operad

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#### Higher algebra
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[[!include higher algebra - contents]]
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#### Categorical algebra
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[[!include categorical algebra -- contents]]
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#Contents#
* table of contents
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## Idea

An **operad** is a gadget used to describe algebraic structures in [[symmetric monoidal category|symmetric monoidal categories]].  It is

* a bunch of abstract _operations_ of arbitrarily many arguments;

* equipped with a notion of how to compose these;

* subject to evident [[associativity]] and [[unitality]] conditions.

Just like a monoid can be seen as a single-object category, an operad is equivalently a single-object multicategory. Multicategories with multiple objects are also called colored operads.

An [[algebra over an operad]] is a concrete realization of these abstract operations: an [[object]] $A$ equipped with $n$-ary operations $A \otimes \cdots \otimes A \to A$ as specified by the operad, subject to the composition relation as specified by the operad.

This way an operad is like a [[Lawvere theory]] in that it can be used to describe algebraic structures having finitary operations obeying equational laws.  However, unlike Lawvere theories, operads can be defined internal to general [[symmetric monoidal categories]] where the [[tensor product]] might not be the [[cartesian product]].

The notion of operad (and allied notions such as [[PROP]], [[club]], [[multicategory]] and so on) come in many flavors. Originally used in [[algebraic topology]] to provide a systematic formalism for describing the internal operations which exist on iterated [[loop space]]s, the basic idea is quite flexible and adaptable to many categorical situations, and the importance of operads continues to grow.

## Definition in components

The original definition is due to [[Peter May]] and was given in his book ([The Geometry of Iterated Loop Spaces](#May1972)). It describes an operad as a collection of operations equipped with a notion of composition and subject to various conditions.

This definition is essentially that of an [[enriched category]], only that the [[hom-object]]s are allowed to go from more than one object to a given output object.

There is a more abstract way to encode all this simply as a [[monoid in a monoidal category|monoid]] in a suitable ambient [[monoidal category]]. This more abstract definition we discuss below in [A detailed conceptual treatment](#Conceptual).

### Symmetric operads {#PedestrianDefOperad}

Let $V$ be a [[symmetric monoidal category]].

{#PermutativeOperad} A (*permutative* or *[[symmetric operad|symmetric]]*) **operad** in $V$ consists of objects $F(n)$ of $V$ indexed over the [[natural number]]s $n = 0, 1, 2, \ldots$ (which we intuitively think of as "objects that parametrize the $n$-ary operations of an algebraic theory") equipped with the following extra structure:

* Right [[actions]] of [[symmetric group]]s $\rho_n: S_n \to \hom(F(n), F(n))$;
* A _unit_ $e: I \to F(1)$ [which we think of as picking out the identity map as unary operation];
* Composition operations
$$F(k) \otimes F(n_1) \otimes F(n_2) \otimes \cdots \otimes F(n_k) \to F(n_1 + \ldots + n_k)$$
(which we think of as the result of plugging the outputs of operations $\theta_1, \ldots, \theta_k$ into a $k$-ary operation $\theta$, to produce a new operation $\theta \circ (\theta_1 \otimes \ldots \otimes \theta_k)$).

These data are subject to obvious identities such as [[associativity]] and [[unitality]] of composition, and compatibility of composition with symmetric group actions. For example, the unit laws say that the evident composite

$$F(n) \cong I \otimes F(n) \stackrel{e \otimes 1}{\to} F(1) \otimes F(n) \stackrel{comp}{\to} F(n)$$

is the identity map, as is

$$F(n) \cong F(n) \otimes I^{\otimes n} \stackrel{1 \otimes e^{\otimes n}}{\to} F(n) \otimes F(1)^{\otimes n} \stackrel{comp}{\to} F(n)$$

Compatibility with symmetric group actions means that for each element $\sigma \in S_k$, the composition operation

$$F(k) \otimes \bigotimes_{i = 1}^k F(n_i) \to F(n_1 + \ldots + n_k)$$

coequalizes a pair of automorphisms

$$\rho(\sigma) \otimes 1, 1 \otimes \lambda(\sigma): F(k) \otimes \bigotimes_{i=1}^k F(n_i) \;\rightrightarrows\; F(k) \otimes \bigotimes_{i=1}^k F(n_i)$$

where $\sigma$ acts on the big tensor product on the left by permuting tensor factors in the obvious way. If $V$ has suitable colimits, this condition could be expressed in terms of tensor products over $S_n$.

The associativity condition will be left for others to fill in.

### Coloured operads {#PedestrianDefColouredOperad}

There is an evident generalization of the above where we allow the operad to have several [[object]]s -- called _colours_ in operad theory. This models algebraic structures where elements of different _types_ may be fed into $n$-ary operations.

Let $C$ be a [[set]], called the _set of colours_ . Then a **coloured operad** is

* for each $n \in \mathbb{N}$ and each $(n+1)$-tuple $(c_1, \cdots, c_n, c)$ an object $P(c_1, \cdots, c_n;c) \in V$;

* for each $c \in C$ a morphism $1_c : I \to P(c;c)$ in $V$ -- the _identity_ on $c$;

* for each $(n+1)$-tuple $(c_1, \cdots, c_n, c)$ and $n$ other tuples

  $(d_{1,1}, \cdots, d_{1,k_1}), \cdots, (d_{n,1}, \cdots, d_{n,k_n})$

  a morphism

  $$
    P(c_1, \cdots, c_n; c) \otimes P(d_{1,1}, \cdots, d_{1,k_1}; c_1)
    \otimes \cdots \otimes P(d_{n,1}, \cdots, d_{n,k_n}; c_n)
    \to
    P(d_{1,1}, \cdots, d_{n,k_n}, c)
  $$

  the _composition_ operation;

* for all $n$, all tuples, and each [[permutation]] $\sigma$ in the [[symmetric group]] $\Sigma_n$ a morphism

  $$
    \sigma^* : P(c_1, \cdots, c_n;c) \to P(c_{\sigma(1)}, \cdots, c_{\sigma(n)};c)
  $$

* subject to the conditions that

  * the $\sigma$s form a [[representation]] of $\Sigma_n$;

  * the composition operation satisfies [[associativity]] and [[unitality]] in the obvious way;

  * and is $\Sigma_n$ equivariant in the evident way.

An equivalent term for a colored operad in $V$ is a [[symmetric multicategory]] which is [[enriched category|enriched]] over $V$.  Which term is used depends on the author and the point of view being taken.  (According to the common terminology of [[horizontal categorification]], one might also call a colored operad an "operad-oid," but thankfully this seems not to be common.  On the other hand one might adapt the terminology backwards and call a [[category]] a "colored [[monoid]]" or call a [[groupoid]] a "colored [[group]]".)

### Algebras

An _[[algebra over an operad]]_  $F$ in $V$ is just a semantics for interpreting the $F(n)$ as objects of actual $n$-ary operations on an object $v$. That is, an $F$-algebra structure on an object $v$ in $V$ consists of a collection of maps

$$F(n) \otimes v^{\otimes n} \to v$$

which intuitively is a mapping like this:

$$\theta \otimes x_1 \otimes \ldots \otimes x_n \mapsto \theta(x_1, \ldots, x_n)$$

so that "elements" of $F(n)$ are interpreted as as $n$-ary operations on $v$. These data are subject to some natural conditions which implement this idea.

Perhaps the quickest way to define it is to suppose that $V$ is symmetric monoidal [[closed monoidal category|closed]], and work by way of parallel to how [[representations]] or [[modules]] work. Just as an $R$-module (over a ring $R$) can be defined as a ring homomorphism

$$R \to \hom(A, A)$$

where the hom here is an internal hom of abelian groups, called an _[[endomorphism ring]]_, so there is such a thing as an _[[endomorphism operad]]_ attached to any object $v$ in a symmetric monoidal closed category, and an $F$-algebra over an operad $F$ is the same thing as an operad morphism

$$F \to \hom(v^{\otimes \bullet}, v)$$

to an endomorphism operad (also called a _tautological operad_).

Now that the clue has been given, the rest is not hard to figure out. The components of the endomorphism operad are defined by

$$End(v)(n) = \hom(v^{\otimes n}, v)$$

Certainly $S_n$ acts on the right (that is, contravariantly) on the [[hom-object]] $\hom(v^{\otimes n}, v)$. And clearly there is a canonical map $e: I \to \hom(v, v)$ to play the role of the unit. The operad composition involves an instance of enriched functoriality of iterated tensor products: there is a map

$$\hom(v^{\otimes n_1}, v) \otimes \ldots \otimes \hom(v^{\otimes n_k}, v) \to \hom(v^{n_1 + \ldots + n_k}, v^{\otimes k})$$

The endomorphism operad composition is obtained by tensoring this last arrow with $\hom(v^{\otimes k}, v)$ on the left, and composing the result with ordinary internal hom-composition

$$\hom(v^{\otimes k}, v) \otimes \hom(v^{\otimes n_1 + \ldots + n_k}, v^{\otimes k}) \to \hom(v^{\otimes n_1 + \ldots + n_k}, v)$$

A closely related way of defining an $F$-algebra is via the monad attached to an operad, which we will describe below.

Note that this definition still makes sense when $v$ lives in any symmetric monoidal $V$-[[enriched category]], not only $V$ itself.

## A detailed conceptual treatment
 {#Conceptual}

We describe here a compact one-sentence definition of operad first worked out by [[Max Kelly]], after a few preliminaries which are important in their own right. The treatment is essentially an exercise in [[enriched category theory]] and the formalism of [[Day convolution]]. We will work this out fully in the case of ordinary [[category theory]] first, that is for categories [[enriched category|enriched in]] $V = Set$; the case for categories enriched in a complete, cocomplete, [[symmetric monoidal category|symmetric monoidal]] [[closed monoidal category|closed]] $V$ is completely parallel.

More details along these lines are add _[[toddtrimble:Towards a doctrine of operads]]_.

### Preparation

Let $\mathbb{P}$ be the [[groupoid]] of finite cardinals with [[bijections]] as morphisms --  the _[[permutation category]]_. Since $\mathbb{P}$ is the [[core]] groupoid of the category $Fin$ of finite cardinals and functions between them, the [[coproduct]] on $Fin$ restricts to a symmetric monoidal product called the cardinal sum on $\mathbb{P}$.

+-- {: .un_remark style="margin-left:2em"}
###### Remark
Under this symmetric monoidal structure, $\mathbb{P}$ may be characterized as the free symmetric strict monoidal category on one generator.
=--

The cardinal sum on $\mathbb{P}$ extends along the [[Yoneda embedding]] to a symmetric monoidal product $F \otimes G$ on the [[presheaf category]] $Psh(\mathbb{P}) \coloneqq [\mathbb{P}^{op},Set]$. This is an instance of the [[Day convolution]].

+-- {: .un_note style="margin-left:2em"}
###### Warning
By abuse of notation, we will also denote the presheaf category $Psh(\mathbb{P})$ equipped with the monoidal structure induced by the cardinal sum by $Psh(\mathbb{P})$.
=--

Since $Psh(\mathbb{P})$ is a presheaf category, it is cocomplete, and since the Day convolution is [[cocontinuous functor|cocontinuous]] in each of its separate arguments we say that $Psh(\mathbb{P})$ is **symmetric monoidally cocomplete**.

+-- {: .un_note style="margin-left:2em"}
###### Note
In addition to the standard [[coend]] formula, the Day convolution product on the $Psh(\mathbb{P})$ may be described by the rule:

$$(F \otimes G)[S] = \sum_{S = T + U} F[T] \times G[U],$$

summing over all partitions of $S$ into two parts (each possibly empty).
=--

According to the yoga of presheaf categories and Day convolution, given a symmetric monoidally cocomplete category $D$, a symmetric monoidal functor

$$X: \mathbb{P} \to D$$

extends uniquely up to isomorphism to a symmetric monoidal _cocontinuous_ functor

$$\hat{X}: Psh(\mathbb{P}) \to D,$$

taking a presheaf $W: \mathbb{P}^{op} \to Set$ to the [[weighted colimit]] $W \cdot X$.

