/abs/math/0206113)

* Hélène Esnault, Phùng Hô Hai, _Gauß--Manin connection and Tannaka duality_, [math.AG/0509111](http://arxiv.org/abs/math/0509111)

* H. Pfeiffer, _Tannaka--Krein reconstruction and a characterization of modular tensor categories_, [arxiv:math.QA/0711.1402](http://arxiv.org/abs/0711.1402)

* [[S. L. Woronowicz]], _Tannaka--Kreĭn duality for compact matrix pseudogroups. Twisted $SU(N)$ groups_, Invent. Math. __93__ (1988), no. 1, 35–76

* [[Michael Müger]], _Abstract duality for symmetric tensor $*$-categories_ (App. to Hans Halvorson: 'Algebraic Quantum Field Theory'.) In: J. Butterfield and J. Earman (eds.): "Handbook of the Philosophy of Physics", p. 865--922. North Holland, 2007; [arxiv:math-ph/0602036](http://arxiv.org/abs/math-ph/0602036); cf. The String Coffee Table, [Müger on Doplicher--Roberts](http://golem.ph.utexas.edu/string/archives/000711.html)

* [[S. Doplicher]], J. E. Roberts, _A new duality theory for compact groups_, Inventiones Math., 98(1):157--218, 1989.

* [[P. Balmer]], _The spectrum of prime ideals in tensor triangulated categories_, J. Reine Angew. Math. __588__ (2005), pp. 149--168 [dvi](http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.dvi) [pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.pdf) [ps](http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.ps).

* [M. Prest](http://www.maths.man.ac.uk/~mprest), [[G. Garkusha]], _Reconstructing projective schemes from Serre subcategories_, J. Algebra __319__ (3) (2008), 1132--1153 ([pdf](http://www-maths.swan.ac.uk/staff/gg/papers/garkpr44.pdf)).

* [[P. Deligne]], J. S. Milne, _Tannakian categories_, Lect. notes in math. 900, 101–228, Springer 1982.

* A. Bruguières, _On a tannakian theorem due to Nori_, [pdf](http://imag.umontpellier.fr/~bruguieres/docs/ntan.pdf); _Théorie tannakienne non commutative_, Comm. Algebra __22__, 5817–5860, 1994

Lex colimits are discussed in:

* [[Richard Garner]] and [[Stephen Lack]]. _Lex colimits_. Journal of Pure and Applied Algebra 216.6 (2012): 1372-1396.

[[!redirects reconstruction theorems]]
[[!redirects reconstruction]]

[[!redirects Tannakian reconstruction]]
[[!redirects Tannakian reconstruction theorem]]
[[!redirects Tannakian reconstruction theorems]]


rectangular band

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#### Algebra
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#Contents#
* table of contents
{:toc}

## Idea

... [[idempotent]] [[semigroup]] ...

## Definition

Recall that a _[[band]]_ is a [[semigroup]] in which every [[element]] is [[idempotent]].

[[commutative magma|Commutative]] bands are usually known as _[[semilattices]]_. This is the semigroup-theoretic definition, but there is also an [[order theory|order theoretic]] definition: given a semilattice $L$ in this semigroup-theoretic sense, it has a canonical [[partial order]] given by $e\precreq f$ when $e f = e$. So semilattices are also [[posets]].

Finitely generated bands are finite: see [Howie 76, Section IV.4](#Howie76).

Now a _rectangular band_ may be described as a [[semigroup]] satisfying the identity $a b a = a$ for all elements $a$ and $b$.

## Properties

* A rectangular band is indeed a [[band]] since the defining identity implies
$ x y z = x z$ for all $x,y,z$ whence by taking $y=z=x$ one gets $x x x = x x$ and from the defining identity $x x x=x$ hence $x x = x$. In order to get the first equation expand $x y z$ by substituting $ x z x$ for $x$: $x y z = (x z x) y z = x (z (xy) z) = x z \; .$

* If $S$ is a rectangular band, then there exist [[inhabited set|non-empty sets]] $I$ and $J$ such that $S$ is [[isomorphism|isomorphic]] as a [[semigroup]] to $I\times J$ equipped with the [[multiplication]]
$(i, j)(p,q) = (i,q)$ for $i,p\in I$ and $j,q\in J$.

