small object argument

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## Idea ##

Quillen's _small object argument_ is a transfinite functorial construction of a [[weak factorization system]] (on some [[category]] $C$) that is **cofibrantly generated** by a set of [[morphisms]] $I \subset Mor(C)$.

This construction is notably used in the theory of [[model category|model categories]] and in particular [[cofibrantly generated model category|cofibrantly generated model categories]] in order to demonstrate the existence of the  required factorization of morphisms into composites of (acyclic) cofibrations following by (acyclic) fibrations, and in order to find such factorization choices [[functorial factorization|functorially]].

The small object argument is the simplest when the underlying category is [[locally presentable category|locally presentable]], in which case the resulting weak factorization system is called [[combinatorial wfs|combinatorial]], and is an ingredient in a [[combinatorial model category]].  A more general notion is that of an [[accessible weak factorization system]], which can be constructed by the [[algebraic small object argument]] from a more general "category of generating maps".

## Statement ##

To say that a weak factorization system is cofibrantly generated by $I$ is to say that the right class $R$ of the system consists of precisely those maps  which have the right [[lifting property]] with respect to $I$ (the _$I$-[[injective morphisms]]_)

$$
  R = rlp(I)
  \,.
$$

The left class $L$ is then necessarily the class of maps who have the left [[lifting property]] with respect to the right class (the _$I$-cofibrations_)

$$
  L = llp(R) = llp(rlp(I))
  \,.
$$

When a weak factorization system is cofibrantly generated, another consequence of Quillen's small object argument is that the left class is the smallest [[saturated class of maps]] containing $I$.

Given that the classes of a cofibrantly generated weak factorization system are determined by lifting properties, the content of the small object argument is to produce the required factorizations. With care, this construction is _functorial_, so the result is a functorial weak factorization system.

If the category $C$ is just assumed to have all [[colimit]]s then the domains of the maps in $I$ are required to satisfy a smallness condition that says that any morphism from these objects to a sufficiently-large-directed colimit will factor through the base of the colimiting diagram. (See the reference by Hovey below.)
If the category is required to be a [[locally presentable category]] then no further condition is required (see the other references below).

+-- {: .un_theorem }
###### Theorem (small object argument)

Let  $I \subset Mor(C)$ be a [[set]] of [[morphism]]s in a [[category]] $C$.

Let $C$ be

* a [[locally presentable category]]

* or, more generally, such that it has all [[colimits]] and each domain of morphisms in $I$ is a [[small object]].

* or, yet more generally, such that it has all [[colimit]]s and each [[domain]] of [[morphisms]] in $I$ is [[small object|small]] relative to [[transfinite composition|transfinite composites]] of [[pushouts]] of maps in $I$.

Then

every morphism $f$ has a [[weak factorization system|factorization]] of the form

$$
  f : \stackrel{g \in cell(I)}{\to}
   \stackrel{h \in rlp(I) }{\to}
$$

where

* $rlp(I)$ is the set of morphisms with [[weak factorization system|right lifting property]] with respect to $I$

* $cell(I)$ is the set of [[transfinite composition]]s of [[pushout]]s of coproducts of morphisms in $I$;

=--

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

One sometimes says (e.g. [Hirschhorn](#ModLoc)) that a collection $I$ of morphisms **admits a small object argument** if all domains are small relative to transfinite composites of pushouts of elements of $I$.

=--

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

Examples of categories where the argument applies that are _not_ [[presentable category|presentable]] include the category [[Top]] of [[topological space]]s.

=--

##The Construction##

Given a morphism $f: X \rightarrow Y$, we would like to factor $f$ as $a : X \rightarrow Z$ followed by $q : Z \rightarrow Y$, where $q$ has the right lifting property with respect to all arrows in $I$. The arrow $a$ will be constructed to be a [[transfinite composition|transfinite composite]] of [[pushouts]] of [[coproducts]] of maps in $i$. The left class of a weak factorization system is closed under all of these constructions, so $a$ will be in the left class cofibrantly generated by $i$.

For convenience, suppose our category is [[locally small category|locally small]]. We can then consider the set $S_1$ of lifting problems between $f$ (on the right) and elements $i \in I$ (on the left), i.e. the set of commuting diagrams

$$
  S_1 =
  \left\{
  \array{
    K &\to& X
    \\
    \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{f}}
    \\
    L &\to& Y
  }
  \;\;
  \vert
  \;\;
   i \in I
  \right\}
  \,.
$$

Form the [[coproduct]] morphism

$$
  (
  K_{(I/f)}
  \to
  L_{(I/f)}
  )
  :=
  \coprod_{i \in S_1} (K \stackrel{i}{\to} L)
$$

over $S_1$ of the corresponding elements of $I$; the squares of $S_1$ then specify a canonical morphism

$$
  K_{(I/f)} \to X =: Z_0
$$

from the [[domain]] of this morphism to $Z_0 \coloneqq X$. The [[pushout]]

$$
 \array{
  K_{(I/f)}
  &\longrightarrow&
  Z_0
  \\
  \downarrow
  &&
  \downarrow^{\mathrlap{a_1}}
  \\
  L_{(I/f)}
  &\to&
  L_{(I/f)}
  \coprod_{K_{(I/f)}}
  Z_0
  =: Z_1
  \\
  &&& \searrow^{\mathrlap{q_1}}
  \\
  &&&& Y
 }
$$

of this diagram defines an object $Z_1$ and morphisms $a_1 : Z_0 \rightarrow Z_1$ and $q_1 : Z_1 \rightarrow Y$ factoring $f$.