+-- {: .un_remark style="margin-left:2em"}
###### Remark
It follows from the earlier remark and the above that we may describe $Psh(\mathbb{P})$ universally up to equivalence as the free symmetric monoidally cocomplete category on a single generator.
=--

+-- {: .un_note style="margin-left:2em"}
###### Digression
Recall that we can describe $W \cdot_{\mathbb{P}} X$ as follows: First note that the functor

$$
  \Lambda_0
  \coloneqq
  Hom_D(X(\cdot),\cdot)
  \colon
  \mathbb{P}^{op}\times D\to Set
  \,,
$$

so $\Lambda_0$ also gives a functor $\Lambda: D\to [\mathbb{P}^{op},Set]$ by [[currying]] through the second coordinate.

Then we define $W \cdot_{\mathbb{P}} X$ to be the object representing the functor

$$\Lambda_W \coloneqq Hom_Psh(\mathbb{P})(W,\Lambda(\cdot)):D\to Set$$

whenever $\Lambda_W$ is [[representable functor|representable]].

In general, weighted colimits may be described explicitly by coend formulas; here

$$W \cdot_{\mathbb{P}} X = \int^{k: \mathbb{P}} F(k) \cdot X(k)$$

where $S \cdot d$ denotes the tensoring of a set $S$ with an object $d$, that is the coproduct of an $S$-indexed set of copies of $d$. The coend here indicates a coequalizer.

$$\sum_k W(k) \cdot S_k \cdot X(k) \rightrightarrows \sum_k W(k) \cdot X(k) \to \int^k W(k) \cdot X(k)$$

where one of the parallel arrows involves right actions of symmetric groups $S_k$ on the $F(k)$, and the other involves left actions of $S_k$ on objects $X$. In other words, the coend in this instance may be described as a sum of tensor products:

$$W \cdot_{\mathbb{P}} X = \sum_k W(k) \otimes_{S_k} X(k).$$
=--

The aforementioned universal property of $Psh(\mathbb{P})$ with its convolution product may be more explicitly described as follows: given a symmetric monoidally cocomplete category $D$ and an object $d$ therein, there exists up to isomorphism a _unique_ symmetric monoidal cocontinuous functor $Psh(\mathbb{P}) \to D$ which sends the presheaf representable by the cardinal $1$, $h_1$, to $d$.

Explicitly, this functor takes a presheaf $F: \mathbb{P}^{op} \to Set$ to the following object of $D$:

$$\sum_k F(k) \otimes_{S_k} d^{\otimes k}.$$

+-- {: .num_example style="margin-left:2em" #analytic}
###### Example
When $D$ is the symmetric monoidally cocomplete category $(Set, \times)$ and $x$ is a set, this formula

$$\hat{F}(x) = \sum_k F(k) \otimes_{S_k} x^k$$

is the value at $x$ of what Joyal calls the _analytic functor_ $\hat{F}: Set \to Set$ associated to a [[combinatorial species|species]] $F$, which has been proposed as the categorification of the theory of exponential [[generating function]]s. The fact that $F \mapsto \hat{F}(x)$ is symmetric monoidal (cocontinuous) means that there is a canonical isomorphism

$$\widehat{(F \otimes_{Day} G)}(x) \cong \hat{F}(x) \times \hat{G}(x)$$

In other words, $F \mapsto \hat{F}$ behaves like a categorified version of Fourier transform, taking convolution products to ordinary (pointwise) products.
=--

+-- {: .un_definition}
For symmetric monoidally cocomplete categories $C, D$, let $\underline{Hom}(C, D)$ denote the category of symmetric monoidal cocontinuous functors $C \to D$. The universal property of $Psh(\mathbb{P})$ means that we have an equivalence

$$\underline{Hom}(Psh(\mathbb{P}), D) \simeq D.$$

Consequently, we have an equivalence $\underline{Hom}(Psh(\mathbb{P}), Psh(\mathbb{P})) \simeq Psh(\mathbb{P}).$

Since symmetric monoidal cocontinuous functors are stable under composition, the category on the left carries a monoidal product given by endofunctor composition. By transport of structure across the equivalence, we  induce a monoidal product on $Psh(\mathbb{P})$ given by endofunctor composition called the _substitution product of species_. The substitution product of species $F, G$ is denoted $F \circ G$.
=--

In detail: a species $G \colon \mathbb{P}^{op} \to Set$ induces a symmetric monoidal cocontinuous functor

$$Psh(\mathbb{P}) \to Psh(\mathbb{P}): F \mapsto F \circ G = \sum_k F(k) \otimes_{S_k} G^{\otimes_{Day} k}$$

The $k$-fold Day tensor power of $G$ is given (in the language of species) by the formula

$$G^{\otimes k}[S] = \sum_{S = T_1 + \ldots + T_k} G[T_1] \times G[T_2] \times \ldots \times G[T_k]$$

where we sum over all ways of breaking up a finite set $S$ into $k$ blocks, some possibly empty. Thus we have an explicit description of the substitution product,

$$(F \circ G)[n] = \sum_k F[k] \otimes_{S_k} (\sum_{[n] = T_1 + \ldots + T_k} G[|T_1|] \times \ldots \times G [|T_k|]),$$

and it is clear from our discussion above that substitution is a monoidal product. The monoidal unit $I$ is a functor $\mathbb{P}^{op} \to Set$ where $I[n]$ is terminal if $n = 1$, else is initial.

### Definition as monoid
 {#DefinitionAsMonoid}

We are at last ready for the one-sentence **definition**:

A ($Set$-based) **operad** is a [[monoid in a monoidal category|monoid]] in the [[monoidal category]] $(Psh(\mathbb{P}), \circ, I)$.

### Remarks

* {#CartesianOperads} We can get different flavors of operads by considering different notions of ambient [[monoidal category]]. For instance, for the theory of monoidal categories, the discrete category $\mathbb{N}$ plays the role of the free (strict) monoidal category on one generator, and $Set^{\mathbb{N}^{op}}$ the free monoidally cocomplete category on one generator. Similarly, for braided monoidal categories, we have the braid category $\mathbb{B}$, and $Set^{\mathbb{B}^{op}}$ is the free braided monoidally cocomplete category on one generator. Again, for cartesian categories, we have $Fin^{op}$ (the opposite of finite sets and functions) as the [[free cartesian category]] on one generator, and $Set^{Fin}$ is the free cartesian monoidally cocomplete category on one generator. In each of these cases we get a corresponding notion of operad by following the above treatment _mutatis mutandis_: nonpermutative operads, braided operads, cartesian operads (better known as [[Lawvere theory|Lawvere theories]]).  These are all special cases of the notion of [[generalized multicategory]].

* All of the above carries over to the enriched setting, where we work over a complete, cocomplete symmetric monoidal closed base category $V$. Here ordinary categories (like $\mathbb{N}, \mathbb{P}, \mathbb{B}, Fin$) are viewed as $V$-enriched by a simple change of base: change from [[hom-set]]s to [[hom-objects]] by applying the change of base functor
$$Set \to V$$
that takes a set $S$ to the $S$-fold coproduct $S \cdot I$, where $I$ is the monoidal unit of $V$.  These can also be defined in the framework of [[generalized multicategories]].

* The notion of generalized muticategories is even more general than this; for instance it also includes [[globular operads]] and [[topological spaces]].  See [[generalized multicategory]] for details.

* In still other directions, there are for example notions of [[cyclic operad]] and [[modular operad]].

* It is sometimes useful to consider an alternative definition of operad based on "partial" composition operations
$$sub_i: C(m) \times C(n) \to C(m+n-1)$$
which encode the idea of substituting an $n$-ary operation into the $i-th$ argument of an $m$-ary operation, to get an $(m+n-1)$-ary operation.
For example, this notion of composition allows the consideration of [[non-unital operad]]s without identity operations (called "pseudo-operads" by [(Markl, Shnider, and Stasheff)](#MarklShniderStasheff)).
On the other hand, in the presence of identity operations, the two forms of operadic composition are mutually definable (see Proposition 13 of [(Markl 2008)](#Markl2008), or Chapter 2 of [(Fresse, HOGTG I)](#FresseHOGTG) for a more detailed discussion).

* Finally, some authors place restrictions on operations of arity zero (a.k.a. nullary operations, or "constants"). May's original definition [(1972)](#May1972) required exactly one nullary operation, while the definition in [(Markl, Shnider, and Stasheff)](#MarklShniderStasheff) considers only operations of positive arity.
Fresse gives a special status to these two restricted forms of operads, referring to them respectively as **unitary operads** and **non-unitary operads** (see 3.1.10 of [(Fresse 2009)](#FresseMOF) and 1.1.19 [(Fresse, HOGTG I)](#FresseHOGTG)).
Beware that non-unitary operads (which have no nullary operations) are _not_ the complement of unitary operads (which have exactly one nullary operation), nor are they the same thing as non-unital operads (which have no identity operation).
As an example, Stasheff's original presentation of the [[associahedra]] [(Stasheff 1963)](#Stasheff63a) implicitly defined an operad which was _both_ non-unitary and non-unital.

## The monad attached to an operad

Each $Set$-based operad $M$ gives rise to a monad $\hat{M}$ on $Set$. Specifically, the monoidal category $(Psh(\mathbb{P}), \circ, I)$ acts on $Set$ in such a way as to give an [[actegory]] structure, and therefore an operad or $\circ$-monoid gives rise to a monad on $Set$.

Here are the details. There is a functor

$$i: Set \to Psh(\mathbb{P})$$

which sends a set $X$ to the functor

$$\hat{X}: \mathbb{P}^{op} \to Set$$

taking $[n]$ to $X$ if $n = 0$, else to $0$. This functor is full and faithful; conceptually, it treats a set $X$ as giving a set of 0-ary operations or constants indexed by itself. Notice that the composite

$$Psh(\mathbb{P}) \times Set \stackrel{1 \times i}{\to} Psh(\mathbb{P}) \times Psh(\mathbb{P}) \stackrel{\circ}{\to} Psh(\mathbb{P})$$

factors through the inclusion $i: Set \to Psh(\mathbb{P})$ (conceptually, when one applies a formal operation to constants, the result is again a constant). This gives an action

$$Psh(\mathbb{P}) \times Set \to Set$$

for an actegory structure; as it is the restriction of the substitution product $\circ$ along the inclusion $i$ in the second argument, we again denote it $\circ$, by abuse of notation. Given $F: \mathbb{P}^{op} \to Set$ and a set $X$, we have

$$F \circ X \cong \sum_{k \geq 0} F(k) \otimes_{S_k} X^k$$

and given $G: \mathbb{P}^{op} \to Set$, we also have coherent natural isomorphisms $(F \circ G) \circ X \cong F \circ (G \circ X)$, $I \circ X \cong X$.

+-- {: .un_def}
######Definition
The **monad** associated with an operad $(M, m: M \circ M \to M, u: I \to M)$ is the functor $\hat{M}: Set \to Set$ taking $X$ to $M \circ X$, equipped with natural transformations

$$\hat{M} \hat{M} X = M \circ (M \circ X) \cong (M \circ M) \circ X \stackrel{m \circ X}{\to} M \circ X = \hat{M} X$$

[ ]

$$X \cong I \circ X \stackrel{u \circ X}{\to} M \circ X = \hat{M} X$$

which provide $\hat{M}$ with the structure of a monad.
=--

This definition of the associated monad carries over with ease to the enriched case, and to variants such as nonpermutative operads, braided operads, and cartesian operads (Lawvere theories).

Notice that an algebra for the operad $\hat{M}$ is a set $X$ equipped with a structure map $\alpha: M \circ X \to X$ which makes $i(X)$ a [[module]] over the monoid $M$ in the monoidal category $Psh(\mathbb{P})$.