* Every [[band]] $S$ has a decomposition as a [[disjoint union]] $\coprod_{x\in L} R_x$ where $L$ is a semilattice, each $R_x$ is a [[subobject|sub]]-semigroup that is a rectangular band, and $R_x R_y \subseteq R_{x y}$ for every $x$ and $y$. This is a bit weaker than saying we have a [[functor]] from the [[poset]] $L$ to the [[category]] of rectangular bands, because we lack connecting morphisms $R_x \to R_y$.

## The category of rectangular bands

Let $Rect$ be the category of rectangular bands with semigroup homomorphisms as morphisms.

* Since rectangular bands are an equationally defined subclass of the class of all semigroups, $Rect$ is a subvariety of the variety $SGr$ of semigroups and hence enjoys all the usual (co)completeness properties of a variety.

* Since $Rect$ is the 2-valued [[collapsed topos|collapse]] of the [[topos]] $Set\times Set$ it is even a [[cartesian closed category|cartesian closed]] variety. Since the [[distributive law]] holds for (finite) coproducts in cartesian closed categories, $Rect$ is a [[distributive category]]. Since it is not [[locally cartesian closed category|locally cartesian closed]] it is neither a topos nor even  an [[extensive category]]. For more on this see Johnstone ([1990](#Johnstone90)).

## Some ramifications

* A band $S$ satisfying the [[graphic category| graphic identity]] $x y x = x y$ for all $x$ and $y$ is said to be _left-regular_. Left-regular bands can arise from hyperplane arrangements and there has been work studying [[random walk|random walks]] on these hyperplane arrangements by analysing the semigroup algebras of the associated bands: see [Brown 00](#Brown00) and [Margolis-Saliola-Steinberg 15](#MargolisSaliolaSteinberg15). Left-regular band monoids are also called _graphic monoids_ which are examples of 1-object [[graphic category|graphic categories]].

## Related concepts

* [[distributive category]]

* [[reader monad]]

* [[graphic category|graphic monoid]]

* [[collapsed topos]]

## References

* {#Howie76} J. Howie, _An introduction to semigroup theory_, Academic Press 1976.

* {#Brown00} K. S. Brown, _Semigroups, Semirings, and Markov Chains_, J. Theor. Prob. **13** no.3 (2000) pp.871-938. ([arXiv:math/0006145](https://arxiv.org/abs/math/0006145))

* {#Johnstone90}[[Peter Johnstone]], _Collapsed toposes and cartesian closed varieties_ , JPAA **129** (1990) pp.446-480.

* N. Kimura, _The structure of idempotent semigroups I_ , Pacific Journal of Mathematics **8** no.2 (1958) pp.257-275. ([pdf](http://msp.org/pjm/1958/8-2/pjm-v8-n2-p07-p.pdf))

* {#MargolisSaliolaSteinberg15}  Stuart Margolis, Franco Saliola, [[Benjamin Steinberg]], _Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry_ ([arXiv:1508.05446](https://arxiv.org/abs/1508.05446))

[[!redirects rectangular bands]]

[[!redirects band]]
[[!redirects bands]]


rectifiable subset

* wikipedia [rectifiable set](http://en.wikipedia.org/wiki/Rectifiable_set)

Related entries: [[geometric measure theory]].

[[!redirects rectifiable set]]
[[!redirects rectifiable sets]]
[[!redirects rectifiable subset]]
[[!redirects rectifiable subsets]]
[[!redirects rectifiable subspace]]
[[!redirects rectifiable subspaces]]
[[!redirects rectifiability]]


rectification

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#Contents#
* table of contents
{:toc}

## Idea

In [[homotopy theory]] and in [[higher category theory]] by _rectification_ one (usually, often) means the equivalent presentation of a [[homotopy]]-[[coherence|coherent]] structure equivalently by apparently more naive "strict" data.