Notice that if, following the example of [[boundary of a simplex|simplicial boundary]] inclusions, we think of the morphisms $i \in I$ as being inclusions of [[spheres]] as [[boundaries]] of closed [[balls]], then we have formed $Z_1$ by [[cell attachments]] for every possible [[attaching map]] from a domain of $I$ into $X=Z_0$.

Now, we iterate this construction with $q_1 : Z_1 \rightarrow Y$ in place of $f$ and taking colimits to construct $Z_{\alpha}$ for limit [[ordinal number|ordinals]] $\alpha$.

This construction does not converge. So we choose instead to stop at a sufficiently large ordinal $\beta$, chosen so that the domains of the maps in $I$ will satisfy the smallness property assumed in the theorem. Define $a$ to be the [[transfinite composition|transfinite composite]] of the $a_{\alpha}$ and $q$ to be the induced map from the colimit $Z_{\beta}$ to $Y$, so that

$$
  (X \stackrel{f}{\to} Y)
  =
  (X \stackrel{a}{\to} Z_\beta \stackrel{q}{\to} Y)
  \,.
$$

It is clear from the construction that $a$ is in the left class of the weak factorization system, so it remains to show that $q$ has the right lifting property with respect to each $i \in I$. Given a lifting problem,

$$
  \array{
    K &\to& Z_\beta
    \\
    \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{a}}
    \\
    L &\to& Y
  }
$$

the map from $K$ to $Z_\beta$ factors through some $Z_{\alpha}$, with $\alpha \lt \beta$, since $Z_\beta$ is a [[filtered colimit]] and using the assumed smallness of $K$ (see _[[compact object]]_). Because $Z_{\alpha+1}$ was defined to be a pushout over squares including this one, we have a map $L \rightarrow Z_{\alpha +1} \rightarrow Z_{\beta} = colim_\alpha Z_{\alpha}$, which is the desired lift:

$$
  \array{
    K &\to& Z_\alpha &\to& Z_\beta
    \\
    \downarrow && \downarrow &\nearrow& \downarrow
    \\
    && Z_{\alpha+1}
    \\
    \downarrow^{\mathrlap{i}} & \nearrow & &&
    \downarrow
    \\
    L &\to& &\to& Y
  }
$$

##A note on Functoriality##

One of the important conclusions of the small object argument is that it is functorial, hence that it produces [[functorial factorizations]].  But since (in its ordinary form) the process does not "converge" (in the up-to-isomorphism sense) but rather is merely stopped when it has gone far enough along, for functoriality we have to take care to terminate the construction at the _same_ ordinal $\beta$ for every input.

Additionally, in an [[enriched category|enriched]] situation, ideally one would like the factorizations to be an _enriched_ functor.  The version of the small object argument given above does not produce an enriched functor, since it takes coproducts over maps in an ordinary category.  It can be modified to produce an enriched functor by replacing these coproducts by [[copower]]s, but the resulting factorizations are only rarely homotopically well-behaved (in a model category, for instance).  One important special case when they are well-behaved is when all objects of the enriching category are cofibrant, as is the case for [[simplicial set]]s and for the [[folk model structure]] on [[Cat]].

## Algebraic variant

A modified version of Quillen's small object argument due to [Garner 2009](#Garner09) produces not just [[functorial factorization]] but those of an [[algebraic weak factorization system]], see at *[[algebraic small object argument]]*. Unlike Quillen's construction, this converges.

## Related concepts

* [[factorization system]], [[anodyne morphism]]

[[!include algebraic model structures - table]]

## References ##

Standard textbook references are for instance

theorem 2.1.14 in

* [[Mark Hovey]], _Model categories_,  volume 63 of Mathematical Surveys and Monographs, American Mathematical Society, (2007),

or section 10.5 in

* {#ModLoc} [[Philip Hirschhorn]], _Model categories and their localization_, volume 99 of Mathematical Surveys and Monographs, American Mathematical Society

A reference with an eye towards [[combinatorial model category|combinatorial model categories]] and [[Smith's theorem]] is

* [[Tibor Beke]], _Sheafifiable homotopy model categories_ ([arXiv](http://arxiv.org/abs/math.CT/0102087))

Based on this a good quick reference is the first two pages of

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

See also the appendix of [[Higher Topos Theory|HTT]].

