exchange law

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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
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## Idea

In [[higher category theory]], the _exchange law_, or _interchange law_, states that the multiple ways of forming the composite of a [[pasting diagram]] of [[k-morphism]]s are equivalent.

## Examples

The first exchange law (often called _the_ exchange law) asserts that for composition of [[2-morphisms]] we have an equivalence

$$
  \array{
    && 1 &&&& 3
    \\
    & \nearrow &\Uparrow& \searrow && \nearrow && \searrow
    \\
    0 &&\to&& 2 &&  && 4
    \\
    &&  &&&& 3
    \\
    & && && \nearrow &\Uparrow& \searrow
    \\
    0 &&\to&& 2 && \to && 4
  }
  \;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;
  \array{
    && 1 &&&& 3
    \\
    & \nearrow && \searrow && \nearrow &\Uparrow& \searrow
    \\
    0 &&&& 2 && \to && 4
    \\
    && 1 &&&&
    \\
    & \nearrow &\Uparrow& \searrow &&  &&
    \\
    0 &&\to && 2 && \to && 4
  }
$$

asserting a compatibility of [[horizontal composition]] and [[vertical composition]] of [[2-morphism]]s.

In a [[bicategory]] this equivalence is an identity. In even higher (and non-[[semi-strict n-category|semi-strict]]) category theory, the interchange law becomes a higher morphism itself: the _exchanger_.

## Combinatorics of exchange laws

One way to capture all exchange laws combinatorially is encoded by the cosimplicial $sSet$-category $S : \Delta \to sSet Cat$ that induces the [[homotopy coherent nerve]]. See there for more details on how this encodes the exchange laws.

## See also

* [[Eckmann-Hilton argument]]

* [[strict 2-category]]

[[!redirects exchange laws]]

[[!redirects interchange law]]
[[!redirects interchange laws]]

[[!redirects interchange axiom]]
[[!redirects interchange axioms]]

[[!redirects exchanger]]
[[!redirects exchangeor]]
[[!redirects exchangor]]
[[!redirects interchanger]]
[[!redirects interchangeor]]
[[!redirects interchangor]]


exchange rule

# The exchange rule
* table of contents
{: toc}

## Idea

In [[logic]], the _exchange rule_ or _permutation rule_ states that the order of the [[premise]]s is irrelevant to the validity of a [[deduction]].  Along with the [[weakening rule]] and the [[contraction rule]], it is one of the most commonly adopted [[structural rules]].

This rule is commonly discarded in [[linguistics|natural language parsers]], such as those built on [[categorial grammar]]s or similar, such as link-grammar. This is because in most natural languages, word-order matters; the proper "deduction" or parse of a sentence depends on the order of of "premises"/words.

## Statements

Exactly how this looks depends on the [[logic]] used.

...

## Related entries

* [[inference rule]]

* [[structural rule]]

* [[contraction rule]]

* [[weakening rule]]

## References

The notion of exchange as a [[structural inference rule]] originates (under the German name *Vertauschung*) with:

* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

On the [[categorical semantics]]:

* {#Jacobs98} [[Bart Jacobs]], p. 122 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

[[!redirects exchange rule]]
[[!redirects exchange rules]]
[[!redirects permutation rule]]
[[!redirects permutation rules]]


exchange structure

# Exchange structure
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## Definitions

Let $\mathcal{C}_{1}$, $\mathcal{C}_2$ be two [[categories]] with the same [[objects]].
Let $\mathcal{D}$ be a [[2-category]].
Let $F_{1}$ $\colon \mathcal{C}_{1} \rightarrow \mathcal{D}$ and $F_{2}\colon \mathcal{C}_{2} \rightarrow \mathcal{D}$ be [[2-functors]] such that $F_{1}(x) = F_{2}(x) = F(x)$ for all objects in $\mathcal{C}_1$ and $\mathcal{C}_2$. Let $\mathcal{E}$ be a fixed class of [[commutative squares]] that are preserved under [[horizontal composition|horizontal]] and [[vertical composition|vertical]] [[composition]] of the form:

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such that:
* $X, X', Y, Y'$ are objects in $\mathcal{C}_1$ and so are clearly in $\mathcal{C}_2$.
* $g$ and $g'$ are arrows of $\mathcal{C}_1$
* $f$ and $f'$ are arrows of $\mathcal{C}_2$.

Commutative squares satisfying these properties are called __mixed squares__.