See also related discussion at [[club]].

## Examples

Ambient categories of relevance in practice are

* [[Top]]/[[sSet]] -- an operad in here is called a [[topological operad]];

* [[category of chain complexes]] -- an operad in here is called a [[dg-operad]]

### Specific examples

* [[endomorphism operad]]

* [[associative operad]]

  * [[A-infinity operad]]

* [[commutative operad]]

  * [[little cubes operad|E-k operad]]

    * [[E-infinity operad]]

    * [[framed little 2-disk operad]]

    * [[framed little n-disk operad]]

    * [[Fulton-MacPherson operad]]

* [[operad for modules over an algebra]]

* [[Lie operad]]

  * [[L-∞ operad]]

* [[Eilenberg-Zilber operad]]

* [[linear operad]]

* The [[Goodwillie derivatives of the identity functor]] form an operad in [[spectra]]

### General

* For $C$ a set, the [[initial object]] $I_C$ of $C$-coloured operads has $I_C(c;c) = I_V$ for all $c$ in $C$ and all other components of $I_C$ are the [[initial object]] of $V$.

* There is also a canonical notion of [[free operad]].

## Properties

### Change of colour

Coloured operads form a [[fibered category]] over the category [[Set]] of colours. The fiber over a set $C$ is the category of $C$-coloured operads.

For $\alpha : D \to C$ a [[function]] of sets, there is an evident pullback functor

$$
  \alpha^* : Oper_C(V) \to Oper_D(V)
$$

given by

$$
  \alpha^*(P)(d_1, \cdots, d_n; d)
  \coloneqq
  P(\alpha(d_1), \cdots, \alpha(d_n); \alpha(d))
  \,.
$$

Together with a morphism $\phi : Q \to \alpha^* P$ of operads this
induces a pair of [[adjoint functor]]s on [[algebra over an operad|algebras over an operad]]

$$
  \big(
    (\alpha,\phi)_! \dashv (\alpha,\phi)^*
  \big)
  \;\colon\;
  Alg_V(P)
  \underoverset
    {\;\;(\alpha,\phi)^*\;\;}
    {\;\;(\alpha,\phi)_!\;\;}
    {\leftrightarrows}
  Alg_V(Q)
  \,.
$$

### Model structures on operads

If the [[symmetric monoidal category]] $V$ that the operads under consideration are enriched in carries the structure of a [[monoidal model category]], then under suitable conditions there is also the structure of a [[model category]] on the category of $V$-operads. This is important for the notion of _homotopy_ [[algebra over an operad]], such as $A_\infty$- and $E_\infty$-algebras.

See

* [[model structure on operads]].

### Koszul duality

On [[quadratic operads]] in chain complexes there is a duality operation called [[Koszul duality]]. See there for details.

### The category of operads

Operads and operad [[homomorphisms]] (for any given flavor of operads, as discussed above) form a [[category]], _[[Operad]]_.

The [[Boardman-Vogt tensor product]] makes the category of symmetric colored operads over [[Set]] into a [[closed monoidal category]].

## Related concepts

* [[symmetric colored sequence]]

* [[algebraic theory]] / [[Lawvere theory]] /  [[(∞,1)-algebraic theory]]

* [[monad]] / [[(∞,1)-monad]]

* **operad** / [[(∞,1)-operad]]

  * [[multicategory]], [[polycategory]]

  * [[cooperad]]

* [[properad]]

* [[PROP]]

* [[cohomology of operads]]

* [[club]]

## References

Stasheff implicitly described the operad of [[associahedra]] in

* {#Stasheff63a} [[James Dillon Stasheff]], _Homotopy associativity of H-spaces I_, Trans. Amer. Math. Soc. 108 (1963), 275--312.  ([web](http://www.jstor.org/stable/1993608))

* [[James Dillon Stasheff]], _Homotopy associativity of H-spaces II_, Trans. Amer. Math. Soc. 108 (1963), 293--312.  ([web](http://www.jstor.org/stable/1993609))

but without making the abstract concept of operad explicit.
The abstract definition (as well as the name) is originally due to

* {#May1972} [[Peter May]], _The geometry of iterated loop spaces_ ([pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))

An earlier version was the notion of [[analyser]] (known usually by French version *analyseur*), introduced in

* M. Lazard, _Lois de groupes et analyseurs_, Ann. École Norm. Sup. __72__  (1955), pp. 299–400.

Monographs:

* [[Igor Kriz]], [[Peter May]], _Operads, algebras, modules and motives_, Astérisque __233__, Société Mathématique de France (1995) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf), [numdam:AST_1995__233__1_0](http://www.numdam.org/item/?id=AST_1995__233__1_0))

* [[Tom Leinster]], _Higher operads, higher categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)

* {#MarklShniderStasheff} [[Martin Markl]], [[Steve Shnider]], [[Jim Stasheff|James D. Stasheff]], _Operads in algebra, topology and physics_, Math. Surveys and Monographs __96__, Amer. Math. Soc. 2002.

* {#Markl2008} [[Martin Markl]], Operads and PROPS, In volume 5 of _Handbook of Algebra_, pages 87–140. Elsevier, 2008. [arXiv:math/0601129](https://arxiv.org/abs/math/0601129)

* V.A. Smirnov, _Simplicial and operad methods in algebraic topology_

* [[Jean-Louis Loday]], [[Bruno Vallette]], _Algebraic operads_, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012), xviii+512 pp. ([pdf](https://www.math.univ-paris13.fr/~vallette/Operads.pdf), [ISBN 978-3-642-30362-3](https://www.springer.com/gp/book/9783642303616))

* {#FresseMOF} [[Benoit Fresse]], _Modules over operads and functors_, Springer LNM __1967__, 2009, x+308 pp. [MR2010e:18009](http://www.ams.org/mathscinet-getitem?mr=2494775)

* {#FresseHOGTG} [[Benoit Fresse]], *Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2)*, Mathematical Surveys and Monographs **217** (2017) [[ISBN:978-1-4704-3480-9](https://bookstore.ams.org/surv-217/), [website](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/), [pdf (draft of Part I)](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.648.9110&rep=rep1&type=pdf)]

See also:

* [[Todd Trimble]], _[[toddtrimble:Notes on operads and the Lie operad]]_ ([pdf](http://math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf))

On [[rectification]] of [[symmetric operad|symmetric]] [[colored operads]]:

* {#PavlovScholbach18} [[Dmitri Pavlov]], [[Jakob Scholbach]], *Admissibility and rectification of colored symmetric operads*, Journal of Topology **11** 3 (2018) 559-601 $[$[doi:10.1112/topo.12008](https://doi.org/10.1112/topo.12008), [arXiv:1410.5675](https://arxiv.org/abs/1410.5675)$]$

On operads as [[double functors]] between certain [[double categories]]:

* [[Claudio Pisani]], _Operads as double functors_ ([arXiv:2208.07028](https://arxiv.org/abs/2208.07028))

[[!redirects operads]]

[[!redirects coloured operad]]
[[!redirects coloured operads]]

[[!redirects colored operad]]
[[!redirects colored operads]]

[[!redirects substitution product]]
[[!redirects substitution products]]


operad for bimodules

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#### Higher algebra
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[operad]] whose [[algebras over an operad]] are triples consistsing of two [[associative algebras]] $A$, $B$ and an $A$-$B$-[[bimodule]] .

## Definition

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

Write $BMod$ for the [[colored operad|colored]] [[symmetric operad]] whose

* [[objects]] are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;

* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

  * if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of [[linear orders]] of $n$ elements;

  * if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of [[linear orders]] of $n$ elements;

  * if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $k \gt k$.

* [[composition]] is given by the composition of linear orders as for the [[associative operad]].

=--

## Properties

### Relation to the associative operad
 {#RelationToTheAssociativeOperad}

There are two canonical inclusions $Assoc \to BMod$ of the [[associative operad]] given by labelling its unique color/object with either $\mathfrak{a}_-$ or $\mathfrak{a}_+$, respectively. For

$$
  (A_1,A_2,N) \colon BMod^\otimes \to \mathcal{C}^\otimes
$$

a morphism to a [[symmetric monoidal category]], there compositions pick the left and the right algebra

$$
  A_i \colon Assoc^\otimes \to BMod^\otimes \stackrel{(A_1, A_2, N)}{\to}
  \mathcal{C}^\otimes
  \,.
$$

There is also a morphism $BMod \to Assoc$ given by forgetting the labels and just remembering the linear orders.

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

This is a [[fibration of (∞,1)-operads]].

=--

In ([Lurie](#Lurie)) this appears as remark 4.3.1.8.

This is such that for

$$
  A \colon Assoc^\otimes \to \mathcal{C}^\otimes
$$

an algebra, the composite

$$
  (A, A, A) \colon BMod^\otimes \to Assoc^\otimes \stackrel{A}{\to} \mathcal{C}^\otimes
$$

exhibits $A$ canonically as a bimodule over itself.

### Relation to the operad for modules over an algebra

Similarly, there is an inclusion of the [[operad for modules over an algebra]]

$$
  LMod \to BMod
$$

etc.

### Relation to bitensoring
 {#RelationToBitensoring}

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

A [[coCartesian fibration of (∞,1)-operads]] $\mathcal{C}^\otimes \to BMod^\otimes$, hence the structure of a  $BMod$-[[monoidal (∞,1)-category]] is a **bi[[tensoring]]** of

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

over the (ordinary) [[monoidal (∞,1)-categories]]

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

=--

([Lurie, def. 4.3.1.17](#Lurie))

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

By the [[microcosm principle]], bitensored $(\infty,1)$-categories are the right context into which to internalize [[bimodules]]. See _[Relation to the category of bimodules](#RelationToCategoriesOfBimodules)_ below.

=--

### Relation to categories of bimodules
 {#RelationToCategoriesOfBimodules}

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

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

is the [[(∞,1)-category of (∞,1)-bimodules]] in $\mathcal{C}$.

## Related concepts

* [[operad for modules over an algebra]]

## References

Section 4.3.1 in

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects operad for bimodules over algebras]]


operad for modules over an algebra

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[operad]] whose [[algebras over an operad]] are pairs consisting of an [[associative algebra]] and a [[module]] over that algebra

## Definition

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

The **operad for modules over an algebra** $LM$ is the [[colored operad|colored]] [[symmetric operad]] whose

* [[objects]] are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;

* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

  * if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of [[linear orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$;

  * if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$;

  * otherwise: the empty set;

* [[composition]] is given by composition of linear orders as for the [[associative operad]].

=--

In ([Lurie](#Lurie)) this appears as def. 4.2.1.1.

## Properties

### Relation to the associative operad

Write $Assoc$ for the [[associative operad]], regarded as a [[symmetric operad]].

There is a canonical inclusion morphism

$$
  Assoc \to LM
$$

given by labeling the unique object/color of $Assoc$ with $\mathfrak{a}$. For $(A,N) \colon LM^\otimes \to \mathcal{C}^\otimes$ a map to a [[symmetric monoidal category]] the composite

$$
  A \colon Assoc^\otimes \to LM^\otimes \stackrel{(A,N)}{\to} \mathcal{C}^\otimes
$$

is the corresponding [[associative algebra]].

There is also a morphism

$$
  LM \to Assoc
$$

given by forgetting the color and just remembering the linear orders. This is such that for

$$
  A \colon Assoc^\otimes \to \mathcal{C}^\otimes
$$

and algebra, the composite

$$
  (A,A) \colon LM^{\otimes} \to Assoc^{\otimes} \to \mathcal{C}^\otimes
$$

is the algebra canonically regarded as a module over itself.

### Relation to the operad for bimodules

Analogous relations exist to the [[operad for bimodules over algebras]].

(...)