For more conceptual background see

* {#Garner09} [[Richard Garner]], *Understanding the small object argument*, Applied Categorical Structures **17**  3 247-285 (2009) [[arXiv:0712.0724](https://arxiv.org/abs/0712.0724), [doi:10.1007/s10485-008-9137-4](https://doi.org/10.1007/s10485-008-9137-4)]

Review in:

* [[Emily Riehl]], Chapter 12 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]


small presheaf

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## Idea

A **small presheaf** on a category $C$ is a [[presheaf]] which is determined by a small amount of data.  If $C$ is itself [[small category|small]], then every presheaf on $C$ is small, but this is no longer true when $C$ is large.  In many cases, when $C$ is large, it is the small presheaves which seem to be more important and useful.

## Definition

Let $C$ be a [[category]] which is [[locally small category|locally small]], but possibly [[large category|large]].  A presheaf $F\colon C^{op}\to Set$ is **small** if it is the left [[Kan extension]] of some functor whose domain is a [[small category]], or equivalently if it is a small [[colimit]] of [[representable functors]].

Of course, if $C$ is itself small, then every presheaf is small.

## Categories of small presheaves

We write $P C$ for the category of small presheaves on $C$.  Observe that although the category of *all* presheaves on $C$ cannot be defined without the assumption of a [[Grothendieck universe|universe]], the category $P C$ can be so defined, using small diagrams in $C$ as proxies for small colimits of representable presheaves.  Moreover $P C$ is locally small, and there is a [[Yoneda embedding]] $C\hookrightarrow P C$.

Of course, if $C$ is small, then $P C$ is the usual category of all presheaves on $C$.

Since small colimits of small colimits are small colimits, $P C$ is [[cocomplete category|cocomplete]].  In fact, it is easily seen to be the [[free cocompletion]] of $C$, even when $C$ is not small.  It is not, in general, complete, but we can characterize when it is (cf. Day--Lack).

+-- {: .un_theorem}
###### Theorem
$P C$ is complete if and only if for every small diagram in $C$, the category of cones over that diagram has a small weakly terminal set, i.e. there is a small set of cones such that every cone factors through one in that set.
=--

+-- {: .un_cor}
###### Corollary
If $C$ is either complete or small, then $P C$ is complete.
=--

We also have:

+-- {: .un_theorem}
###### Theorem
If $C$ and $D$ are complete, then a functor $F\colon C\to D$ preserves small limits if and only if the functor $P F\colon P C \to P D$ (induced by left Kan extension) also preserves small limits.
=--

These results can all be generalized to [[enriched categories]], and also relativized to limits in some class $\Phi$ (which, for some purposes, we might want to assume to be "saturated").  See the paper by Day and Lack.

## References

* [[B. J. Day]], [[Steve Lack]], _Limits of small functors_, JPAA, [pdf](http://www.maths.usyd.edu.au/u/stevel/papers/small.pdf)

* [[Boris Chorny]], [[William Dwyer]], _The homotopy theory of small diagrams over large categories_,  Forum Math. 21 (2009), no. 2, 167--179; [preprint](http://math.haifa.ac.il/chorny/publications/Chorny,%20Dwyer%20-%20SmallFunctors.pdf)

* [[Paolo Perrone]], [[Walter Tholen]], *Kan extensions are partial colimits*, Applied Categorical Structures, 2022. ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531))

The results of Chorny--Dwyer are cited by Rosický in

* [[Jiří Rosický]], _Accessible categories and homotopy theory_, lecture notes for the Summer School on Contemporary Categorical Methods in Algebra and Topology (2007) [pdf](https://www.math.muni.cz/~rosicky/papers/homacc3.pdf)

See also

* [[Boris Chorny]], David White, _Homotopy theory of homotopy presheaves_. [arXiv:1805.05378](https://arxiv.org/abs/1805.05378).

On when [[finitely continuous]] [[presheaves]] are small:

* [[Jiří Adámek]], V. Koubek, and V. Trnková. *How large are left exact functors?*. Theory and Applications of Categories 8.13 (2001): 377-390. ([pdf](http://www.tac.mta.ca/tac/volumes/8/n13/n13.pdf))

[[!redirects small presheaves]]
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[[!redirects small functors]]
[[!redirects small presheaf category]]
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small set

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## Idea

In some choices of [[foundations]], one says __small set__ in order to amplify that one really means a [[set]] and not a [[proper class]].  Strictly speaking, in this case one could just say "set".

However, in other choices of [[foundations]], such as [[Grothendieck universes]], there exist both "small sets" (sets that belong to the universe) and "large sets" (sets that do not, such as the universe itself).  In this case the adjective really is necessary.  We also gain an intermediate notion of __moderate set__: a subset of the universe, which may be small or large.  (Every small set is moderate, but not conversely; again, the universe itself is the standard counterexample.)

Since in many cases the choice of foundations is irrelevant, it makes sense to always say "small set" for emphasis even if one has in mind a foundation where all sets are small.

Similarly, a __small family__ is a [[family]] indexed by a small set; the [[axiom of replacement]] (if it applies in the foundation in question) then says that the [[image]] of the family is also small.