Many or most rectification statements are examples of the general _[Rectification theorem for algebras over an operad](model+structure+on+algebras+over+an+operad#RectificationOfAlgebras)_ which states sufficient conditions for the category of [[algebras over an operad]] to already be [[Quillen equivalence|Quillen equivalent]] to that of the corresponding [[∞-algebras over an (∞,1)-operad]].

This includes notably [[Vogt's theorem]] on the rectification of [[homotopy coherent diagrams]].

## Examples

* [[relation between L-infinity algebras and dg-Lie algebras|relation between $L_\infty$-algebras and dg-Lie algebras]]

## Related concepts

* [[coherence]]

* [[semi-strict infinity-category]]

* [[rectified infinity-stack]]

[[!redirects rectifications]]


rectified infinity-stack

An [[∞-stack]] on a [[(∞,1)-category]]-domain $C$ that happens to be an ordinary [[category]] (i.e. not a [[derived stack]]) is **[[rectification|rectified]]** if it is an ordinary [[functor]]
$
 C^{op} \to  $ [[SSet]] instead of a general [[(∞,1)-functor]] (i.e. [[pseudofunctor]]).

A central theorem about the [[model structure on simplicial presheaves]] says that rectified [[∞-stack]]s are _sufficient_: they already [[presentable (∞,1)-category|present]] the full [[(∞,1)-category of (∞,1)-sheaves]] (= [[∞-stacks]]).

Notice that, by a result recalled at [[descent for simplicial presheaves]], a rectified $\infty$-stack $A$ is an [[∞-groupoid]] [[internalization|internal to]] (pre)sheaves satisfying a [[descent]] condition.

#Examples#

...

[[!redirects rectified infinity-stacks]]
[[!redirects rectified ∞-stack]]
[[!redirects rectified ∞-stacks]]


Rector completion theorem

#Contents#
* table of contents
{:toc}

## Idea

[Rector 74](#Rector74) proved the analog of the [[Atiyah-Segal completion theorem]] with [[topological K-theory]] replaced by the [[algebraic K-theory]] $K \mathbb{F}_p$ of a [[prime field]] $\mathbb{F}_p$.

This theorem further amplifies the analogy to the [[Segal-Carlsson completion theorem]] for [[stable cohomotopy]] and the interpretation of the latter as the algebraic K-theory over the "[[field with one element]]" ([Guillot 06](#Guillot06)):

[[!include Segal completion -- table]]

## References

The result is due to

* {#Rector74} [[David L. Rector]], _Modular characters and K-theory with coefficients in a finite field_, Journal of Pure and Applied Algebra **4** 2 (1974) 137-158 $[$doi:10.1016/0022-4049(74)90019-X$]$

Its interpretation as amplifying the identification of [[stable cohomotopy]] with the algebraic K-theory of the "[[field with one element]]" is highlighted in

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](https://arxiv.org/abs/math/0612327))

[[!redirects Rector completion theorems]]


recursion

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#### Induction
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# Recursion
* table of contents
{: toc}

## Idea

The traditional notion of _recursion_ over the [[natural numbers]] $\mathbb{N}$ is a way of defining a [[function]] out of $\mathbb{N}$ by specifying the image of *[[zero|$0$]]* (the "initial value") along with a way to obtain each *[[successor|successive]]* value from the previous one(s).
The study of functions on the [[natural numbers]] which can or cannot be defined using only recursion -- *[[recursive functions]]* -- is the topic of [[computability theory]] and has deep connections with the [[formal logic]] of [[Peano arithmetic]].

More generally, recursion is a way of defining a function on any mathematical object which is "defined inductively" (in a way analogous to how the [[natural numbers]] are characterized by [[zero]] and [[successor]]).  In place of the "initial value" and "successor step", a general definition by recursion consists of giving one "clause" for each "constructor" of the inductively defined object.