## Related concepts

* [[associative operad]], [[commutative operad]]

* [[operad for bimodules over algebras]]

* [[model structure on modules over an algebra over an operad]]

## References

Section 4.2.1 of

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

[[!redirects (∞,1)-operad for (∞,1)-modules over an A-∞ algebra]]

[[!redirects operad for modules]]


operad of set operations

Let $X$ be a [[set]]. The usual operations [[intersection]], [[union]], [[symmetric difference]], [[complement]]... on the [[subsets]] of $X$ can be presented in an homogeneous way using an operation $\multiscripts{_^n}{\Delta}{_i^j}:\mathcal{P}(X)^{\times n} \rightarrow \mathcal{P}(X)$ for every $n \ge 1$ and $0 \le i \le j \le n$. The [[operad]] thus generated gives much more operations.

For every $n \ge 1$, $A_{1},...,A_{n}$ subsets of $X$, and $0 \le i \le j \le n$, define the set:
$
\multiscripts{_^n}{\Delta}{_i^j}(A_{1},...,A_{n}) = \{x \in A_{1} \cup ... \cup A_{n}, i \le |k \in [1,n], x \in A_{k}| \le j\}
$

We then have:

* $A_{1} \cap ... \cap A_{n} = \multiscripts{_^n}{\Delta}{_n^n}(A_{1},...,A_{n})$
* $A_{1} \cup ... \cup A_{n} = \multiscripts{_^n}{\Delta}{_1^n}(A_{1},...,A_{n})$
* $A_{1} \Delta A_{2} = \multiscripts{_^2}{\Delta}{_1^1}(A_{1}, A_{2})$
* $\emptyset = \multiscripts{_^2}{\Delta}{_1^1}(A_{1},A_{1})$
* $X = \multiscripts{_^n}{\Delta}{_0^n}(A_{1},...,A_{n})$
* $X - A_{1} = \multiscripts{_^1}{\Delta}{_0^0}(A_{1})$
* $A_{1} = \multiscripts{_^1}{\Delta}{_1^1}(A_{1})$

We also have:

* $X - (A_{1} \cup ... \cup A_{n}) = \multiscripts{_^n}{\Delta}{_0^0}(A_{1},...,A_{n})$
* $X - (A_{1} \cap ... \cap A_{n}) = \multiscripts{_^n}{\Delta}{_0^{n-1}}(A_{1},...,A_{n})$
* $\multiscripts{_^n}{\Delta}{_1^1}(A_{1},...,A_{n})$ is the set of elements which are in exactly one $A_{i}$.

We get an operad by starting with the operations $\multiscripts{_^n}{\Delta}{_i^j}$ and composing them together.


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#### Higher algebra
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[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _operadic (∞,1)-Grothendieck construction_ is the generalization of the [[(∞,1)-Grothendieck construction]] from [[(∞,1)-categories]] to [[(∞,1)-operads]].

## Properties

Notice that where in the categorical context we had [[pseudofunctors]]

$$
  C \to Cat
$$

and then in the [[(∞,1)-category theory|(∞,1)-category theoretic]] context [[(∞,1)-functors]]

$$
  C \to (\infty,1)Cat
  \,,
$$

as input if the Grothendieck construction, in the [[(∞,1)-operad|(∞,1)-operadic]] context such morphisms

$$
  P \to (\infty,1)Cat^{\times}
$$

have the interpretation of [[∞-algebra over an (∞,1)-operad]].

### Equivalence between $\infty$-Algebras and fibrations

For $P$ an [[(∞,1)-operad]], there is an [[equivalence of (∞,1)-categories]] between [[∞-algebra over an (∞,1)-operad|∞-algebras]] over $P$ in [[(∞,1)Cat]] and opCartesian fibrations into $P$.

This is the central theorem in ([Heuts](#Heuts)).

## Related concepts

* [[Cartesian fibration of dendroidal sets]]

## References

A construction modeled on [[dendroidal sets]] is discussed in

* [[Gijs Heuts]], _Algebras over infinity-operads_ ([arXiv:1110.1776](http://arxiv.org/abs/1110.1776))
  {#Heuts}

In section 2.1.3 of

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

the statement of the above equivalence is essentially taken as a definition.

[[!redirects operadic Grothendieck construction]]

[[!redirects operadic (infinity,1)-Grothendieck construction]]


operadic Dold-Kan correspondence

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###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

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

## Idea

The _monoidal Dold-Kan correspondence_ relates [[algebras over an operad]] in abelian [[simplicial group]]s with algebras over an operad in [[chain complex]]es.

This generalizes the [[monoidal Dold-Kan correspondence]].

## Statements

* In ([Mandell](#Mandell)) a Quillen equivalence between [[E-infinity algebra]]s in chain complexes and in simplicial abelian groups is demonstrated.

* In ([Richter](#Richter)) it is shown that for any reduced [[operad]] $\tilde \mathcal{O}$ in $Ch_\bullet^+(Mod)$, which is a [[resolution]] of an operad in $Mod$, the inverse map $\Gamma : Ch_\bullet^+ \to sAb$ of the [[Dold-Kan correspondence]] lifts to a  [[Quillen adjunction]] between homotopy $\mathcal{O}$-algebras in $Ch_\bullet(Mod)$ and in $Mod^{\Delta^{op}}$. (Around therem 5.5.5). It is not shown yet if or under which conditions this is a Quillen equivalence.

## Related concepts

* [[Dold-Kan correspondence]]

* [[monoidal Dold-Kan correspondence]]

* **operadic Dold-Kan correspondence**

## References ##

A Quillen equivalence between $E_\infty$ dg-algebras and $E_\infty$ simplicial algebras is given in

* [[Michael Mandell]], _Topological André-Quillen Cohomology and $E_\infty$ André-Quillen Cohomology_ Adv. in Math., Adv. Math. 177 (2) (2003) 227–279
{#Mandell}

where the [[Moore complex]] functor is the right adjoint.

A construction that allows its inverse to be part of the adjunction is in

* [[Birgit Richter]], _Homotopy algebras and the inverse of the normalization functor_ ([pdf](http://www.math.uni-hamburg.de/home/richter/invjpaa.pdf))
{#Richter}.

[[!redirects operadic Dold--Kan correspondence]]


operation

In a very common sense of the word, "operation" connotes a [[map]] between [[sorts]] or [[types]], especially when interpreting [[algebraic theories]] such as the theory of [[groups]], which involves a set or type $X$ and prescribed operations such as a binary multiplication map $X \times X \to X$. In such cases, an operation is considered as having an assigned [[arity]], for example arity $2$ for a binary operation, or arity $1$ for an unary operation such as the power set operation $P: V \to V$ on the [[universe]] of sets.

There is however no single global definition which precisely delineates all usages of the word "operation" in mathematics; thus, "operation" functions as something of a catch-all term with varying connotations. Some pages within the nLab in which the word appears include

* [[operad]]

* [[cohomology operation]]

* [[six operations]]

* [[quantification]]

among others, many of which describe concepts related to the general notion of algebraic theory (such as [[magma]], [[Lawvere theory]], [[Mal'cev theory]], etc.). See also _[[operator]]_.

One might stop short of synonymizing "operation" simply with the word "[[function]]", but just in what way is hard to say; suffice it to say there may be shades of meaning or context where one is definitely inclined to choose one word over the other.

Compare related terms and language use described in [[function]].

[[!redirects operations]]


operational calculus

__Operational methods/calculus__ are (systematic) methods to reduce some problems in analysis, typically solving differential equations, by reducing them to algebraic problems (and often also algebraic combinatorics). Most well known are methods involving [[Laplace transform]] and [[umbral calculus]]. Though it would fit the above description, this term usually does not refer to the theory of D-modules (where [[algebraic analysis]] is sometimes used term) nor differential algebra with [[Picard-Vessiot theory]] ([wikipedia](https://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory)).

* wikipedia [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus)
* Jan Mikusiński, _Sur les fondements du calcul opératoire_,  Studia Mathematica 11 (1950) 41-70 [doi](https://doi.org/10.4064/sm-11-1-41-70)
* G. Dattoli, E. Palma, , E. Sabia, K. Górska, A. Horzela, K. A. Penson, _[[operational calculus|Operational]] versus umbral methods and the Borel transform_, Int. J. Appl. Comput. Math (2017) 3: 3489 [doi](https://doi.org/10.1007/s40819-017-0315-7)
* Gian-Carlo Rota, D. Kahaner, A. Odlyzko, _On the foundations of combinatorial theory. VIII. Finite operator calculus_,
J. Math. Anal. Appl. 42 (1973), 684–760 [MR345826](https://mathscinet.ams.org/mathscinet-getitem?mr=345826) doi

category: analysis


operational semantics

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[semantics]]

  * [[categorical semantics]]

  * [[denotational semantics]]

  * [[distributional semantics]]

## References

Textbook accounts:

* [[Glynn Winskel]], §2 of: *The Formal Semantics of Programming Languages*, MIT Press (1993) [[ISBN:9780262731034](https://mitpress.mit.edu/9780262731034/the-formal-semantics-of-programming-languages/), [pdf](https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf)]

See also:

* Wikipedia, _[Operational semantics](http://en.wikipedia.org/wiki/Operational_semantics)_


operational tangent space

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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

To define the [[tangent space]] of a [[smooth manifold]], one may equivalently use [[curves]] or [[derivations]] of the [[algebra of functions|algebra]] of [[smooth functions]].  For more general [[smooth spaces]], these are no longer equivalent.

The _operational tangent space_ of a [[generalized smooth space]] (at some point $c$) is the space of [[derivations]] of the [[algebra of functions|algebra]] of [[germs]] (at $c$) of [[smooth functions]].

The alternative notion (using curves) is that of the [[kinematic tangent space]].

## Related concepts

* [[synthetic tangent bundle]], [[kinematic tangent bundle]]

* [[Kähler differentials]]

[[!redirects operational tangent space]]
[[!redirects operational tangent spaces]]

[[!redirects operational tangent bundle]]
[[!redirects operational tangent bundles]]


operator

> This entry is about operators in the sense of [[higher-order logic]], which is the original sense and which has only that name.  For operators in the sense of [[functional analysis]], see _[[linear operator]]_.  For the relation between these, see under Examples [below](#Examples). For yet other kinds of operators see at _[[operation]]_.

# Operators
* table of contents
{: toc}

## Definition

In a [[type theory]] with [[function types]], given a [[type]] $X$, an __operator__ of base type $X$ is a [[term]] of type $(X^X)^{X^X}$ aka $(X \to X) \to (X \to X)$.  This should be distinguished from a [[functional]], which is a term of type $X^{X^X}$ aka $(X \to X) \to X$, and a [[endofunction|function]], which (in this context) is a term of type $X^X$ aka $X \to X$.

More generally, any term whose type has the form $(A^B)^{C^D}$ aka $(D \to C) \to (B \to A)$ may be called an operator, although usually not if any of these types is very trivial (since any type has this form, up to equivalence, if $B,C,D \coloneqq 1$).

Although one typically interprets type theory within [[set theory]] so that operations between types become [[functions]], one may also use [[partial functions]], which is necessary for the connection to [[linear operators]] in [[functional analysis]].

An operator could also be interpreted within [[category theory]] as a [[globe|globular]] [[2-morphism]] in the [[2-category|2-]][[category of sets]].

## Examples
 {#Examples}

The [[Church numeral]]s are operators in the (possibly typed) [[lambda-calculus]]:

* $0 \coloneqq (f \mapsto (x \mapsto x))$,
* $1 \coloneqq (f \mapsto (x \mapsto f x))$,
* $2 \coloneqq (f \mapsto (x \mapsto f f x))$,
* etc.