An **exchange structure** with respect to $\mathcal{E}$ on a pair $(F_{1},F_{2})$ for all mixed squares $(C)$ in $\mathcal{E}$ is a 2-morphism e(C) of $\mathcal{D}$ (called a 2-morphism __associated with the exchange__ of mixed $(C)$):

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The direction of the 2-morphisms is constant (that is independent of the mixed square). This family of 2-morphisms must satisfy the following conditions:

* They must be compatible with the horizontal composition of mixed squares, that is for every [[horizontal composition]] of mixed squares $\mathcal{C}_1$, $\mathcal{C}_1$:

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  </g>
  <line id="svg_81238_193" y2="314" x2="182" y1="323" x1="173" stroke-width="4" stroke="#000000" fill="none"/>
  <line id="svg_81238_195" y2="332" x2="182" y1="323" x1="173" stroke-width="4" stroke="#000000" fill="none"/>
  <g id="svg_81238_198">
   <line id="svg_81238_196" y2="114" x2="182.5" y1="123" x1="173.5" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_197" y2="132" x2="182.5" y1="123" x1="173.5" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <g id="svg_81238_199">
   <line id="svg_81238_200" y2="114" x2="452.499972" y1="123" x1="461.499972" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_201" y2="132" x2="452.499972" y1="123" x1="461.499972" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <line id="svg_81238_202" y2="147" x2="169" y1="138" x1="180" stroke-width="4" stroke="#ffffff" fill="none"/>
  <g id="svg_81238_203">
   <line id="svg_81238_204" y2="314" x2="453.499972" y1="323" x1="462.499972" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_205" y2="332" x2="453.499972" y1="323" x1="462.499972" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <g id="svg_81238_206">
   <line id="svg_81238_207" y2="136.215383" x2="168.027487" y1="147.563816" x1="168.027487" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_208" y2="135.26968" x2="179.961593" y1="135.26968" x1="166.038468" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <g id="svg_81238_209">
   <line id="svg_81238_210" y2="335.799367" x2="167.027487" y1="347.147801" x1="167.027487" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_211" y2="334.853665" x2="178.961593" y1="334.853665" x1="165.038468" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <g id="svg_81238_212">
   <line id="svg_81238_213" y2="137.799367" x2="465.238426" y1="149.147801" x1="465.238426" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_214" y2="136.853665" x2="453.86617" y1="136.853665" x1="467.133803" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <g id="svg_81238_215">
   <line id="svg_81238_216" y2="337.797414" x2="466.738426" y1="349.145847" x1="466.738426" stroke-width="4" stroke="#000000" fill="none"/>
   <line id="svg_81238_217" y2="336.851711" x2="455.36617" y1="336.851711" x1="468.633803" stroke-width="4" stroke="#000000" fill="none"/>
  </g>
  <text xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" id="svg_81238_218" y="195" x="217" stroke-width="0" stroke="#000000" fill="#000000">F (g')</text>
  <text id="svg_81238_219" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="196" x="410" stroke-width="0" stroke="#000000" fill="#000000">F (h')</text>
  <text font-weight="normal" id="svg_81238_221" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="108" x="313" stroke-width="0" stroke="#000000" fill="#000000">F (h'∘g')</text>
  <line id="svg_81238_228" y2="336" x2="312" y1="308" x1="312" stroke-width="4" stroke="#ffffff" fill="none"/>
  <line id="svg_81238_229" y2="338" x2="320" y1="310" x1="320" stroke-width="4" stroke="#ffffff" fill="none"/>
  <text xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" id="svg_81238_230" y="113" x="293" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  <text id="svg_81238_231" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="200.5" x="208.5" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  <text id="svg_81238_232" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="200.5" x="402.5" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  <text id="svg_81238_233" font-weight="normal" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="308.5" x="233" stroke-width="0" stroke="#000000" fill="#000000">F (h∘g)</text>
  <text id="svg_81238_234" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="314.5" x="215" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  <text id="svg_81238_235" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="280" x="345" stroke-width="0" stroke="#000000" fill="#000000">F (f')</text>
  <text id="svg_81238_236" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="285.5" x="339.5" stroke-width="0" stroke="#000000" fill="#000000">2</text>
  <text id="svg_81238_237" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="227.25" x="504" stroke-width="0" stroke="#000000" fill="#000000">F (f)</text>
  <text id="svg_81238_238" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="232.75" x="499.5" stroke-width="0" stroke="#000000" fill="#000000">2</text>
  <text id="svg_81238_239" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="230.25" x="129" stroke-width="0" stroke="#000000" fill="#000000">F (f'')</text>
  <text id="svg_81238_240" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="235.75" x="120.5" stroke-width="0" stroke="#000000" fill="#000000">2</text>
  <text id="svg_81238_241" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="392.25" x="213" stroke-width="0" stroke="#000000" fill="#000000">F (g)</text>
  <text id="svg_81238_242" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="397.75" x="205.5" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  <text id="svg_81238_243" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="18" y="395.25" x="415" stroke-width="0" stroke="#000000" fill="#000000">F (h)</text>
  <text id="svg_81238_244" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="12" y="400.75" x="407.5" stroke-width="0" stroke="#000000" fill="#000000">1</text>
  </g>
  </svg>