We also have related notions of small [[ordinals]], [[small categories]], etc. In type theory, we also have a notion of small [[type]].

## See also

* [[class]]

* [[large set]]

[[!redirects small set]]
[[!redirects small sets]]

[[!redirects small type]]
[[!redirects small types]]

[[!redirects small family]]
[[!redirects small families]]

[[!redirects moderate set]]
[[!redirects moderate sets]]

[[!redirects moderate family]]
[[!redirects moderate families]]


small site

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By a "small site" is typically meant a [[site]] whose underlying [[category]] is a [[small category]]. This is the default case, but may be used, for emphasis, in contrast to _[[large site]]_.

But beware that there is also the, unrelated, concept of a _[[little site]]_.

[[!redirects small sites]]


small violations of choice

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## Statement

The axiom of **small violations of choice**, or **SVC**, introduced by [Blass (1979)](#Blass79) asserts that

* There exists a set $S$ such that every set is a [[subquotient]] of $A\times S$ for some [[choice set]] $A$.

Note that if $S=1$, then every set is a subquotient of a choice set; but choice sets are closed under subsets and quotients, so this means that every set is choice, i.e. that the [[axiom of choice]] holds.  Thus, SVC is a weakened version of AC; intuitively, it says that "the failure of AC to hold is parametrized by a mere set" (rather than a [[proper class]]).

At least assuming [[classical logic]], it is equivalent to say that every set is a quotient of $A\times S$, for we can extend any function defined on a subset of $A\times S$ to be constant on the complement of this subset.

## Implications

The following statements are all consequences of SVC (some requiring [[excluded middle]]).

* Assuming SVC with $S$, AC holds as soon as $S$ is choice.

* The [[regular extension axiom]] (REA).

* The [[axiom of multiple choice]] (AMC).

* There are "enough highly filtered ordinals," in the sense that for any set $A$, there is a limit [[ordinal]] $\alpha$ such that there is no cofinal function $A\to \alpha$.  (This is actually a consequence of REA.)

* There are enough [[injective object|injective]] [[abelian groups]].

* The category of [[anafunctors]] between two [[small categories]] is essentially small.

* The (local) [[axiom of small cardinality selection]].

## Models

SVC holds in all "ordinary" [[permutation model|permutation]] and [[symmetric model]]s of [[classical logic|classical]] (material) [[set theory]], as well as in hereditarily-ordinal-definable and [[constructible universe|relatively-constructible]] submodels.  However, it can fail in permutation models over [[proper classes]].

When we move to [[constructive mathematics]], however, SVC quickly becomes unreasonable, because in the absence of LEM there are usually very few choice objects.  In particular (see [MO:a/339314](https://mathoverflow.net/a/339314/49)) [[Simon Henry]] argues that any Grothendieck topos satisfying both SVC and $\neg LEM$ (i.e. $\neg \forall P, P\vee \neg P$, which holds in many toposes such as $Sh([0,1])$) is trivial.

## SVC and COSHEP

SVC can be regarded as a sort of "dual" or "complement" to [[COSHEP]], since it deals with choice sets and COSHEP deals with projective sets.  And while COSHEP implies the existence of projective resolutions, SVC implies the existence of (at least some) injective resolutions.

Moreover, at least assuming classical logic, SVC + COSHEP implies AC.  First note that if SVC holds with $S$, then it also holds with $S'$ for any surjection $S'\to S$; thus under SVC + COSHEP we may assume that SVC holds with a projective set $S$.  Let $Z$ be any set, and let $g\colon A\times S \twoheadrightarrow Z^S$ be a surjection, where $A$ is choice.  For each $s\in S$, let $Z_s = \{g(a,s)(s) \vert a\in A\} \subseteq Z$.  Then each $Z_s$ is a surjective image of $A$, and hence choice.  If each $Z\setminus Z_s$ is nonempty, then (since $S$ is projective) we have a function $f\colon S\to Z$ such that $f(s)\notin Z_s$ for all $s$.  But $g$ is surjective, so there exist $s_0\in S$ and $a_0\in A$ with $g(a_0,s_0) = f$, whence $g(a_0,s_0)(s_0) = f(s_0) \notin Z_{s_0}$, a contradiction.  Thus some $Z_s$ is all of $Z$, and hence $Z$ is choice; so AC holds.

## References

SVC was introduced in the paper

* {#Blass79} [[Andreas Blass]], _Injectivity, projectivity, and the axiom of choice_,  Trans. Amer. Math. Soc. **255** (1979) 31-59 [[doi:10.1090/S0002-9947-1979-0542870-6](https://doi.org/10.1090/S0002-9947-1979-0542870-6)]

where most of the above results were proven.  Some others can be found in

* [[Michael Rathjen]], _Choice principles in constructive and classical set theories_, in: Logic Colloquium '02, pp 299-326, Lect. Notes Log., **27**, (2006) ([pdf](http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf))

category: foundational axiom

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[[!redirects small violations of choice]]
[[!redirects axiom of small violations of choice]]


small-generated site

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## Definition

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###### Definition

A (possibly) [[large site]] is a **small-generated site** if it has a [[small category|small]] [[dense sub-site]].