Recursion is formalized in [[type theory]] by the notion of _[[inductive type]]_ (and the corresponding [[elimination rule]])  and, equivalently, in [[category theory]] by the notion of _[[initial algebra of an endofunctor]]_.  For $F$ an [[endofunctor]], a [[morphism]] of the form $F(X) \to X$ determines a collection of _constructors_ and the _recursion principle_ is the statement that there is a (unique) morphism $f : A \to X$ from the [[initial object|initial]] such structure $F(A) \to A$.  This $f$ is the corresponding _recursively defined function_.

Viewed from just a slightly different angle, this state of affairs is the _[[induction principle]]_ on non-[[dependent types]].

## Examples

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

In the theory of [[Peano arithmetic]], we define $x + y$ recursively in terms of the [[successor]] operation $s$ as follows:

1. $x + 0 = x$

2. $x + s(y) = s(x + y)$

The definition above is taken as an [[axiom]] in [[Peano arithmetic]].

More generally, given a (parametrizable) [[natural numbers object]] $\mathbb{N}$, its universal property guarantees that there is a unique (!) map $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ with the above properties.
=--

## Recursion theorem

### In general

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

Let $X$, $Y$, and $Z$ be [[set|sets]], and suppose $\rightsquigarrow$ is a [[well-founded relation]] on $X$. Let $h\colon X \times Y \times \mathcal{P}(Z) \to Z$ be a given function. Then, there is a unique function $f\colon X \to Z$ satisfying
$$f (x', y) = h (x', y, S)$$
for all $y$ in $Y$, where
$$S = \{ z \in Z : \exists x . \, ( x \rightsquigarrow x' \wedge z = f (x, y) ) \}$$
=--

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

By the principle of well-founded induction. (Details to be added.)
=--

### For a natural numbers object

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

Let $\mathbb{N}$ be a (parametrizable) [[natural numbers object]] in a [[cartesian monoidal category|category with finite products]], with zero $0\colon 1 \to \mathbb{N}$ and successor $s\colon \mathbb{N} \to \mathbb{N}$. For any morphisms $f_0\colon Y \to Z$ and $h\colon \mathbb{N} \times Y \times Z \to Z$, there is a unique morphism $f\colon \mathbb{N} \times Y \to Z$ such that
$$f(0, -) = f_0$$
and
$$f(s(x), y) = h(s(x), y, f(x, y))$$
where $x\colon \mathbb{N} \times Y \to \mathbb{N}$ is the first projection and $y\colon \mathbb{N} \times Y \to Y$ is the second projection.
=--

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

Recall that the universal property for $\mathbb{N}$ states that for data $g_0\colon Y \to A$, $k\colon Y \times A \to A$, there is a unique morphism $u\colon \mathbb{N} \times Y \to A$ such that $u \circ (0 \times \mathrm{id}_Y) = g_0$ and $u \circ (s \times \mathrm{id}_Y) = k \circ u$. We take $A = \mathbb{N} \times Z$, $g_0 = 0 \times f_0$, and $k(y, n, z) = \langle n, h(n, y, z) \rangle$, where $n\colon Y \times \mathbb{N} \times Z \to \mathbb{N}$, $y\colon Y \times \mathbb{N} \times Z \to Y$, and $z\colon Y \times \mathbb{N} \times Z \to Z$ are the obvious projections. The $f$ we seek is then obtained as $p_2 \circ u$, where $p_2 : \mathbb{N} \times Z \to Z$ is the second projection.
=--

## Related concepts

* [[recursive mathematics]]

* [[recursive function]]

* [[partial recursive function]], [[recursive subset]]

* [[guarded recursion]]

Dually, there is a notion of [[corecursion]] on a [[coinduction|coinductive structure]].