* They must be compatible with the vertical composition of mixed squares, that is for every [[vertical composition]] of mixed squares $\mathcal{C}$, $\mathcal{C}'$:

  <svg width="480" height="640" xmlns="http://www.w3.org/2000/svg">
  <g>
  <title>Layer 1</title>
  <g id="svg_1" transform="rotate(90, 241.625, 322.25)">
   <line fill="none" stroke="#000000" stroke-width="4" x1="13" y1="447.530912" x2="212.002513" y2="447.530912" id="svg_5"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="9.498047" y1="201.657357" x2="208.500559" y2="201.657357" id="svg_8"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="-24" y1="418.359473" x2="-24" y2="224.577773" id="svg_9"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="244" y1="418.359473" x2="244" y2="224.577773" id="svg_11"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="200.244141" y1="210.779536" x2="209.757289" y2="200.868385" id="svg_14"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="-33.755859" y1="235.783626" x2="-24.242711" y2="225.872475" id="svg_16"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="233.244141" y1="235.783626" x2="242.757289" y2="225.872475" id="svg_18"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="201.876953" y1="438.026183" x2="210.123164" y2="446.617391" id="svg_20"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="199.876953" y1="193.194465" x2="208.123164" y2="201.785674" id="svg_21"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="244.244141" y1="225.873956" x2="253.757289" y2="235.785107" id="svg_27"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="-22.755859" y1="225.873969" x2="-13.242711" y2="235.785119" id="svg_28"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="201.244141" y1="456.65309" x2="210.757289" y2="446.74194" id="svg_32"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="-24" cy="445.447237" r="14.015625" id="svg_67"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="244" cy="447.530912" r="14.015625" id="svg_69"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="243" cy="198.531846" r="14.015625" id="svg_70"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="-23" cy="200.61552" r="14.015625" id="svg_71"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="276.25" y1="445.968156" x2="475.252513" y2="445.968156" id="svg_88"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="272.748047" y1="200.094601" x2="471.750559" y2="200.094601" id="svg_89"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="507.25" y1="416.796717" x2="507.25" y2="223.015017" id="svg_90"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="463.494141" y1="209.21678" x2="473.007289" y2="199.305629" id="svg_92"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="496.494141" y1="234.22087" x2="506.007289" y2="224.30972" id="svg_93"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="465.126953" y1="436.463427" x2="473.373164" y2="445.054636" id="svg_94"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="463.126953" y1="191.63171" x2="471.373164" y2="200.222918" id="svg_95"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="507.494141" y1="224.311217" x2="517.007289" y2="234.222367" id="svg_96"/>
   <line fill="none" stroke="#000000" stroke-width="4" x1="464.494141" y1="455.090335" x2="474.007289" y2="445.179184" id="svg_98"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="507.25" cy="445.968156" r="14.015625" id="svg_102"/>
   <circle fill="#000000" stroke="#000000" stroke-width="0" cx="506.25" cy="196.96909" r="14.015625" id="svg_103"/>
  </g>
  <text transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" id="svg_2" y="190.25" x="79" stroke-width="0" stroke="#000000" fill="#000000">f'</text>
  <text id="svg_4" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="192.374023" x="302.626518" stroke-width="0" stroke="#000000" fill="#000000">f</text>
  <text id="svg_6" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="379.874489" x="80.625977" stroke-width="0" stroke="#000000" fill="#000000">e'</text>
  <text id="svg_7" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="380.624489" x="304.126522" stroke-width="0" stroke="#000000" fill="#000000">e</text>
  <text id="svg_10" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="69.373724" x="197.626264" stroke-width="0" stroke="#000000" fill="#000000">g''</text>
  <text id="svg_12" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="265.124214" x="199.876281" stroke-width="0" stroke="#000000" fill="#000000">g'</text>
  <text id="svg_15" transform="matrix(1.33333, 0, 0, 1.33333, -22, -48)" font-weight="bold" xml:space="preserve" text-anchor="middle" font-family="Fantasy" font-size="24" y="460.874701" x="196.876264" stroke-width="0" stroke="#000000" fill="#000000">g</text>
  </g>
  </svg>

  the following solid arrow diagram commutes:

  <svg width="640" height="480" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="81238">
  <g>
  <title>Layer 1</title>
  <g transform="rotate(90, 317.367, 256.375)" id="svg_81238_3">
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m191.5,326c0,1 -1.346191,2.705444 -3,5c-1.307434,1.813995 -2.714111,4.21167 -5,7c-1.793213,2.187347 -5,4 -6,8l-3,2l-2,2" id="svg_81238_138"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m215.5,339c0,1 -0.149353,1.474274 -1,2c-1.9021,1.175568 -2,4 -4,6c-2,2 -4.21167,5.714111 -7,8c-2.187347,1.793213 -5,7 -8,12l-2,4l-2,7" id="svg_81238_139"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m241.5,355c-1,1 -1.0979,2.824432 -3,4c-0.850647,0.525726 -1.61731,1.076111 -2,2c-0.541199,1.30658 -2,2 -4,4c-2,2 -4.610657,5.159271 -6,8c-1.584106,3.238922 -5,4 -6,6l-2,2l-1,1" id="svg_81238_140"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m263.5,367c0,1 0.38269,2.076111 0,3c-1.082397,2.613129 -2.692566,4.186005 -4,6c-1.653809,2.294556 -4,4 -5,5l-2,2l-1,2l-2,1" id="svg_81238_141"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m286.5,381c-2,2 -6,6 -9,9l-6,5l-3,3l-3,1" id="svg_81238_142"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m177.5,325c0,1 0.307434,2.186005 -1,4c-0.826904,1.147278 -1,2 -3,4l-1,1l-3,0" id="svg_81238_143"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m198.5,337c0,1 -0.346191,1.705444 -2,4c-1.307434,1.813995 -4,4 -6,6c-2,2 -4,5 -5,6l-2,2l-2,1" id="svg_81238_144"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m223.5,350c0,1 0,2 -1,3c-2,2 -3.852722,3.173096 -5,4c-1.813995,1.307434 -4,2 -6,4l-2,2l-1,2" id="svg_81238_145"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m252.5,361c0,1 1.414215,2.585785 0,4c-0.707092,0.707092 -2,1 -4,3c-2,2 -3.186005,3.692566 -5,5c-1.147278,0.826904 -4,3 -6,5l-1,2" id="svg_81238_146"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m276.5,375c-1,2 -2.173096,2.852722 -3,4c-1.307434,1.813995 -2.692566,4.186005 -4,6c-2.480713,3.441803 -5,6 -8,7l-3,2l-1,1l-3,2" id="svg_81238_147"/>
   <path fill="none" stroke="#ffffff" stroke-width="4" d="m292.5,385c0,1 -0.076111,1.61731 -1,2c-1.30658,0.541199 -2.692566,4.186005 -4,6c-1.653809,2.294556 -3.190277,3.881531 -5,7c-1.122345,1.93399 -2,2 -3,3l-1,0" id="svg_81238_148"/>
   <g id="svg_81238_161">
    <line fill="none" stroke="#000000" stroke-width="4" x1="464.499999" y1="129.918299" x2="332.499995" y2="197.058954" id="svg_81238_149"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m441.988373,128.959152c0,0.959152 1.377472,2.594925 3.069794,4.795761c1.337799,1.739899 2.777222,4.039627 5.116241,6.714066c1.8349,2.097992 5.116272,3.836594 6.139557,7.673203l3.069763,1.918335l2.046509,1.918259" id="svg_81238_150"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m417.430206,141.428131c0,0.959167 0.152863,1.414032 1.023285,1.918304c1.94635,1.127533 2.046509,3.836578 4.093018,5.754913c2.046539,1.918304 4.309631,5.480682 7.162811,7.673203c2.23819,1.719971 5.116272,6.714081 8.186005,11.509827l2.046509,3.836624l2.046539,6.714066" id="svg_81238_151"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m390.825592,156.774551c1.023285,0.959137 1.123383,2.709061 3.069763,3.836609c0.870422,0.504257 1.654907,1.032196 2.046509,1.91832c0.553772,1.253174 2.046509,1.918289 4.093018,3.836594c2.046539,1.918304 4.717896,4.948517 6.139526,7.673218c1.620941,3.106613 5.116302,3.836609 6.139557,5.754929l2.046509,1.918289l1.023224,0.959167" id="svg_81238_152"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m368.313934,168.284378c0,0.959167 -0.391571,1.991333 0,2.877457c1.107574,2.506363 2.755188,4.01503 4.093018,5.754913c1.692322,2.200821 4.093048,3.836639 5.116302,4.795761l2.046509,1.91832l1.023285,1.918304l2.046448,0.959152" id="svg_81238_153"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m344.779083,181.712509c2.046509,1.91832 6.139496,5.754944 9.20929,8.632355l6.139526,4.795776l3.069763,2.877457l3.069794,0.959152" id="svg_81238_154"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m456.313965,128c0,0.959152 -0.314606,2.09671 1.023285,3.836594c0.846069,1.100418 1.023224,1.91832 3.069702,3.836594l1.023285,0.959167l3.069763,0" id="svg_81238_155"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m434.825592,139.509827c0,0.959152 0.354218,1.635773 2.046478,3.836609c1.33786,1.739899 4.093079,3.836578 6.139557,5.754913c2.046539,1.918304 4.093018,4.795746 5.116272,5.754898l2.046509,1.918304l2.046509,0.959137" id="svg_81238_156"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m409.244202,151.978775c0,0.959167 0,1.91832 1.023285,2.877472c2.046478,1.918304 3.942261,3.043488 5.116211,3.836655c1.856201,1.254013 4.093048,1.918259 6.139526,3.836578l2.04657,1.918289l1.023254,1.918304" id="svg_81238_157"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m379.569763,162.52948c0,0.959152 -1.447113,2.480148 0,3.836594c0.723541,0.678207 2.046539,0.959183 4.092987,2.877472c2.046539,1.918289 3.260132,3.541733 5.116302,4.795746c1.17395,0.793121 4.093048,2.877457 6.139526,4.795776l1.023285,1.91832" id="svg_81238_158"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m355.011627,175.957611c1.023224,1.918289 2.223633,2.736176 3.069794,3.836609c1.337799,1.739899 2.755157,4.015015 4.092987,5.754913c2.538391,3.301208 5.116272,5.754898 8.186066,6.71405l3.069733,1.918304l1.023285,0.959152l3.069763,1.91832" id="svg_81238_159"/>
    <path fill="none" stroke="#ffffff" stroke-width="4" d="m338.639496,185.549133c0,0.959152 0.077911,1.551239 1.023285,1.91832c1.336975,0.519043 2.755219,4.014984 4.093018,5.754898c1.692261,2.200806 3.264496,3.722946 5.116302,6.714066c1.148438,1.85498 2.046478,1.91832 3.069763,2.877457l1.023224,0" id="svg_81238_160"/>
   </g>
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The exchange on $(F_{1},F_{2})$ by this family of exchanges on 2-morphisms is sometimes denoted as $(e(C))_{C \in \mathcal{E}}$.