=--

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

In ([Johnstone, p. 548](#Johnstone)) this is called an **essentially small site**. But notice that the underlying [[category]] of such need not be an [[essentially small category]].

=--

## Examples

* The category of [[paracompact manifold]]s with its standard [[open cover]] [[coverage]] is a [[large site]] that is small-generated: a dense sub-site is [[CartSp]]${}_{top}$.

* Similarly the category [[Diff]] of smooth manifolds with the [[open cover]] [[coverage]] is a [[large site]] that is small-generated: a dense sub-site is [[CartSp]]${}_{smooth}$.

## References

Page 548 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
{#Johnstone}

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smart contract

In the wide sense, one can use the definition from English wikipedia: A smart contract is a computer protocol intended to digitally facilitate, verify, or enforce the negotiation or performance of a contract.

The main variety are the smart contracts in the context of the distributed ledger technology and [[blockchain]] in particular.

The smart contract on the blockchain typically consists of a code written for the blockchain-supported virtual machine. In the dominant scenario, the miners are the only parties which strictly execute the smart contract code. The main function of the code is handling assets in a deterministic way, e.g. cryptocurrencies or other tokens, as well as triggering or controlling the operations on off-chain assets (the latter being harder to enforce). Part of the true content of the smart contract is however off-chain. Various software and other agents verify the milestones happening off-chain or trigger changes in off-chain state. Sometimes an external legal action may be necessary, thus some smart contracts contain legal prose which explains both the code contained in the manner understandable to legal system as well as various contract externalities.

### Blockchain virtual machine and supported languages

In most reincarnations the virtual machine supports a Turing complete language.
In practice, there are limits on the level of complexity of the code and computer resources it may require, and the spending of resources is controlled by cryptoassets ("gas") held by the smart contract viewed as an account on blockchain. The efficiency is important because the same code has to be executed/verified by virtually all nodes in the blockchain protocol.

Regarding that the smart contracts on blockchain by their nature are likely to often handle additional cryptographic steps, and 256-bit hashes are common in use most blockchain virtual machines (Ethereum Virtual Machine (EVM), TON Virtual Machine etc.). This is elegant but nonefficient, as 256-byte register is expensive for too small operations and does not map directly to most machines. Thus some machines pack several "bytecode" operations into a very long instruction word. Substrate project (and there are some plans by Ethereum community to go the same way) rather implements WebAssembly (Wasm) bytecode which is very efficient and the same code is used on non-blockchain platforms (emerging browser for efficient common virtual machine of web browsers!), what gives access to quality compiler, debugger and other development tools. Ethereum community plan to establish a WebAssembly virtual machine on Ethereum in future (eWasm, for Ethereum Wasm).

In all case above, higher level languages are used, which compile to the blockchain virtual machine language. For example, Solidity (introduced by Gavin Wood in Aug 2014) is the common choice for a higher language on Ethereum blockchain, and compiles to EVM bytecode. Solidity is specially designed for blockchain use. Its design (which is intended to be Javascript-like, with some influence of C and Python; with later specific improvements/versions) is not considered very successful. Some later languages include (Python-like) Serpent and Viper. For WebAssembly,
there are now compilers from C, C++, Rust, Lua, while the compiler for GoLang is in the development. Substrate blockchain project is itself mainly written in [[Rust]], which is a relatively low level language designed for clarity and security, what makes it suitable for critical applications like blockchain. Moreover both WebAssembly is a quite well performing language, and Rust creates rather small runtime and has no need for garbage collection, the features which make them very suitable for resource expensive blockchain execution.

[Tezos](https://tezos.com) uses OCaml (a functional language). A major functional language on Cardano is [plutus](https://cardanodocs.com/technical/plutus/introduction), Haskell like functional language with eager rather than laisy execution.

[Accordproject](https://www.accordproject.org) develops open source framework to generate the blockchain smart contracts from source which is closer to the working needs of legal teams and is accompanied with careful creation protocols and sufficient legal prose to make the off-chain effects legally enforceable.

[[zoranskoda:Hyperledger]] Fabric supports smart contracts in Go programming language and in Javascript. Hyperledger Sawtooth allows adding new language frameworks and it included [[Rust]] in December 2018.

"[Scilla](https://scilla-lang.org), short for Smart Contract Intermediate-Level Language, is an intermediate-level smart contract language being developed for Zilliqa. Scilla has been designed as a principled language with smart contract safety in mind"..."developed hand-in-hand with formalization of its semantics and its embedding into the [[nLab:Coq]] proof assistant."

[[Nikolai Durov]] has recently designed a Forth-like stack based efficient language Fift for low level access and debugging to [[TON]] VM.

### Literature

* wikipedia [smart contract](https://en.wikipedia.org/wiki/Smart_contract)

A much longer list of references is at [[zoranskoda:smart contract]] (at zoranskoda).