[[!include computable mathematics -- table]]

## References

See also:

* Wikipedia, Recursion (computer science)

Discussion in [[type theory]]:

* [[Frank Pfenning]], *Recursive types*, Section 4.6 in:  *Linear Logic* (1998) [[pdf](https://www.cs.cmu.edu/~fp/courses/98-linear/handouts/notes.pdf), [webpage](https://www.cs.cmu.edu/~fp/courses/98-linear/handouts.html), [[Pfenning-LinearLogic98.pdf:file]]]

and in view of [[homotopy type theory]]:

* [[Univalent Foundations Project]], §1.10 *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects recursion]]
[[!redirects recursive]]
[[!redirects recursive definition]]
[[!redirects recursive definitions]]

[[!redirects recursion principle]]


recursion scheme

[[!redirects recursion schemes]]

#Contents#
* table of contents
{:toc}

## Idea

It is a well-known fact in [[computability]] that it is not possible to write a decision procedure that tells whether another program is terminating or not. This fact is know as the [[halting theorem]].

For this reason most [[functional programming]] languages are endowed with an unrestricted [[fixed-point operator]] at all [[types]], which simplifies the implementation of the language, but allows the user to write functions that are not-well defined mathematically.

Inspired by [[category theory]], a _recursion scheme_ is a structured [[recursion]] operator that ensures that the definitions of [[recursive]] functions are well-defined.

For example,

1. A [[catamorphism]] (a.k.a. a fold) is the unique $F$-algebra homomorphism from an [[initial algebra]] for the functor $F$.

2. An [[anamorphism]] (a.k.a.  unfold) is the unique $G$-coalgebra homomorphism into the [[final coalgebra]] for the functor $G$.

3. An [[hylomorphism]] corresponds to the idea of a recursive coalgebra or [[well-founded coalgebra]]. In words, a recursive $F$-coalgebra $(c, X)$ has a unique mapping $f : X \to Y$ into any other $F$-algebra $(Y, a)$ such that $f = a \circ F(f) \circ c$.

A hylomorphism gives a simple example of an opportunity for optimization: an anamorphism builds up a tree structure and a catamorphism consumes it, so if this hylomorphism structure is recognized in the program, then an implementation need not build the intermediate tree, and so the [[computational complexity|space usage]] of the program can be reduced.

## Recursion Schemes via Adjunctions

Catamorphisms and their variants are basic and rather restricted forms of recursion, i.e. where the recursive call is called on smaller input.

Many other patterns of recursion exists that are equally well-defined, but they cannot be expressed *directly* in terms of catamorphisms. However, most recursion schemes are adjoint folds ([Hinze, Wu and Gibbons, 2013](#HinzeWG13)), i.e. every complex recursion scheme can be reduced to a basic one via an adjunction. Moreover,  *every* recursion scheme is an instance of a conjugate hylomorphism ([Hinze, Wu and Gibbons, 2015](#HinzeWG15)) .

A comprehensive list of recursion schemes can be found in [Yang and Wu, 2022](#YangW22)

## References

* Erik Meijer, Maarten M. Fokkinga, Ross Paterson, _Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire_, FPCA 1991, ([doi](https://doi.org/10.1007/3540543961_7))

* {#HinzeWG13} Ralf Hinze, Nicolas Wu and Jeremy Gibbons, _Unifying structured recursion schemes, ACM SIGPLAN International Conference on Functional Programming, ICFP'13, [doi](https://doi.org/10.1145/2500365.2500578)

* {#HinzeW16} Ralf Hinze, Nicolas Wu, _ Unifying structured recursion schemes - An Extended Study, J. Funct. Program., 2016, [doi](https://doi.org/10.1017/S0956796815000258)

* {#HinzeWG15} Ralf Hinze, Nicolas Wu and Jeremy Gibbons, Conjugate Hylomorphisms - Or: The Mother of All Structured Recursion Schemes, Symposium on Principles of Programming Languages, POPL 2015, [doi](https://doi.org/10.1145/2676726.2676989)

* {#YangW22} Zhixuan Yang and Nicolas Wu, Fantastic Morphisms and Where to Find Them - A Guide to Recursion Schemes, Mathematics of Program Construction, MPC 2022, [doi](https://doi.org/10.1007/978-3-031-16912-0_9)


recursive function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A recursive function is any [[function]] defined by [[recursion]] (i.e. non-dependent [[induction]]).