If we interpret $X$ as the [[real line]], then $X^X$ consists of real-valued maps of a real variable, which form a [[vector space]].  The [[partial function|partially defined]] [[linear maps]] from $X^X$ to itself are the original [[linear operators]].  In [[functional analysis]], we now replace $X^X$ with an arbitrary [[topological vector space]] $V$ (originally but no longer necessarily taken to be a [subspace](subspace#vector_subspaces) of $X^X$) and consider partial linear maps from $V$ to itself instead; so these linear operators are actually functions (meaning [[endofunctions]]) in a type-theoretic sense.  In [[operator theory]], we go further and replace $V^V$ with an arbitrary [[operator algebra]] (originally but no longer necessarily taken to be a [[subalgebra]] of $V^V$); so these operators are unstructured terms in a type-theoretic sense.

A [[differential operator]] is a good example of something that is both an operator in the logical sense (since it turns functions into functions) and a linear operator in the sense of functional analysis (since [[differentiable functions]] form a vector space and [[differentiation]] acts linearly).

## Related concepts

* [[operator product]]

[[!redirects operator]]
[[!redirects operators]]


operator algebra

+-- {: .rightHandSide}
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

An __operator algebra__ is any subalgebra of the algebra of continuous [[linear operators]] on a [[topological vector space]], with composition as the multiplication. In most cases, the space is a separable [[Hilbert space]], and most attention historically has been paid to algebras of [[bounded linear operator|bounded linear operators]]. The operator algebras themselves are often equipped with their own topologies (e.g. norm topology, weak topology, weak$^*$ topology and so on) and sometimes involution. __$C^*$-[[C-star algebra|algebras]]__ are (in the complex case) norm-closed complex $*$-[[subalgebras]] of the algebra of the [[bounded linear operator|bounded linear operators]] on a [[Hilbert space]]. They can be however characterized structurally as a $*$-[[representation]] of a [[Banach algebra]] with involution satisfying some compatibility conditions; the latter could be called abstract $C^*$-algebras (or sometimes $B^*$-algebras). Every _commutative_ unital $C^*$-algebra can be realized as an algebra of functions on a Hausdorff compact space, which is obtained as its [[Gel'fand spectrum]].

Special class of $C^*$-algebras are __[[von Neumann algebra|von Neumann]]__ or $W^*$-algebras (see [wikipedia](http://en.wikipedia.org/wiki/Von_Neumann_algebra)), which are (in the complex case) _weakly closed_ (closed in the weak topology) $*$-subalgebras of the algebra of bounded linear operators on a Hilbert space. In structural theory, an important role among [[von Neumann algebra|von Neumann algebras]] is played by so-called [[von Neumann algebra factor|factors]], characterized by the property that their center is trivial ($1$-dimensional, consisting of "scalars").

Operator algebras play major roles in [[functional analysis]], [[representation theory]], [[noncommutative geometry]] and [[quantum field theory]].

## Topics of interest for the understanding of AQFT

This paragraph will collect some facts of interest for the aspects of [[AQFT]] in the nLab. Unless stated otherwise all operator algebras will be assumed to be unital, i.e. having an identiy denoted by $\mathbb{1}$.

###States {#statesOfOperatorAlgebras}

See [[states]].

## Related concepts

* [[operator]], [[linear operator]]

* [[bounded linear operator]] / [[unbounded linear operator]]

  * [[compact operator]], [[Calkin algebra]]

* [[operator norm]]

* [[C*-algebra]]

  * [[Gelfand-Naimark theorem]]

  * [[Hilbert C*-module]], [[Hilbert C*-bimodule]]

  * [[KK-theory]], [[nonabelian topology]]

* [[von Neumann algebra]]

## References

Textbook:

* [[Richard V. Kadison]], [[John R. Ringrose]], _Fundamentals of the theory of operator algebras_

  * Vol I *Elementary Theory*, Graduate Studies in Mathematics **15**, AMS 1997 ([ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0888.46039&format=complete))

  * Vol II *Advanced Theory*, Graduate Studies in Mathematics **16**, AMS 1997 ([ISBN:978-0-8218-0820-7](https://bookstore.ams.org/gsm-16))

  * Vol III, *Special Topics, Elementary Theory -- An Exercise Approach*, Springer 1991 ([doi:10.1007/978-1-4612-3212-4](https://link.springer.com/book/10.1007/978-1-4612-3212-4), )

* [[Gerard Murphy]], _$C^\ast$-algebras and Operator Theory_, Academic Press 1990 ([doi:10.1016/C2009-0-22289-6](https://doi.org/10.1016/C2009-0-22289-6))

* [[Masamichi Takesaki]], _Theory of operator algebras I, II and III_ ([pdf I](http://www.math.ucla.edu/~mt/Corrections1.pdf) [pdf II](http://www.math.ucla.edu/~mt/Corrections2.pdf) [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0990.46034&format=complete)).

* [[Kehe Zhu]], *An Introduction to Operator Algebras*, CRC Press 1993 ([ISBN:9780849378751](https://www.routledge.com/An-Introduction-to-Operator-Algebras/Zhu/p/book/9780849378751))

* {#Schmuedgen90} [[Konrad Schmüdgen]], _Unbounded  operator  algebras  and  representation  theory_, Operator  theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) ([doi:10.1007/978-3-0348-7469-4](https://link.springer.com/book/10.1007/978-3-0348-7469-4))

* [[Bruce Blackadar]], *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

Lecture notes with an eye towards application in [[quantum physics]]:

* [[Klaas Landsman]], _Lecture notes on operator algebras_ (2011) ([pdf](http://www.math.ru.nl/~landsman/oa2011.pdf))

For application to [[operator K-theory]] and [[KK-theory]]:

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_, Cambridge University Press 1986, second ed. 1999 ([pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf))

See also

* Wikipedia, _[Operator algebras] (http://en.wikipedia.org/wiki/Operator_algebras)_

category: analysis, noncommutative geometry

[[!redirects operator algebra]]
[[!redirects operator algebras]]

[[!redirects algebra of operators]]
[[!redirects algebras of operators]]


operator K-theory

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## Idea

A variant of [[algebraic K-theory]] for [[operator algebras]] ([[Banach algebras]] and in particular [[C-star algebras]]) is _operator K-theory_: it generalizes [[topological K-theory]] via the [[Serre-Swan theorem]] from commutative [[C-star algebras]] to general $C^\ast$-algebras.

## Related concepts

[[!include noncommutative motives - table]]

## References

A standard textbook account is in section 2 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_ ([pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf))

Other introductions include

* M. Rørdam, F. Larsen, N. Laustsen, _An introduction to K-theory for $C^\ast$-algebras_, London Mathematical Society Student Textx

[[!redirects K-theory of C-star algebras]]

[[!redirects algebraic K-theory of operator algebras]]


operator norm

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## Idea

(...)

## Definition

(...)

## Related concepts

* [[operator algebra]]

* [[operator spectrum]]

## References

See also

* Wikipedia, *[Operator norm](https://en.wikipedia.org/wiki/Operator_norm)*

[[!redirects operator norms]]


operator product

An _operator product_ is the [[composition]] of [[linear operators]]. This is the product in an [[operator algebra]].

A [[formal power series]] expansion of an operator product is called an _[[operator product expansion]]_.

[[!redirects operator products]]


operator product expansion

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## Idea

In [[quantum field theory]] the term _operator product expansion_ (OPE) refers to [[Taylor expansion]] of products in the [[algebra of quantum observables]] of [[field observables]] $\mathbf{\Phi}^a(x)$ in the form

$$
  \label{SchematicOPE}
  \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)
  \;\sim\;
  C^{a b}{}_c(x) \mathbf{\Phi}^c(y)
  \,.
$$

Here "$\sim$" means something like "as $x$ gets close to $y$ the [[expectation values]] in the given [[vacuum state]] of [[observables]] proportional to the left hand side approach those of observables proportional to the right hand side", schematically:

$$
  \left\langle
    \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)
    A
  \right\rangle
  \underset{x \to y}{\longrightarrow}
  C^{a b}{}_c(x)
  \left\langle
    \mathbf{\Phi}^c(y)
    A
  \right\rangle
$$

If the [[algebra of observables]] is [[representation|represented]] via [[operator-valued distributions]] [[action|acting]] on some [[Hilbert space]], as commonly assumed, then this becomes an expansion of ([[product of distributions|distributional]]) [[operator products]], whence the name _operator product expansion_ ([Wilson 69](#Wilson69), [Wilson-Zimmermann 72](#WilsonZimmermann72)).

For [[conformal field theory]] and specifically for [[2d CFT]] the operator product expansion is well understood, is neatly captured by the concept of [[vertex operator algebras]]. Via the _[[conformal bootstrap]]_ the equation (eq:SchematicOPE) defines or at least constrains the conformal field theory ([Ferrara-Grillo-Gatto 73](#FerraraGrilloGatto73), [Polyakov 74](#Polyakov74), [Belavin-Polyakov-Zamolodchikov 84](#BelavinPolyakovZamolodchikov84)).

Attempts to similarly use the concept of the operator product expansion to _define_ more general [[perturbative quantum field theory]] (and possibly [[non-perturbative quantum field theory]]) are discussed in ([Hollands-Olbermann 09](#HollandsOlbermann09), [Holland-Hollands 13](#HollandHollands13)). The concept of [[factorization algebras]], rooted as it is in that of [[vertex operator algebras]], may be thought of as meaning to axiomatize operator product expansions in the generality of Euclidean ([[Wick rotation|Wick rotated]]) field theory.

Apparently [Kontsevich 10](#Kontsevich10) suggested that [[perturbative QFT]] should be thought of as the [[deformation theory]] controlled by a certain [[L-infinity algebra]] which is constructed from OPEs.

## See also

* [[Vertex operator algebra]]

## References

The general concept is due to

* {#Wilson69} [[Kenneth Wilson]], _Nonlagrangian models of current algebra_, Phys. Rev., 179:1499–1512, 1969.

* {#WilsonZimmermann72} [[Kenneth Wilson]], [[Wolfhart Zimmermann]], _Operator product expansions and composite field operators in the general framework of quantum field theory_, Commun. Math. Phys., 24:87–106, 1972

and is further developed in

* {#HollandsOlbermann09} [[Stefan Hollands]], Heiner Olbermann, _Perturbative Quantum Field Theory via Vertex Algebras_, J. Math. Phys. 50:112304, 2009 ([arXiv:0906.5313](https://arxiv.org/abs/0906.5313))

* {#HollandHollands13} Jan Holland, [[Stefan Hollands]], _Operator product expansion algebra_,  	J. Math. Phys. 54, 072302 (2013) ([arXiv:1205.4904](https://arxiv.org/abs/1205.4904))

Historical recollections regarding [[non-perturbative effects]] in [[QCD]] ([[hadrodynamics]]) via operator product expansion:

* [[Mikhail Shifman]], *OPE-based Methods in Nonperturbative QCD*, Eur. Phys. J. C (2022) [[arXiv:2208.10600](https://arxiv.org/abs/2208.10600)]

Development for [[AQFT on curved spacetime|QFT on curved spacetimes]]:

* [[Stefan Hollands]], [[Robert Wald]], _Axiomatic quantum field theory in curved spacetime_, Commun. Math. Phys. 293:85-125, 2010 ([arXiv:0803.2003](https://arxiv.org/abs/0803.2003))

* [[Stefan Hollands]], [[Robert M. Wald]], *The Operator Product Expansion in Quantum Field Theory* [[arXiv:2312.01096](https://arxiv.org/abs/2312.01096)]

Brief survey:

* _[Operator Product Expansion](http://home.uni-leipzig.de/tet/?page_id=288)_

The relevance of the concept of OPEs to [[conformal field theory]] ([[conformal bootstrap]]) was noticed in

* {#FerraraGrilloGatto73} S. Ferrara, A. F. Grillo, and R. Gatto, _Tensor representations of conformal algebra and conformally covariant operator product expansion_, Annals Phys. 76 (1973) 161–188

* {#Polyakov74} [[Alexander Polyakov]], _Nonhamiltonian approach to conformal quantum field theory_, Zh. Eksp. Teor. Fiz. 66 (1974) 23–42