Related pages at $n$Lab include [[distributed computing]], [[arithmetic cryptography]]

The idea of a smart contract is from

* N. Szabo,The idea of smart contracts, 1997 [html](http://szabo.best.vwh.net/idea.html)

and a full-fledged [[blockchain]] implementation started with the Ethereum blockchain.

* [[Vitalik Buterin]], _Ethereum: A next-generation smart contract and decentralized application platform_, [pdf](https://github.com/ethereum/wiki/wiki/White-Paper.pdf)

category: computer science, applications

[[!redirects smart contracts]]


smash product

> This article is about smash products in [[topology]]/[[homotopy theory]]. For the notion of _Hopf smash product_ see at *[[crossed product algebra]]*.

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

The _smash product_ is the canonical [[tensor product]] of [[pointed objects]] in an ambient [[monoidal category]]. It is essentially given by taking the tensor product of the underlying objects and then identifying with a new basepoint all pieces that contain the base point of either factor.

An archetypical special case is the smash product of [[pointed topological spaces]] and hence of pointed [[homotopy types]]. Under [[stabilization]] this induces the important [[smash product of spectra]] in [[stable homotopy theory]].

## Definition

### For pointed sets
 {#ForPointedSets}

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

The __smash product__ $A \wedge B$ of two [[pointed sets]] $A$ and $B$ is the [[quotient set]] of the [[cartesian product]] $A \times B$ where all points with the basepoint as a coordinate (the one from $A$ or the one from $B$) are identified.

The subset that is 'smashed' here can be identified with the [[wedge sum]] $A \vee B$, so the definition of the smash product can be summarised as follows:
$$ A \wedge B = \frac{A \times B}{A \vee B} $$

=--

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

The smash product is the [[tensor product]] in the [[closed monoidal category]] of [[pointed sets]].
That is, it is characterized by the existence of [[natural isomorphisms]]

$$
  Fun_*(A \wedge B, C) \cong Fun_*(A, Fun_*(B, C))
$$

where $Fun_*(A,B)$ is the set of basepoint-preserving [[functions]] from $A$ to $B$, itself made into a pointed set by taking as basepoint the [[constant function]] from all of $A$ to the basepoint in $B$.

=--

This is a special case of the general discussion [below](#ForGeneralPointedObjects).

### For general pointed objects
 {#ForGeneralPointedObjects}

Let $(\mathcal{C}, \otimes, 1_{\mathcal{C}})$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with ([[finite limit|finite]]) [[limits]] and [[colimits]]. Write $\ast \in \mathcal{C}$ for the [[terminal object]] of $\mathcal{C}$. Write $\mathcal{C}^{\ast/}$ for the category of [[pointed objects]] in $\mathcal{C}$.

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

For $X,Y \in \mathcal{C}^{\ast/}$ two [[pointed objects]] in $\mathcal{C}$, their _smash product_ is given by the following [[pushout]] of [[pushouts]] and [[tensor products]] (all formed in $\mathcal{C}$)

$$
  X \wedge Y
  \coloneqq
  \ast \underset{(X \otimes \ast)\coprod (Y \otimes \ast)}{\coprod} (X \otimes Y)
  \,,
$$

regarded as a [[pointed object]] via the induced [[co-projection]] from $\ast$.

=--

In this generality this appears as ([Elmendorf-Mandell 07, construction 4.19](#ElmendorfMandell07)).

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

The smash product of def. \ref{GeneralSmashProduct} makes $\mathcal{C}^{\ast/}$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with ([[finite limit|finite]]) [[limits]] and [[colimits]].

=--

A proof appears as ([Elmendorf-Mandell 07, lemma 4.20](#ElmendorfMandell07)). For more of these details see at _[Pointed object -- Closed and monoidal structure](pointed+object#ClosedMonoidalStructure)_. For [[base change]] functoriality of these structures see at _[Wirthmüller context -- Examples -- On pointed objects](Wirthmüller+context#PointedObjects)_.

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

The formula for the smash product in def. \ref{GeneralSmashProduct} can be considered in any [[category]] $\mathcal{C}$ with [[finite limits]] and [[colimits]], but unless $\mathcal{C}$ is closed symmetric monoidal, it will not have all these properties.

If finite [[products]] in $C$ distribute over finite colimits, then the smash product is [[associativity|associative]], and if $C$ is also [[cartesian closed category|cartesian closed]], then it makes the category of pointed objects in $C$ [[closed monoidal category|closed monoidal]].  However, if finite products in $C$ do not distribute over finite colimits, the smash product can fail to be associative.

=--

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

Examples of closed symmetric monoidal categories $(\mathcal{C}, \otimes 1_{\mathcal{C}})$ include in particular [[toposes]] with their [[cartesian monoidal category|cartesian monoidal structure]]. For the topos $\mathcal{C} = $ [[Set]] the general discussion here reduces to that [above](#ForPointedSets).