## References

J. Ayoub, _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique_. I., Astérisque No. 314 (2007), x+466 pp. (2008) [MR2009h:14032](http://www.ams.org/mathscinet-getitem?mr=2423375); II. Astérisque No. 315 (2007), vi+364 pp. (2008) [MR2009m:14007](http://www.ams.org/mathscinet-getitem?mr=2438151); also a file at K-theory archive [THESE.pdf](http://www.math.uiuc.edu/K-theory/0761/THESE.pdf)

[[!redirects exchange structure]]
[[!redirects exchange structures]]


excision

## Related entries

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[Blakers-Massey theorem]]

* [[Mayer-Vietoris sequence]]

## References

* Wikipedia, _[Excision](http://en.wikipedia.org/wiki/Excision_theorem)_

* [[Moritz Groth]], _Excision property and Mayer-Vietoris sequence_ ([pdf](http://www.math.ru.nl/~mgroth/teaching/algtopI14/Lecture06.pdf), this link is currently broken)

[[!redirects excision axiom]]


excisive (∞,1)-functor

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

#Contents#
* table of contents
{:toc}

## Idea

An _excisive (∞,1)-functor_ $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is an [[(∞,1)-functor]] which sends [[homotopy pushout]] squares to [[homotopy pullback]] squares. If the [[(∞,1)-category]] $\mathcal{C}$ is [[pointed object|pointed]] [[finite homotopy types]] and $\mathcal{D}$ is [[spectra]], then this condition is the [[axiom]] of [[excision]] in [[generalized (Eilenberg-Steenrod) cohomology|generalized homotopy]], whence the name.

Moreover, if here $\mathcal{D}$ is instead [[∞Grpd]] (i.e. [[homotopy types]]), or more generally any [[(∞,1)-topos]] $\mathbf{H}$, then excisive functors that send the point to the point (up to equivalence) are still equivalent to [[spectra]] ([[spectrum objects]]) -- essentially by the [[Brown representability theorem]] -- and those without such restriction are equivalent to [[parameterized spectra]], hence form the [[tangent (∞,1)-topos]] $T \mathbf{H}$. (See also [here](n-excisive+functor#nExcisiveFunctorsFormInfinityTopos) at *[[n-excisive functor]]* and at *[[Joyal locus]]*.)