* {#BelavinPolyakovZamolodchikov84} [[Alexander Belavin]], [[Alexander Polyakov]], and [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two-dimensional quantum field theory_, Nucl. Phys. B241 (1984) 333–380

There was also a talk

* {#Kontsevich10} [[Maxim Kontsevich]], talk at _[Renormalization: algebraic, geometric and probabilistic aspects](http://math.univ-lyon1.fr/~calaque/Colloque-schedule.htm)_,  Institut Camille Jordan (Lyon) June 16 - 18, 2010

On [[operator product expansions]] in 2D [[conformal field theories]] (via [[holography as Koszul duality]]):

* [[Kevin Costello]], [[Natalie Paquette]], _On the associativity of one-loop corrections to the celestial OPE_ ([arXiv:2204.05301](https://arxiv.org/abs/2204.05301))

* [[Natalie Paquette]], _4d physics from 2d chiral correlators_ ([slides](https://indico.cern.ch/event/1085701/contributions/4940742/attachments/2481322/4259658/Natalie%20Paquette%20Monday-Research-Paquette.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=967d7949-dc4e-47e8-befb-8aa53c589ba9))

[[!redirects OPE]]
[[!redirects OPEs]]
[[!redirects operator product expansions]]


Operator theory in the C*-algebra framework

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This page collects pointers related to the article

* [[Stanisław Woronowicz]], K. Napiórkowski, _Operator theory in the $C^\ast$-algebra framework_, Reports on Mathematical Physics Volume 31, Issue 3, June 1992, Pages 353–371 ([publisher](http://www.sciencedirect.com/science/article/pii/003448779290025V), [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/OP.pdf))

on [[operator algebra]] and [[C*-algebras]], based on

* [[Stanisław Woronowicz]], _Unbounded elements affiliated with $C^\ast$-algebras and
non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991)

category: reference

[[!redirects www.fuw.edu.pl/~slworono/PDF-y/OP.pdf]]


operator topology

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# Contents
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{: toc}

## Idea

An __operator topology__ is an abbreviation of a __[[topology]] on a space of (continuous linear) [[linear operator|operator]]s__ between [[topological vector space]]s over a fixed [[field]] $k$ of [[real number|reals]] or [[complex number|complexes]] (possibly also p-adics, skewfield of [[quaternions]] etc.). In other words the [[hom-sets]] in the category of topological vector spaces as objects and continuous linear operators as morphisms are equipped with an operator topology.

There are many widely used topologies, some with standard names. Let $L(V,W) = Hom_{TVS}(V,W)$ be the set of continuous linear operators.

* __weak operator topology__ on $L(V,W)$ is given by the basis of open neighborhoods of zero given by sets of the form $U(x,f) = \{A\in L(V,W) : |f(A(x)) \lt 1 \} $ where $x\in V$ and $f\in W^* = Hom_{TVS}(W,k)$. A sequence $(A_n)$ converges to $A$ in weak operator topology iff the sequence $(A_n(x))$ converges to $A(x)$ in the weak topology on $W$. We write $A_n\stackrel{w}\longrightarrow A$ or $w-lim A_n = A$.

* __strong operator topology__: the basis of neighborhoods of zero is given by sets
$N(x,U) = \{A\in L(V,W) \,|\, A v \in U\}$, where $v\in V$ and $U$ is a neighborhood of zero in $W$. For convergence of sequences, we write $A_n\stackrel{s}\longrightarrow A$ or $s-lim A_n= A$.

* __uniform operator topology__: here we assume that $V,W$ are normed spaces with norms $p_V$, $p_W$. Then $L(V,W)$ has a uniform operator topology induced by the norm given by the formula

$$
p(A) = sup_{v\neq 0} \frac{p_W(A v)}{p_V (v)}
$$

* __ultraweak operator topology__ ...

## Properties

### Relation to norm topology

The reason that in the definition of a [[unitary representation]], the strong operator topology on $\mathcal{U}(\mathcal{H})$ is used and not the [[norm topology]], is that only few [[homomorphism]]s turn out to be [[continuous map|continuous]] in the norm topology.

Example: let $G$ be a [[compact topological space|compact]] [[Lie group]] and $L^2(G)$ be the [[Hilbert space]] of square integrable [[measurable function]]s with respect to its [[Haar measure]]. The right [[regular representation]] of $G$ on $L^2(G)$ is defined as

$$
      R: G \to \mathcal{U}(L^2(G))
$$

$$
         g \mapsto (R_g: f(x) \mapsto f(x g))
$$

and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.

## Related entries

* [[U(ℋ)]]

## References

* A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

[[!redirects operator topologies]]

[[!redirects topology on a space of operators]]
[[!redirects topologies on a space of operators]]

[[!redirects strong operator topology]]
[[!redirects weak operator topology]]
[[!redirects uniform operator topology]]


operator-valued distribution

[[!redirects operator-valued distributions]]

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## Idea

An _operator valued distribution_ is a [[distribution]] with values in a space of [[linear operators]].

In mathematically rigorous constructions of [[perturbative quantum field theory]] -- such as via the [[Wightman axioms]] or [[causal perturbation theory]] -- the [[quantum observables]] which naively would be represented by would-be [[linear operators]] on some [[Hilbert space]] assigned to single points of [[spacetime]], such as the field operator "$\Phi(x)$" in [[scalar field theory]], are understood to yield such operators only after "smearing" or "[[adiabatic switching]]", namely after "integrating" them against a [[bump function]] on [[spacetime]]. This realizes these objects as [[continuous linear functionals]] on the space of bump functions with values in [[linear operators]], and hence as [[distributions]] with values not in the real numbers (as for ordinary distributions) but with values in operators. Where an ordinary distribution may be thought of as a "[[generalized function]] with values in the real numbers", such an "operator valued distribution" may be thought of as a "generalized function with values in linear operators". For example the object denoted $\mathbf{\Phi}(x)$ is understood as really being a distribution which to a [[bump function]] $b$ assigns the (well-defined) operator traditionally written $\int_X b(x) \mathbf{\Phi}(x) dvol$.

In [[causal perturbation theory]]/[[locally covariant perturbative AQFT]] these operator-valued distributions are just a means to an end: the actual [[algebra of observables]] is the [[Wick algebra]] of the [[free fields]] and its deformation to the [[interacting field algebra]], both of which are ([[Fedosov deformation quantization|Fedosov-]])[[formal deformation quantizations]]  of algebras of ordinary [[smooth functions]] on [[phase space]]. But since the [[phase space]] in field theory is itself a [[mapping space]] (for the [[scalar field]]) or more generally a [[space of sections]], such functions are conveniently expressed or constructed in terms of distributions.

Indeed the realization of these algebras via distributions makes transparent that

1. the restriction of the [[free field]]'s [[Wick algebra]] to [[microcausal functionals]] is governed by the [[wave front set]] condition for the [[multiplication of distributions]];

1.  the [[renormalization]] freedom in defining the [[S-matrix]]/[[interacting field algebra]] is governed by the freedom of performing [[point-extensions of distributions]] .

This way the theory of [[distributions]] serves to give a mathematically rigorous construction of [[perturbative quantum field theory]]. The notorious "infinities" that allegedly "plague" perturbative quantum field theory in other approaches may be argued (vividly so in [Scharf 95, p. 181-182](#Scharf95)) to be just mathematical errors in handling distributions properly.

Incidentally, in the context of [[causal perturbation theory]]/[[locally covariant perturbative AQFT]] the "operator-valued distributions" are in general just _algebra-valued distributions_, since a [[representation]] of these algebra elements ([[quantum observables]]) as actual [[linear operators]] on some [[Hilbert space]] is not required, in fact it may not generally exists in a suitably invariant sense on [[AQFT on curved spacetimes|on curved spacetimes]] and if it exists it will not be unique, and even if a choice is made, most quantities of physical interest may be computed more directly from the algebra and a algebraic choice of [[state on an operator algebra|state]].

## Definition
 {#Definition}

Let $V$ be a [[topological vector space]] and $X$ a [[smooth manifold]]. A _$V$-valued distribution_ on $X$ is a [[continuous linear functional]] from the space $C^\infty_c(X)$ of [[bump functions]] to $V$.

If instead of $C^\infty_c(X)$ one uses the space $C^\infty(X)$ of all [[smooth functions]] then these are _[[compactly supported distributions]]_, or if one uses the [[Schwartz space]] $\mathcal{S}(X)$, then these are [[tempered distributions]].

## References

An early use of the concept of operator-valued distributions in [[perturbative quantum field theory]] is in the foundations of [[causal perturbation theory]] in

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.

* [[Nikolay Bogoliubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, _General principles of quantum field theory_, Kluwer 1990

For review see

* Pierre Ca Grange, Ernst Werner, _Quantum fields as operator valued distributions and causality_ ([arXiv:math-ph/0612011](https://arxiv.org/abs/math-ph/0612011))

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001


opetope

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## Idea ##

_Opetopes_ are one of the [[geometric shapes for higher structures|geometric shapes]] of cells in the approach to the [[higher category theory]] of [[n-category|n-categories]] and [[∞-categories]] put forward in ([Baez-Dolan 97](#BaezDolan97)) and developed by ([Makkai](#Makkai)) and others: _[[opetopic ∞-categories]]_.

A [[syntax|syntactic]] formalization of [[opetopic ∞-categories]] in the [variant by Palm](#opetopic+omega-category#DefinitionByPalm) is _[[opetopic type theory]]_ ([Finster 12](#Finster12)).

## References ##

An overview is in [chapter 4](http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf#page=63) of

* [[Eugenia Cheng]], [[Aaron Lauda]], _Higher dimensional categories: an illustrated guidebook_ ([pdf](http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf))

and in chapter 7 of

* [[Tom Leinster]], _Higher operads, higher categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)

Opetopes were introduced here:

* {#BaezDolan97} [[John Baez]], [[James Dolan]], Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, _Adv. Math._ **135** (1998), 145--206. ([arXiv:q-alg/9702014](http://arxiv.org/abs/q-alg/9702014))

Some mistakes were corrected in subsequent papers:

* [[Eugenia Cheng]], The category of opetopes and the category of opetopic sets,
_Th. Appl. Cat._ **11** (2003), 353--374.
[arXiv](http://arxiv.org/abs/math/0304284))

* [[Tom Leinster]], Structures in higher-dimensional
category theory.   ([arXiv](http://arxiv.org/abs/math/0109021))

Makkai and collaborators introduced a slight variation they called 'multitopes':

* [[Claudio Hermida]], [[Michael Makkai]], [[John Power]], _On weak higher-dimensional categories I, II_ _Jour. Pure Appl. Alg._ **157** (2001), 221--277 ([journal](http://www.sciencedirect.com/science/article/pii/S0022404999001796), [ps.gz files](http://www.math.mcgill.ca/makkai/multitopicsets/))

* {#Makkai} [[Michael Makkai]], The multitopic $\omega$-category of all multitopic $\omega$-categories.
([web](http://www.math.mcgill.ca/makkai/mltomcat04/))

Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic $n$-categories:

* [[Eugenia Cheng]], Weak $n$-categories: opetopic and multitopic foundations, _Jour. Pure Appl. Alg._ **186** (2004), 109--137.([arXiv](http://arxiv.org/abs/math/0304277))

* [[Eugenia Cheng]], Weak $n$-categories: comparing opetopic foundations, _Jour. Pure Appl. Alg._ **186** (2004), 219--231.
 ([arXiv](http://arxiv.org/abs/math/0304279))

She has also shown that opetopic [[bicategories]] are "the same" as the ordinary kind:

* [[Eugenia Cheng]], Opetopic bicategories: comparison with the classical theory.  ([arXiv](http://arxiv.org/abs/math/0304285))

A higher dimensional [[string diagram]]-notation for opetopes was introduced (as "zoom complexes" in section 1.1) in

* {#KockJoyalBataninMascari07} [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], _Polynomial functors and opetopes_ ([arXiv:0706.1033](http://arxiv.org/abs/0706.1033))

Animated exposition of this higher-dimensional string-diagram notation is in

* [[Eric Finster]], _Opetopic Diagrams 1 - Basics_ ([video](http://www.youtube.com/watch?v=OANwLohwJqk))

* [[Eric Finster]], _Opetopic Diagrams 2 - Geometry_ ([video](http://www.youtube.com/watch?v=E7OvuA1jRKM))

The variant of [Palm opetopic omega-categories](opetopic+omega-category#DefinitionByPalm) is due to

* [[Thorsten Palm]], ...