=--

There is a general abstract way to obtain this smash product monoidal structure:

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

The category of [[pointed objects]] is the [[Eilenberg-Moore category]] of [[algebras over a monad]] for the "[[maybe monad]]", $X \mapsto X \coprod \ast$. This being a suitably [[monoidal monad]] it canonically induces a monoidal structure on its EM-category, and that is the smash product.

=--

For more on this see at _[maybe monad -- EM-Category and Relation to pointed objects](maybe+monad#EMCategoryAndRelationToPointedObjects)_.

## Examples

### Of pointed topological spaces

The most common case when $\mathcal{C}$ is a category of [[topological spaces]].  In that case, the [[natural transformation|natural]] [[continuous function|map]] $A \wedge (B \wedge C) \to (A\wedge B)\wedge C$ is a [[homeomorphism]] provided $C$ is a [[locally compact Hausdorff space]]. Thus if both $A$ and $C$ are locally compact Hausdorff, then we have the [[associativity]] $A\wedge(B\wedge C)\cong (A\wedge B)\wedge C$.

Associativity fails in general for the category [[Top]] of all topological spaces; however, it is satisfied for pointed objects in any [[convenient category of topological spaces]], since such a category is cartesian closed.  In particular, the smash product is associative for pointed [[compactly generated spaces]].

+-- {: .num_prop #OnePointCompactificationAndSmashProduct}
###### Proposition
**([[one-point compactification intertwines Cartesian product with smash product]])

On the [[subcategory]] $Top_{LCHaus}$ in [[Top]] of [[locally compact Hausdorff spaces]] with [[proper maps]] between them, the [[functor]] of [[one-point compactification]] (Prop. \ref{OnePointCompactificationFunctor})

$$
  (-)^{cpt}
  \;\colon\;
  Top_{LCHaus}
  \longrightarrow
  Top^{\ast/}
$$

1. sends [[coproducts]], hence [[disjoint union topological spaces]], to [[wedge sums]] of [[pointed topological spaces]];

1. sends [[Cartesian products]], hence [[product topological spaces]], to [[smash products]] of [[pointed topological spaces]];

hence constitutes a [[strong monoidal functor]] for both [[monoidal category|monoidal]] structures of these [[distributive monoidal categories]] in that there are [[natural transformation|natural]] [[homeomorphism]]

$$
  \big(
    X \sqcup Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \vee Y^{cpt}
  \,,
$$

and

$$
  \big(
    X \times Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \wedge Y^{cpt}
  \,.
$$

=--

This is briefly mentioned in [Bredon 93, p. 199](#Bredon93).
The argument is spelled out in: [MO:a/1645794](https://math.stackexchange.com/a/1645794/58526), [Cutler 20, Prop. 1.6](#Cutler20).

### Of spectra

See at _[[symmetric smash product of spectra]]_.

## Properties

[[!include smash-monoidal diagonals -- section]]

## Related concepts

* [[pointed object]]

* [[wedge sum]]

* [[pointed mapping space]]

* While the smash product is not cartesian, it does [[monoidal category with diagonals|admit diagonals]].

## References

Basic accounts:

* {#Bredon93} [[Glen Bredon]], p. 199 of: _Topology and Geometry_, Graduate Texts in Mathematics 139, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

Review:

* {#Cutler20} Tyrone Cutler, _The category of pointed topological spaces_, 2020 ([pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), [[CutlerPointedTopologicalSpaces.pdf:file]])

On the general definition of smash products via [[closed monoidal category]] structure on [[pointed objects]]:

* {#ElmendorfMandell07} [[Anthony Elmendorf]], [[Michael Mandell]], _Permutative categories, multicategories, and algebraic K-theory_, Algebraic & Geometric Topology 9 (2009) 2391-2441 ([arXiv:0710.0082](http://arxiv.org/abs/0710.0082))

On commutativity of smashing with [[homotopy limits]]:

* [[Wolfgang Lueck]], Holger Reich, Marco Varisco, _Commuting homotopy limits and smash products_, K-Theory, 30 (2): 137--165, 2003 ([arXiv:math/0302116](http://arxiv.org/abs/math/0302116))

[[!redirects smash product]]
[[!redirects smash products]]
[[!redirects smash produc]]


smash product of spectra

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###Context###
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Higher linear algebra
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

### General

The [[smash product]] on [[pointed topological spaces]] induces a smash product on [[spectra]].

This is the canonical [[tensor product]] in the [[symmetric monoidal (infinity,1)-category of spectra]]. There are various [[model category]] [[presentable (infinity,1)-category|presentations]] for which which are symmetric [[monoidal model categories]] (such as the [[highly structured spectra]]: [[S-modules]], [[symmetric spectra]] and [[orthogonal spectra]], but also for instance [[excisive functors]]). See at _[[symmetric monoidal category of spectra]]_ for more on this. Passing to the [[stable homotopy category]], it become a [[symmetric monoidal category]] under the smash product. Under mild assumptions, this is the essentially unique symmetric tensor product with [[unit]] the [[sphere spectrum]] ([Shipley 01](#Shipley01)).