As such, excisive functors are the lowest nontrivial stage in the [[Goodwillie-Taylor tower]] that approximates the [[classifying (∞,1)-topos]] $\mathbf{H}[X_\bullet]$ for [[pointed objects]]. The higher stages of this tower are given by the [[n-excisive (∞,1)-functors]].

## Definition

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

An [[(∞,1)-functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] is **excisive** if it takes [[(∞,1)-pushout]] squares in $\mathcal{C}$ to [[(∞,1)-pullback]] squares $\mathcal{D}$.

This is the $n=1$ case of the concept of _[[n-excisive (∞,1)-functor]]_.

=--

(e.g. [HigherAlg, def. 1.4.2.1.](#HigherAlg))

## Examples

### Spectrum objects
 {#SpectrumObjects}

Write $\infty Grpd_{fin}$ for the [[(∞,1)-category]] of [[finite homotopy types]], hence those freely generated by [[finite (∞,1)-colimits]] from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the [[pointed object|pointed]] finite homotopy types.

+-- {: .num_defn #SpectraAsExcisiveFunctors}
###### Proposition

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-limits]]. Then
**[[spectrum objects]]** in $\mathcal{C}$ are equivalently  [[reduced (infinity,1)-functor|reduced]] [[excisive (∞,1)-functor]] of the form

$$
  \infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C}
  \,.
$$

=--

(Sometimes, e.g. in [Lurie, def. 1.4.2.8](#HigherAlg), this is taken as the very definition of spectrum objects).

A proof of prop. \ref{SpectraAsExcisiveFunctors} passing through [[model category]] [[presentable (infinity,1)-category|presentations]] for excisive $\infty$-functors and of the [[Bousfield-Friedlander model structure]] for [[sequential spectra]] is due to ([Lydakis 98](#Lydakis98)), see at _[[model structure for excisive functors]]_ at _[Relation to BF-model structure on sequential spectra](model+structure+for+excisive+functors#ModelStructureForSpectra)_.

The idea of the equivalence is as follows. Let $E$ be a reduced excisive functor.
For each $n \in \mathbb{N}$, write $S^n \in \infty Grpd_{fin}^{\ast/}$ for the [[n-sphere]] and write $E_{n} \coloneqq E(S^n)$. We have the homotopy pushout squares

$$
  \array{
    S^n &\longrightarrow& \ast
    \\
    \downarrow & & \downarrow
    \\
    \ast &\longrightarrow& S^{n+1}
  }
$$

and since $E$ sends them to homotopy pullbacks with the point going to the point, this gives equivalences

$$
  E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1}
  \,.
$$

This makes $E_\bullet$ have the structure of an  [[Omega spectrum]]. The idea then is that as such it represents a [[generalized homology theory]] and the value of the excisive functor on any finite homotopy type $X$ is then $\Omega^\infty(E \wedge X)$ (see [Lurie, remark 1.4.3.3](#HigherAlg)).

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

The traditional definition of a [[generalized homology theory]] is as a functor on (finite) homotopy types with values in [[graded abelian groups]]. The [[Brown representability theorem]] says that these all arise from [[spectra]] $E$ via  taking [[stable homotopy groups]] of [[smash products]]:  $X \mapsto E_\bullet(X) \coloneqq \pi_\bullet(E \wedge X)$. But due to the existence of [[phantom maps]], this does not quite yield an [[equivalence of (infinity,1)-categories|equivalence]] between spectra and generalized homology theories.

In view of this, the proof of prop. \ref{SpectraAsExcisiveFunctors} may be thought of saying that this mismatch is fixed by refining homotopy groups by full [[homotopy types]] $X \mapsto \Omega^\infty(E\wedge X)$.

Notice also that spectra realized as excisive functors this way are in the spirit of [[coordinate-free spectra]].

=--

## Properties

For the moment we focus on properties of the case of excisive functors $\mathcal{C} \to \mathcal{D}$ for $\mathcal{C} = \infty Grpd_{fin}^{\ast/}$ and $\mathcal{D} = \mathbf{H}$, hence on

$$
  Exc(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq T \mathbf{H} \hookrightarrow \mathbf{H}[X_\ast]
  \,.
$$

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

Write $\infty Grpd_{fin}$ is the [[(∞,1)-category]] of [[finite homotopy types]].
For $\mathbf{H}$ a given [[base (∞,1)-topos]], write

$$
  \begin{aligned}
    \mathbf{H}[X_\ast]
    & \simeq
    [\infty Grpd_{fin}^{\ast/}, \mathbf{H}]
    \\
    & \simeq
    PSh((\infty Grpd_{fin}^{\ast/})^{op}, \mathbf{H})
  \end{aligned}
$$

for the [[classifying (∞,1)-topos]] (over $\mathbf{H}$) for [[pointed objects]].