A [[syntax|syntactic]] formalization of [[opetopic omega-categories]] in terms of [[opetopic type theory]] is in

* {#Finster12} [[Eric Finster]], _Type theory and the opetopes_, talk at HDACT Ljubljana, June 2012 ([pdf](http://sma.epfl.ch/~finster/opetope/types-and-opetopes.pdf))

Something like an implementation of aspects of opetopic type theory _within_ [[homotopy type theory]] is described in

* {#Finster18} [[Eric Finster]], _[[homotopytypetheory:Eric Finster, Towards Higher Universal Algebra in Type Theory|Towards Higher Universal Algebra in Type Theory]]_, [Homotopy Type Theory Electronic Seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html) 2018 ([recording](https://www.youtube.com/watch?v=hlCVHVtAlqQ), [[Agda]] [code](https://github.com/ericfinster/higher-alg))

[[!redirects opetopic set]]
[[!redirects opetopic n-category]]
[[!redirects opetopes]]
[[!redirects opetopic sets]]
[[!redirects opetopic n-categories]]
[[!redirects multitope]]
[[!redirects multitopes]]
[[!redirects multitopic set]]
[[!redirects multitopic n-category]]
[[!redirects multitopic n-categories]]


opetopic omega-category

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## Idea

In [[higher category theory]], a concept of _opetopic omega-category_ is one of _[[weak omega-category]]_ modeled on [[opetope|opetopic]] [[geometric shape for higher structures|shapes]].

There are various flavors of the definition. Typically they all say that an opetopic omega-category is an [[opetopic set]] equipped with certain [[structure]] and [[property]].

## Definitions

### By Baez-Dolan

([Baez-Dolan 97](#BaezDolan97))

### By Makkai

([Hermida-Makkai-Power 01](#HermidaMakkaiPower01), [Makkai](#Makkai))

The actual definition of opetopic $\omega$-categories, called "multitopic $\omega$-categories" here, appears on p.57 of ([Makkai](#Makkai)).

### By Palm
 {#DefinitionByPalm}

In the definition by ([Palm](#Palm)) of opetopic $\omega$-category, the extra [[structure]] on an [[opetopic set]] is that some cells are labeled as "thin" (as [[equivalences]]) and the extra [[property]] are two classes of higher dimensional [[horn]]-filler conditions:

1. given a [[horn]] ("niche", "nook") obtained from an [[opetope]] by discarding the [[codimension]]-0 interior and the outgoing [[codimension]]-1 face, then there is a filler opetope whose codimension-0 interior is marked as an equivalence. The filling outgoing codimension-1 face is marked an equivalence if all the other codimension-1 faces are.

1. given a [[horn]] ("niche", "nook") obtained from an [[opetope]] by discarding the [[codimension]]-0 interior and an incoming codimension-1 face such that all the remaining codimension-1 faces are marked as equivalences, then there exists a filler both whose codimension-1 and codimension-0 cell are marked as equivalences.

This definition has been given a [[syntax|syntactic]] formulation by ([Finster 12](#Finster12)) in terms of _[[opetopic type theory]]_.

However, there is no saturation condition in this definition.

## References

An overview is in [chapter 4](http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf#page=63) of

* [[Eugenia Cheng]], [[Aaron Lauda]], _Higher dimensional categories: an illustrated guidebook_ ([pdf](http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf))

and in chapter 7 of

* [[Tom Leinster]], _Higher operads, higher categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)

Opetopes were introduced here:

* {#BaezDolan97} [[John Baez]], [[James Dolan]], Higher-dimensional algebra
III: $n$-categories and the algebra of opetopes, _Adv. Math._ **135** (1998), 145--206. ([arXiv:q-alg/9702014](http://arxiv.org/abs/q-alg/9702014))

Some mistakes were corrected in subsequent papers:

* [[Eugenia Cheng]], The category of opetopes and the category of opetopic sets,
_Th. Appl. Cat._ **11** (2003), 353--374.
[arXiv:0304284](http://arxiv.org/abs/math/0304284))

* [[Tom Leinster]], _Structures in higher-dimensional
category theory_, ([arXiv:0109021](http://arxiv.org/abs/math/0109021))

Makkai and collaborators introduced a slight variation they called 'multitopes':

* {#HermidaMakkaiPower01} [[Claudio Hermida]], [[Michael Makkai]], [[John Power]], _On weak higher-dimensional categories I, II_ _Jour. Pure Appl. Alg._ **157** (2001), 221--277 ([journal](http://www.sciencedirect.com/science/article/pii/S0022404999001796), [ps.gz files](http://www.math.mcgill.ca/makkai/multitopicsets/), [[HermidaMakkaiPower01.pdf:file]])

* {#Makkai} [[Michael Makkai]], The multitopic $\omega$-category of all multitopic $\omega$-categories.
([pdf](http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf), [web](http://www.math.mcgill.ca/makkai/mltomcat04/))

Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic $n$-categories:

* {#Cheng03} [[Eugenia Cheng]], _Weak $n$-categories: comparing opetopic foundations_, Jour. Pure Appl. Alg. **186** (2004), 219--231.
 ([arXiv:0304279](http://arxiv.org/abs/math/0304279))

* {#Cheng04} [[Eugenia Cheng]], _Weak $n$-categories: opetopic and multitopic
foundations_, Jour. Pure Appl. Alg._**186** (2004), 109--137.([arXiv:0304277](http://arxiv.org/abs/math/0304277))

She has also shown that opetopic [[bicategories]] are "the same" as the ordinary kind:

* [[Eugenia Cheng]], Opetopic bicategories: comparison with the classical theory.  ([arXiv](http://arxiv.org/abs/math/0304285))

A higher dimensional [[string diagram]]-notation for opetopes was introduced (as "zoom complexes" in section 1.1) in

* {#KockJoyalBataninMascari07} [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], _Polynomial functors and opetopes_ ([arXiv:0706.1033](http://arxiv.org/abs/0706.1033))

Animated exposition of this higher-dimensional string-diagram notation is in

* [[Eric Finster]], _Opetopic Diagrams 1 - Basics_ ([video](http://www.youtube.com/watch?v=OANwLohwJqk))

* [[Eric Finster]], _Opetopic Diagrams 2 - Geometry_ ([video](http://www.youtube.com/watch?v=E7OvuA1jRKM))

The variant of [Palm opetopic omega-categories](opetopic+omega-category#DefinitionByPalm) is due to

* {#Palm} [[Thorsten Palm]], ...

A [[syntax|syntactic]] formalization of [[opetopic omega-categories]] in terms of [[opetopic type theory]] is in

* {#Finster12} [[Eric Finster]], _Type Theory and the Opetopes_, talk at [Polish Seminar on Category Theory and its Applications, June 2012](http://www.mimuw.edu.pl/~zawado/SemTK/OSTKA.html)  ([pdf](https://ncatlab.org/nlab/files/FinsterTypesAndOpetopes2012.pdf))

[[!redirects opetopic omega-categories]]

[[!redirects opetopic ∞-category]]
[[!redirects opetopic ∞-categories]]


opetopic type theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Directed Type Theory
+-- {: .hide}
[[!include directed homotopy type theory - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Opetopic type theory_ ([Finster 12](#Finster12)) is a higher dimensional [[directed homotopy type theory|directed]] [[homotopy type theory|homotopy]] [[type theory]] for [[omega-categories]], i.e. of [[infinity-categories]] in the full sense of [[(infinity,infinity)-categories|$(\infty,\infty)$-categories]].  Specifically, it realizes the higher-dimensional [[horn]]-filler conditions in the definition of [[opetopic omega-categories]] [due to Palm](opetopic+omega-category#DefinitionByPalm) as [[inference rules]] of [[term introduction]] for a higher-dimensional kind of [[formal logic]] (but it presently ignores the saturation condition on the thin cells).

Where in [[homotopy type theory]] there is an [[identity type]] of any type axiomatizing the [[infinity-groupoid]] structure of that type in, effectively, the style of a [[globular omega-groupoid]], in opetopic type theory there is axiomatized instead a type of [[k-morphisms]] for all $k$ in the [[geometric shape for higher structures|shape]] of [[opetopes]].

Hence given a base type $\mathbf{B}\mathcal{U}$ -- and a priori there is just one, to be thought of as the categorically delooped [[type universe]], see below  -- a type is thus effectively identified with the shape of an [[opetope]] and thought of as the type of [[k-morphisms]] in $\mathbf{B}\mathcal{U}$, the only other data being a possible marking that identifies it as just the sub-type of $k$-[[equivalences]] of the given opetopic shape.

The only [[natural deduction|deduction rule]] is opetopic [[type formation]] and a [[term introduction]] rule which expresses the evident [[Kan complex|Kan filler-like]] condition saying that if a term of given opetopic shape is an outer [[horn]] (here: a "nook") of $k$-cells, then a $k$-dimensional filler term is deduced, and if an outer or inner [[horn]] has a boundary by equivalences then a filler term marked as an equivalence is deduced.

Due to the genuinely [[infinity-category|omega-categorical]] nature of the setup, it makes sense to think of the (a priori) unique base type $\mathbf{B}\mathcal{U}$ as the categorical [[delooping]] of a [[type universe]] $\mathcal{U}^\times$ being a [[monoidal (infinity,n)-category]] [[omega-category]], and hence of the 1-[[endomorphisms]] (hence the terms of shape the [[interval category|directed interval]]  $(\mathbf{B}\mathcal{U}\to \mathbf{B}\mathcal{U})$ ) as being the [[types]] in that universe. Composition of 1-morphism hence implies a [[type formation]] for [[multiplicative conjunction]]-types in a directed kind of [[linear homotopy type theory]].

Introducing an [[axiom]] to this system just means postulating terms of (the type of) the shape of prescribed [[opetopes]]. For example:

* **$n$-truncation** -- For the language inside an [[(infinity,n)-category]] one demands that all [[k-morphisms]] for $k \geq n+1$ are marked as [[equivalences]];

* **$k$-adjoints** -- To axiomatize that two $k$-morphism are [[left adjoint]] and [[right adjoint]] to each other, respectively, one postulates the existence of $k+1$-morphisms serving as [[unit of an adjunction]] and [[counit of an adjunction]] and of a [[3-morphism]] marked as [[equivalence]] which axiomatizes the [[zig-zag identities]].

Imposing $n$-trunction and adjoints for all [[k-morphisms]] for $0 \leq k \leq n$ therefore axiomatizes the language for a [[free construction|free]] [[(infinity,n)-category with adjoints]] on a single object. Categorical looping (which is immediate and primitive in the system, as above) hence gives the [[free construction|free]] [[(infinity,n)-category with duals]] on a single object.

This is the structure to which the [[cobordism hypothesis]] applies. Of course the proof of the cobordism hypothesis is not formulated in opetopic type theory and one would have to show that the language it is formulated in is suitably equivalent to that of opetopic type theory, but inspection in low dimension shows that the [[higher dimensional traces]] that opetopic type theory produces are of just the right kind.