Recursive functions could be contrasted with inductively defined dependent functions, where one uses terms of a general [[dependent product type]] rather than a [[function type]]. Alternatively, they could be considered as a special case of an inductively defined dependent function where the target type family does not actually depend upon the source.

Similar to the case with induction, recursive functions could be generalized to *higher recursive functions*, where functions are recursively defined not only on the [[term]] constructors, but also the [[identity]] constructors of an [[higher inductive type]].

Recursive functions defined on the [[natural numbers]] are particularly important for [[computability]]. For more see also at *[[partial recursive function]]*.

## Example

\begin{example}
The [[negation]] function

$$
  \neg \colon \mathrm{Bool} \to \mathrm{Bool}
$$

on the [[booleans type]] is recursively defined by sending  [[true|$\top$]] to [[false|$\bot$]] and [[false|$\bot$]] to [[true|$\top$]]:

$$
  \begin{array}{l}
    \neg \top =_{\mathrm{Bool}} \bot
    \\
    \neg \bot =_{\mathrm{Bool}} \top
  \end{array}
$$

\end{example}

\begin{example}
The double function

$$
  \mathrm{double} \colon \mathbb{N} \to \mathbb{N}
$$

on the [[natural numbers type]] is recursively defined by sending $0$ to $0$ and $s(n)$ to $s(s(\mathrm{double}(n)))$:

$$
  \begin{array}{l}
    \mathrm{double}(0) =_{\mathbb{N}} 0
    \\
    n:\mathbb{N} \vdash \mathrm{double}(s(n)) =_{\mathbb{N}} s(s(\mathrm{double}(n)))
  \end{array}
$$

\end{example}

\begin{example}
This last example is an example of a *higher recursive function*. Given a type $A$, elements $a:A$ and $b:A$, and identity $q:a =_A b$, there is a higher recursive function $f:\mathbb{I} \to A$ from an [[interval type]] $\mathbb{I}$ to $A$, recursively defined by

$$
  \begin{array}{l}
    \beta_f^0:f(0) =_A a
    \\
    \beta_f^1:f(1) =_A b
    \\
    \beta_f^p:\mathrm{ap}_f(p) =_{f(0) =_A f(1)} \mathrm{trans}_{f(0), b, f(1)}(\mathrm{trans}_{f(0), a, b}(\beta_f^0, q), \mathrm{sym}_{f(1), b}(\beta_f^1))
  \end{array}
$$

where $\mathrm{ap}_f:(0 =_\mathbb{I} 1) \to (f(0) =_A f(1))$ is the [[function application to identities]], $\mathrm{trans}_{a, b, c}:(a =_A b) \times (b =_A c) \to (a =_A c)$ is [[transitivity]], and $\mathrm{sym}_{a, b}:(a =_A b) \to (b =_A a)$ is [[symmetry]] of identities.
\end{example}

## Related concepts

* [[function]]

* [[recursion]]

* [[partial recursive function]]

* [[proof theory]]

## References

See also:

* Wikipedia, *[Recursive function](https://en.m.wikipedia.org/wiki/Recursive_function)

[[!redirects recursive functions]]


recursive mathematics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Related concepts

* [[recursion]]

* [[partial recursive function]]

[[!include computable mathematics -- table]]

## References

* _Handbook of recursive mathematics_ ([pdf](http://home.gwu.edu/~harizanv/HandbookCurrent.pdf))

* [[Pieter Hofstra]], [[Philip Scott]], _Aspects of categorical recursion theory_, ([arXiv:2001.05778](https://arxiv.org/abs/2001.05778))


recursive set

## References

* Wikipedia, _[Recursive set](http://en.wikipedia.org/wiki/Recursive_set)_

[[!redirects recursive sets]]


recursive subset

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[concrete set|set]] $S$ of [[natural numbers]] is **recursive** if there is an [[algorithm]] which will decide in finitely many steps whether a given natural number belongs to $S$.