=--

### Reflection and excisive approximation
 {#ReflectionAndExcisiveApproximation}

+-- {: .num_defn #P1}
###### Definition

Write

$$
  T_1 \colon \mathbf{H}[X_\ast] \longrightarrow \mathbf{H}[X_\ast]
$$

for the functor given by

$$
  E \mapsto \Omega_{E(\ast)} E(\Sigma(-))
  \,.
$$

Write

$$
  P_1 \coloneqq \underset{\longrightarrow}{\lim}_{n} T_1^{n}
$$

for the [[homotopy colimit]] of the iterations of this functor, with respect to the canonical comparison map.

=--

Unwinding the definition and using that [[suspension]] is equivalently the [join with the 0-sphere](join%20of%20topological%20spaces#JoinWith0Sphere), this is indeed the functor of the same name in  ([Goodwillie 91, p. 657 (13 of 67)](#Goodwillie03)).

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

The inclusion of excisive functors into $\mathbf{H}[X_*]$ is a [[reflective sub-(∞,1)-category]] with reflector given by $P_1$ from def. \ref{P1}:

$$
  T \mathbf{H}
   \stackrel{\overset{P_1}{\longleftarrow}}{\hookrightarrow}
  \mathbf{H}[X_\ast]
$$

=--

This is due (in the generality of _[n-excisive functor -- n-Excisive Approximation and reflection](https://ncatlab.org/nlab/show/n-excisive+functor#nExcisiveApproximation)_) to ([Goodwillie 91, theorem 1.8](#Goodwillie91)). See also ([Lurie, theorem 6.1.1.10, construction 6.1.1.27](#HigherAlg)).

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

Let $\mathcal{C}$ have finite colimits and a terminal object and let $\mathcal{D}$ be differentiable.

The excisive approximation of a [[reduced functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is the [[(infinity,1)-colimit]]

$$
  P_1 F
    \simeq
  \underset{\longrightarrow_{\mathrlap{k}}}{\lim}
  ( \Omega^k \circ F \circ \Sigma^k)
$$

(where $\Omega$ and $\Sigma$ denote [[looping]] and [[suspension]] in $\mathcal{D}$ and in $\mathcal{C}$, respectively).

=--

([Lurie, example 6.1.1.28](#HigherAlg))

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

Under the equivalence to [[sequential spectra]] in prop. \ref{SpectraAsExcisiveFunctors}, the formula in prop. \ref{ExcisiveApproximation} is the standard formula for [[spectrification]] of [[prespectra]].

=--

### Characterization via a generic stable object
 {#CharacterizationViaAGenericStableObject}

We discuss a characterization of excisive functors on $\infty Grpd_{fin}^{\ast/}$, hence of [[parameterized spectra]], as the result of [[forcing]] a generic pointed object to become a [[stable homotopy type]]. This general perspective is being highlighted by [Anel-Biederman-Finster-Joyal](#AnelBiedermanFinsterJoyal17b).

For a slick formulation, we use a generalization of [[powering]] to pointed powers:

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

For $X$ an object in an [[(∞,1)-category]] $\mathcal{C}$ with [[finite (∞,1)-limits]], and for $S_\ast \in \infty Grpd_{fin}^{\ast/}$ a pointed [[finite ∞-groupoid]], then the _pointed power_

$$
  X^{S_\ast} \in \mathcal{C}
$$

is the object which is the image of $S$ under the essentially unique [[(∞,1)-functor]]

$$
  (\infty Grpd_{fin}^{\ast/})^\op \longrightarrow \mathcal{C}
$$

which preserves [[finite (∞,1)-limits]] and sends $S^0 \leftarrow \ast$ to $X_\ast \to \ast$.

=--

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

Excisive functors $\infty Grpd_{fin}^{\ast/} \longrightarrow \mathbf{H}$, def. \ref{ExcisiveFunctor}, are the [[localization of an (∞,1)-category|localization]] of $\mathbf{H}[X_\ast]$, def. \ref{ClassifyingToposForPointedObjects}, at the set of morphisms

$$
  \left\{
    \Sigma \Omega (X_\ast^{S_\ast}) \longrightarrow X_\ast^{S_\ast}
  \right\}_{S \in \infty Grpd_{fin}^{\ast/}}
  \,,
$$

(where $X_{\ast}^{S_\ast}$ is the pointed power, def. \ref{PointedPower}, of the generic pointed object $X_\ast \in \mathbf{H}[X_\ast]$):

$$
  T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ]
  \,.
$$

In other words, the [[parameterized spectra]] are those objects in $\mathbf{H}[X_\ast]$ which regard each finite pointed power of the generic pointed object $X_\ast$ as a [[stable homotopy type]].