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References
 {#References}

Opetopic type theory is due to

* {#Finster12} [[Eric Finster]], _Type Theory and the Opetopes_, talk at [Polish Seminar on Category Theory and its Applications, June 2012](http://www.mimuw.edu.pl/~zawado/SemTK/OSTKA.html) ([[FinsterTypesAndOpetopes2012.pdf:file]])

It [[syntax|syntactically]] implements the definition of [[opetopic omega-category]] due to

* {#Palm} [[Thorsten Palm]], ...

by interpreting the higher-dimensional [[horn]]-filler conditions given there as [[inference rules]].

([Palm](#Palm) shows that any opetopic set satisfying these filler conditions satisfies the property of [[Michael Makkai]]'s definition of a opetopic set that qualifies as an [[opetopic omega-category]]. The converse is presently unknown.)

The higher dimensional [[string diagram]]-notation used here for opetopes was introduced (as "zoom complexes" in section 1.1) in

* {#KockJoyalBataninMascari07} [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], _Polynomial functors and opetopes_ ([arXiv:0706.1033](http://arxiv.org/abs/0706.1033))

Animated exposition of this higher-dimensional string-diagram notation is in

* [[Eric Finster]], _Opetopic Diagrams 1 - Basics_ ([video](http://www.youtube.com/watch?v=OANwLohwJqk))

* [[Eric Finster]], _Opetopic Diagrams 2 - Geometry_ ([video](http://www.youtube.com/watch?v=E7OvuA1jRKM))

A computer [[proof assistant]] for working with opetopic type theory is

* [[Eric Finster]], _Orchard_ ([GitHub repository](https://github.com/ericfinster/orchard))

* [[Eric Finster]], _[Opetope!](http://sma.epfl.ch/~finster/opetope/opetope.html)_ (online opetopic type theory system)

A video tutorial for Orchard is

* [[Eric Finster]], _Orchard 1 - Introduction_ ([video](http://www.youtube.com/watch?v=sEVJJCwohW0&list=PLWAw3zSOqFVKZ-LzAq4hQaKIVj8zjtfCs))

* [[Eric Finster]], _Orchard 2 - Modifying the shape_ ([video](http://www.youtube.com/watch?v=sEVJJCwohW0&list=PLWAw3zSOqFVKZ-LzAq4hQaKIVj8zjtfCs))

* [[Eric Finster]], _Orchard 3 - Composition and associativity_ ([video](http://www.youtube.com/watch?v=Vxnf_3Bxj2o&index=5&list=PLWAw3zSOqFVKZ-LzAq4hQaKIVj8zjtfCs))

* [[Eric Finster]], _Orchard 4 - Identities and invertibility_ ([video](http://www.youtube.com/watch?v=DRFjUSaNexQ&list=PLWAw3zSOqFVKZ-LzAq4hQaKIVj8zjtfCs&index=6))

Apparently Orchard is discontinued in favour of an online version of similar functionality:

* [[Eric Finster]], _Opetopic_ ([GitHub repository](https://github.com/ericfinster/opetopic)) ([Website](http://opetopic.net/))

Something like an implementation of aspects of opetopic type theory _within_ [[homotopy type theory]] is described in

* {#Finster18} [[Eric Finster]], _[[homotopytypetheory:Eric Finster, Towards Higher Universal Algebra in Type Theory|Towards Higher Universal Algebra in Type Theory]]_, [Homotopy Type Theory Electronic Seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html) 2018 ([recording](https://www.youtube.com/watch?v=hlCVHVtAlqQ), [[Agda]] [code](https://github.com/ericfinster/higher-alg))

* {#AlliouxFinsterSozeau21} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_, 2021 ([hal:03133144](https://hal.inria.fr/hal-03133144), [pdf](https://hal.inria.fr/hal-03133144/document))

* [[Eric Finster]], *The $(\infty,1)$-category of Types*, talk at [[CQTS]] (Dec 2022) [slides: [[Finster-CQTS2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/share/MFJT2B2SX8XUsB0n4z-hwMKb-mxyenQhE5HgBC72qIdHb-ixm7FqQ_KN8hDa0YWl.tEp5kHPzkef1ZDHU)]

  > Abstract: A major outstanding difficulty in [[homotopy type theory|Homotopy Type Theory]] is the description of [[coherence law|coherent]] [[nLab:higher algebra|higher algebraic structures]]. As an example, we know that the algebraic structure possessed by the collection of types and functions between them is *not* a traditional 1-category, but rather an [[(infinity,1)-category|(∞,1)-category]]. In this talk, I will describe how the addition of a finite collection of additional definitional equalities designed to render the notion of "[[opetopic type theory|opetopic type]]" definable in fact allows one to construct the [[(∞,1)-category]] structure on the [[universe of types]].


oplax monoidal functor

+-- {: .rightHandSide}
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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

If $C$ and $D$ are monoidal categories, an **oplax monoidal functor** $F : C \to D$ is defined to be a [[lax monoidal functor]] $F: C^{op} \to D^{op}$.  So, among other things, tensor products are preserved up to morphisms of the following sort in $D$:

$$\Delta_{c,c'} : F(c \otimes c') \to F(c) \otimes F(c')$$

which must satisfy a certain [[coherence law]].

## Properties

An oplax monoidal functor sends [[comonoids]] in $C$ to comonoids in $D$, just as a lax monoidal functor sends [[monoids]] in $C$ to monoids in $D$.  For this reason an oplax monoidal functor is sometimes called a **lax comonoidal functor**.  The other obvious terms, __colax monoidal__ and __lax opmonoidal__, also exist (or at least are attested on Wikipedia).

Note that a *strong* opmonoidal functor --in which the morphisms $\phi$ are required to be [[isomorphisms]]--- is the same thing as a [[strong monoidal functor]].

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

A functor with a [[right adjoint]] is oplax monoidal if and only if that right adjoint is a [[lax monoidal functor]].

=--

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

This is a special case of the statement of [[doctrinal adjunction]] for the case of the [[2-monad]] whose algebras are monoidal categories,

=--

Here is the explicit construction of the oplax monoidal structure from a lax monoidal structure on a right adjoint:

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a pair of [[adjoint functor]]s and let $(C,\otimes)$ and $(D,\otimes)$ be structures of [[monoidal categories]].

Then if $R$ is a [[lax monoidal functor]] $L$ becomes an oplax monoidal functor with oplax unit

$$
  L(I_D) \to I_C
$$

the [[adjunct]] of the lax unit $I_D \to R(I_D)$ of $R$ and with oplax monoidal transformation

$$
    (L (x \otimes y) \stackrel{\Delta_{x,y}}{\to} L(x) \otimes L(y))
$$

given by the [[adjunct]] of

$$
  x \otimes y \stackrel{\eta_x \otimes \eta_y}{\to}
  R L x \otimes R L y
  \stackrel{\nabla_{L x, L y}}{\to}
  R(L x \otimes L y)
  \,.
$$

By the formula for [[adjuncts]] in terms of the [[adjunction counit]] ([this prop.](adjunct#AdjunctionCoUnitGiveesAllAdjuncts)) this adjunct is the composite

$$
  L(x \otimes y)
    \stackrel{L(\eta_x \otimes \eta_y)}{\longrightarrow}
  L(R L x \otimes R L y)
    \stackrel{L(\nabla_{L x, L y})}{\longrightarrow}
  L R(L x \otimes L y)
    \stackrel{\epsilon_{L x \otimes L y}}{\longrightarrow}
  L x \otimes L y
  \,.
$$

This appears for instance on [p. 17](http://arxiv.org/PS_cache/math/pdf/0209/0209342v2.pdf#page=17) of ([SchwedeShipley](#SchwedeShipley)).

## Related concepts

* [[monoidal functor]]

* [[strong monoidal functor]]

* [[lax monoidal functor]]

* **oplax monoidal functor**

* [[bilax monoidal functor]]

* [[Frobenius monoidal functor]]

## References

The construction of oplax monoidal functors from right adjoint lax monoidal functors is considered for instance around page 17 of

* [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_ , Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv](http://arxiv.org/abs/math.AT/0209342))
{#SchwedeShipley}

[[!redirects oplax monoidal functors]]

[[!redirects lax comonoidal functor]]
[[!redirects colax monoidal functor]]
[[!redirects comonoidal functor]]

[[!redirects lax comonoidal functors]]
[[!redirects colax monoidal functors]]
[[!redirects comonoidal functors]]


opposite (infinity,1)-category

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

#Contents#
* table of contents
{:toc}

## Idea

The analog of the notion of [[opposite category]] for [[(∞,1)-categories]].

## Definition

For $C$ an [[(∞,1)-category]] regard it in its incarnation as a [[∞-groupoid]]-[[enriched category]]. Then its opposite is the $(\infty,1)$-category $C^{op}$ with the same objects and with [[hom-object]]s given by

$$
  C^{op}(X,Y) := C(Y,X)
$$

with the obvious composition law.

With $C$ incarnated as a [[quasi-category]], the [[simplicial set]] $C^{op}$ is that obtained by reversing the order of all the face and degeneracy maps. See [[opposite quasi-category]].

## Properties

The operation extends to an [[automorphism|automorphic]] [[(∞,1)-functor]]

$$
  op : (\infty,1)Cat \to (\infty,1)Cat
$$

from [[(∞,1)Cat]] to itself. Up to equivalence, this is the only nontrivial such automorphism. For more on this see [[(∞,1)Cat]].

## Related concepts

* [[opposite model category]]

[[!redirects opposite quasi-category]]
[[!redirects opposite (∞,1)-category]]


opposite 2-category

+-- {: .rightHandSide}
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[2-category]] $K$ has three different [[opposite category|opposites]] (two independent):

* The **$1$-cell dual** $K^{op}$, in which the $1$-cells but not the $2$-cells are reversed.
* The **$2$-cell dual** $K^{co}$, in which the $2$-cells but not the $1$-cells are reversed.
* The **bidual** $K^{co op}$ in which both $1$-cells and $2$-cells are reversed.

## Related pages

* [[opposite category]]

* [[contravariant 2-functor]]

* [[3-category with contravariance]]

* [[opposite adjunction]]

* [[reverse monoidal category]]

[[!redirects opposite 2-category]]
[[!redirects opposite 2-categories]]

[[!redirects co]]
[[!redirects coop]]


opposite adjunction

+-- {: .rightHandSide}
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###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a pair of [[adjoint functors]]

$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{C}
$$

there is an induced [[adjunction]] of [[opposite functors]]  between their [[opposite categories]] of the form

$$
  \mathcal{C}^{op}
    \underoverset
      {\underset{L^{op}}{\longrightarrow}}
      {\overset{R^{op}}{\longleftarrow}}
      {\;\;\;\;\bot\;\;\;\;}
  \mathcal{D}^{op}
  \,.
$$

Hence where $L$ was the [[left adjoint]], its [[opposite functor|opposite]] becomes the [[right adjoint]], and dually for $R$.

This is immediate from the definition of [[opposite categories]] and the characterization of adjoint functors via the corresponding [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism).

The [[adjunction unit]] of the opposite adjunction has as components the components of the original [[adjunction counit]], regarded in the opposite category, and dually:

$$
  \epsilon^{R^{op} L^{op}}_{d}
  \;\colon\;
  R^{op}\circ L^{op}(d)
  \xrightarrow{\;\;  \big( \eta^{R L}_d \big)^{op}  \;\;}
  d
  \,,
  {\phantom{AAAAAA}}
  \eta^{L^{op} R^{op}}_{c}
  \;\colon\;
  c
  \xrightarrow{\;\;  \big( \epsilon^{L R}_c \big)^{op}  \;\;}
  L^{op} \circ R^{op}(c)
  \,.
$$

## Related concepts

* [[opposite Quillen adjunction]]

* [[opposite category]], [[opposite functor]]

* [[adjunction]]

* [[opposite 2-category]]

[[!redirects opposite adjunctions]]

[[!redirects opposite adjoint functor]]
[[!redirects opposite adjoint functors]]

[[!redirects adjoint opposite functor]]
[[!redirects adjoint opposite functors]]