## Definition

A [[subset]] $S$ of the set of [[natural numbers]] $\mathbb{N}$, or more generally of $\mathbb{N}^k$ with $k$ finite, is _recursive_ if there is a [[computable function]] (a [[total function|total]] [[partial recursive function|recursive]] function) $f: \mathbb{N}^k \to \mathbf{2} = \{0, 1\} \subseteq \mathbb{N}$ such that $S = f^{-1}(1)$. Recursive subsets are a proper subclass of the class of _[[recursively enumerable set|recursively enumerable]]_ sets, which are [[domains]] of [[partial recursive functions]] $f: \mathbb{N}^k \to \mathbb{N}$, or equivalently [[images]] of total recursive functions.

## Properties

Recursive sets form a [[Boolean algebra|Boolean subalgebra]] of the [[power set]] algebra $P(\mathbb{N})$ (whereas recursively enumerable subsets do not form a Boolean subalgebra).  Even in [[constructive mathematics]] (where the power set algebra may be only a [[Heyting algebra]]), the recursive sets form a Boolean algebra (a Boolean subalgebra of the algebra of [[decidable subsets]]).

## Examples and counterexamples

* One can encode [[proofs]] in the formal theory $PA$ ([[Peano arithmetic]]) as natural numbers, via a process of [[Gödel-numbering]]. The set of codes of such formal proofs is a recursive set. In colloquial language: it is possible to program a computer to detect whether or not a string of symbols represents a valid proof in $PA$.

* It follows that the set of codes of [[theorems]] (provable propositions) is recursively enumerable. However, it is **not** recursive. This is one way of saying that provability in PA cannot be decided by an algorithm, which is closely related to Gödel's [[incompleteness theorem]]s.

## Related concepts

* [[definable set]], [[constructible set]]

[[!redirects recursive subset]]
[[!redirects recursive subsets]]
[[!redirects recursive set]]
[[!redirects recursive sets]]


red herring principle

# The red herring principle
* table of contents
{: toc}

## Idea

The mathematical **red herring principle** is the principle that in [[mathematics]], a "red herring" need not, in general, be either red or a herring.

Frequently, in fact, it is conversely true that all herrings are red herrings.  This often leads to mathematicians speaking of "non-red herrings," and sometimes even to a redefinition of "herring" to include both the red and non-red versions.

## Notes

* Clearly, the term "red herring" here is not to be confused with the usual meaning, which typically refers to a deliberate attempt to divert or throw one off track, as for example a rhetorical tactic for this purpose, or a novelistic device.

* It should also not be thought that "red herring" as used here signals a pejorative, indicating for example ineptitude or lack of care in naming. For example, it may be that "foo" once meant a "bar", but over time the meaning of "foo" changed, so that while "weak $*$-foo" might seem strange if "foo" is taken in the modern sense, it made much better sense under the older (but mostly forgotten) meaning of "foo".

Thus, "red herring" as used here is to be interpreted neutrally: it refers to a name of a concept which might throw the reader off-track, by accident as it were.

## Examples

* A [[manifold with boundary]] is not a [[manifold]].  This leads to the use of "manifold without boundary."

* In [[linear algebra]], an [[associative algebra|algebra]] is usually defined to be associative, so a [[nonassociative algebra]] is not an algebra in this sense.  This leads to the technically redundant use of "associative algebra".  Furthermore, a "nonassociative algebra" *might* happen to be associative; it just doesn\'t *have* to be!

* Similarly, [[noncommutative geometry]] is really about _not necessarily commutative_ or _possibly non-commutative_ geometry. In fact, many tools developed in "non-commutative geometry" are also useful tools in ordinary commutative geometry.