=--

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

For $K \in \infty Grpd_{fin}^{\ast/}$, write $R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast]$ for its [[formal dual]] under [[(∞,1)-Yoneda embedding]]. Since the [[(∞,1)-Yoneda embedding]] preserves [[(∞,1)-limits]], we have

$$
  R(K)^S \simeq R(S \cdot K)
  \,.
$$

Observe that the generic pointed object in $\mathbf{H}[X_\ast]$ is that [[representable functor|represented]] by the [[0-sphere]]:

$$
  X_\ast = R(S^0)
  \,.
$$

Hence

$$
  X_\ast^S \simeq R(S)
  \,.
$$

Now using the [[(∞,1)-Yoneda lemma]] we have  for each $E \in \mathbf{H}[X_\ast]$ that

$$
  \begin{aligned}
    Hom( \Sigma \Omega R(K), E )
    &\simeq
    \Omega_{E(\ast)} Hom(\Omega R(K),E)
    \\
    & \simeq
    \Omega_{E(\ast)} Hom( R(\Sigma K), E )
    \\
    & \simeq
    \Omega_{E(\ast)} E(\Sigma K)
  \end{aligned}
  \,.
$$

Hence for all $K \in \infty Grpd_{fin}^{\ast/}$

$$
  \begin{aligned}
    Hom(\Sigma \Omega R(K) \to R(K), E)
    & \simeq
    ( E(K) \longrightarrow  \Omega(\Sigma K) )
    \\
    & =
    (E \to T_1 E)(K)
  \end{aligned}
  \,,
$$

where in the last line we observe that the expression is that for
the comparison map in def. \ref{P1}.

This means that the [[local objects]] are precisely those $E$ for which the morphism

$$
  E \longrightarrow T_1 E
$$

from def. \ref{P1} is an equivalence. With this the statement follows from theorem \ref{PnLocalizes}.

=--

## Related concepts

* [[exact (∞,1)-functor]]

* [[n-excisive (∞,1)-functor]]

* [[model structure for excisive functors]]

* [[spectrum object]]

## References

The notion of [[n-excisive functors]] was introduced in

* {#Goodwillie91} [[Thomas Goodwillie]], _Calculus II, Analytic functors_, K-Theory 5 (1991/92), no. 4, 295-332

The [[Taylor tower]] formed by $n$-excisive functors was then studied in

* {#Goodwillie03} [[Thomas Goodwillie]], _Calculus III: Taylor Series_, Geom. Topol. 7(2003) 645-711 ([arXiv:math/0310481](http://arxiv.org/abs/math/0310481))

A discussion in the general abstract context of [[(∞,1)-category theory]] is in

* {#HigherAlg} [[Jacob Lurie]], section 6.1.1 of _[[Higher Algebra]]_

Review includes:

* {#Harpaz2013} [[Yonatan Harpaz]], section 5 of _Introduction to stable $\infty$-categories_, October 2013 ([[HarpazStableInfinityCategory2013.pdf:file]])

On excisive 1-functors for [[genuine G-spectra]] (i.e.  in [[proper equivariant stable homotopy theory|proper]] [[equivariant stable homotopy theory]]):

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], Section 3.1 of: _Proper equivariant stable homotopy theory_, Memoirs of the AMS [[arXiv:1908.00779](https://arxiv.org/abs/1908.00779)]

A [[model structure for excisive functors]] was given in

* {#Lydakis98} Lydakis, _Simplicial functors and stable homotopy theory_ Preprint, available via Hopf archive, 1998 ([pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf))

Discussion in terms of [[stable homotopy types]] is due to

* {#AnelBiedermanFinsterJoyal17b} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _Goodwillie's Calculus of Functors and Higher Topos Theory_ ([arXiv:1703.09632](https://arxiv.org/abs/1703.09632))

[[!redirects excisive (∞,1)-functors]]

[[!redirects excisive (infinity,1)-functor]]
[[!redirects excisive (infinity,1)-functors]]

[[!redirects excisive ∞-functor]]
[[!redirects excisive ∞-functors]]

[[!redirects excisive infinity-functor]]
[[!redirects excisive infinity-functors]]

[[!redirects excisive functor]]
[[!redirects excisive functors]]

[[!redirects pre-excisive functor]]
[[!redirects pre-excisive functors]]

[[!redirects preexcisive functor]]
[[!redirects preexcisive functors]]

[[!redirects reduced excisive functor]]
[[!redirects reduced excisive functors]]


excited state

## Idea

a [[quantum state]] of [[energy]] above the [[ground state]]

## Related concepts

[[!include states and observables -- content]]

[[!redirects excited states]]