* A [[star-autonomous category|*-autonomous category]] is not an [[autonomous category]], but the reverse is almost true: a *symmetric* autonomous (a.k.a. [[compact closed category|compact closed]]) category is a fairly special case of a $*$-autonomous one.  This red herring is apparently an accident of history: at the time $*$-autonomous categories were invented, "autonomous category" was sometimes used to mean a [[closed monoidal category]], of which $*$-autonomous categories are indeed a special case, but nowadays that usage has mostly disappeared.

* A [[linearly distributive category]] has essentially nothing to do with a [[distributive category]].

* If a [[localizer]] refers to a class of maps in a [[presheaf category]], as it sometimes does, then a [[basic localizer]], being a class of maps in [[Cat]], is not a localizer.

* A [[multivalued function]] is not a [[function]], but a function is a special case of a multivalued function.

* A [[planar ternary ring]] is not a [[ring]].  A general ring is not a planar ternary ring either; their "intersection" is the class of [[division rings]].

Some adjectives are almost universally used as "red herring adjectives," i.e. placing that adjective in front of something makes it *more* general in some way.  Some red herring adjectives almost always have the same meaning, such as "pseudo" and "lax," but others, such as "weak," have different meanings in different contexts.

* A [[weak factorization system]] is not a factorization system, as originally defined, but rather the reverse.  This has led some people to the use of [[orthogonal factorization system]] in place of the classical term "factorization system."

* A [[weak limit]] is not a [[limit]] in a similar way: it satisfies only existence but not uniqueness.

* Originally, [[n-category]] referred only to the strict version, so that a "weak $n$-category" was not an $n$-category.  Nowadays some people (including many authors of the nLab)  are trying to reverse this, so that $n$-category means the weak version, while the strict version needs an adjective added.  Likewise for related concepts such as [[2-limit]] and [[strict 2-limit]].

Sometimes a red herring is only a red herring in degenerate cases, perhaps only one case:

* Fixing a [[natural number]] $n$, a [[homogeneous polynomial]] of degree $n$ is almost always a [[polynomial]] of degree $n$, but there is one exception: the [[zero]] polynomial has no degree (or is degree $-1$ or degree $-\infty$, depending on conventions), yet it is homogeneous of any degree.

## Non-examples

Some uses of terminology are similar in some ways, but don't quite fall under the same category.  For instance, in a number of cases mathematicians working in a particular field tend to omit [[dichotomy between nice objects and nice categories|niceness]] adjectives, e.g.:

* The generally accepted *definition* of a [[ring]] does not include commutativity, but in [[algebraic geometry]] and [[commutative algebra]] it is an almost universal convention that all rings are taken to be commutative.  However, almost every book concerning the subject announces its conventions early on.

* Algebraic topologists usually work with a [[convenient category of topological spaces]] rather than the category of all [[topological spaces]], but usually they simply say "space" to mean an object of their chosen convenient category.  Some algebraic topologists (particularly of the MIT school) take this even further and say "space" to mean [[simplicial set]]. In the Chicago school, this is occasionally stretched even further, with "space" meaning [[spectrum]] (although this is arguably more of a joke than a common usage).

These terminological uses can create situations that appear similar to actual red herrings, such as the use of "noncommutative ring" by people who are familiar with using "ring" to mean "commutative ring."  However, since the actual definitions of terms like "ring" and "topological space" is generally accepted to be unchanged (as opposed to the commonly used abbreviations), these are not true red herrings.

### Semi-non-examples

Sometimes, adding an adjective does not destroy the property that the noun has, but the noun now has its property in a different way, e.g.:

* A [[contravariant functor]] from $C$ to $D$ _is_ still a functor, however not from $C$ to $D$, but from $C^{op}$ to $D$ (or equivalently from $C$ to $D^{op}$).

[[!redirects red herring]]
[[!redirects red herring principle]]
[[!redirects the red herring principle]]