### History

Historically the discussion proceeded in the opposite direction: available variants of the construction of smash products on [[sequential spectra]] (the "handicrafted" or "naive" smash products [Boardman 65](#Boardman65), [Adams 74, part III, section 4](#Adams74)) were found to yield a [[symmetric monoidal category]] structure _only_ after passage to the [[stable homotopy category]]. Then models via non-sequential [[highly structured spectra]] were discovered which do admit a [[symmetric smash product of spectra]] in the sense of 1-category theory, as do [[excisive functors]], see at _[[model structure on excisive functors]]_.

### As Day convolution spectra

Many versions of the smash product of spectra, [[symmetric smash product of spectra|symmetric or not]], arise as [[Day convolution]] products on a suitable [[enriched category|enriched]] [[monoidal category|monoidal]] version of a [[category]] of "index spaces".

The [[symmetric smash product of spectra]] on, in particular,  [[symmetric spectra]] and  [[orthogonal spectra]] is the Day convolution product for [[Top]]-[[enriched functors]] on monoidal categories of [[symmetric groups]] of [[orthogonal groups]], respectively ([MMSS 00, theorem 1.7 and section 21.](#MMSS00)).

Similarly the [[symmetric smash product of spectra]] on the [[model structure for excisive functors]] is Day convolution for [[sSet]]-[[enriched functors]] on the plain [[smash product]] of finite pointed [[simplicial sets]] ([Lydakis 98](#Lydakis98)).

See also at _[[functor with smash products]]_.

## Properties

### Graded commutativity
 {#GradedCommutativity}

The smash product of spectra exhibits a certain graded commutativity akin to the graded commutativity in the [[tensor product of chain complexes]] (in fact, under the [[stable Dold-Kan correspondence]] the latter maps to the former).

This comes down to the following basic fact about the [[smash product]] of [[pointed topological spaces]]:

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

There are [[homeomorphisms]] between [[n-spheres]] and their [[smash products]]

$$
  \phi_{n_1,n_2}
  \;\colon\;
  S^{n_1} \wedge S^{n_2}
  \stackrel{\simeq}{\longrightarrow}
  S^{n_1 + n_2}
$$

such that in [[Ho(Top)]] there are [[commuting diagrams]] like so:

$$
  \array{
    (S^{n_1} \wedge S^{n_2}) \wedge S^{n_3}
    &&\stackrel{\simeq}{\longrightarrow}&&
    S^{n_1} \wedge (S^{n_2} \wedge S^{n_3})
    \\
    {}^{\mathllap{\phi_{n_1,n_2} \wedge id}}\downarrow
    && && \downarrow^{\mathrlap{id \wedge \phi_{n_2,n_3}}}
    \\
    S^{n_1+n_2} \wedge S^{n_3}
    && &&
    S^{n_1}\wedge S^{n_2 + n_3}
    \\
    & {}_{\mathllap{\phi_{n_1+n_2, n_3}}}\searrow && \swarrow_{\mathrlap{\phi_{n_1,n_2+n_3}}}
    \\
    &&
    S^{n_1+n_2 + n_3}
  }
  \,.
$$

and

$$
  \array{
    S^{n_1} \wedge S^{n_2}
    &\stackrel{b_{n_1,n_2}}{\longrightarrow}&
    S^{n_2} \wedge S^{n_1}
    \\
    {}^{\mathllap{\phi_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\phi_{n_2,n_1}}}
    \\
    S^{n_1 + n_2}
    &\stackrel{(-1)^{n_1 n_2}}{\longrightarrow}&
    S^{n_1 + n_2}
  }
  \,,
$$

where here $(-1)^n \colon S^n \to S^n$ denotes the homotopy class of a [[continuous function]] of [[degree of a continuous function|degree]] $(-1)^n \in \mathbb{Z} \simeq [S^n, S^n]$.

=--

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

With the [[n-sphere]] $S^n$ realized as the [[one-point compactification]] of the [[Cartesian space]] $\mathbb{R}^n$, then $\phi_{n_1,n_2}$ is given by the identity on [[coordinates]] and the [[braiding]] homeomorphism

$$
  b_{n_1,n_2}
  \;\colon\;
  S^{n_1} \wedge S^{n_2}
  \stackrel{\sigma}{\longrightarrow}
  S^{n_2} \wedge S^{n_1}
$$

is given by permuting the [[coordinates]]:

$$
  (x_1, \cdots, x_{n_1}, y_1, \cdots, y_{n_2})
  \mapsto
  (y_1, \cdots, y_{n_2}, x_1, \cdots, x_{n_1})
  \,.
$$

This has [[degree of a continuous map|degree]] $(-1)^{n_1 n_2}$ .

=--

This statement passes to the [[suspension spectra]] $\Sigma^\infty S^n$ of the spheres ([Adams 74, part III, prop. 4.8](#Adams74), [Schwede 12, chapter II.4, prop. 4.4](#Schwede12)).

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

The phenomenon in prop. \ref{GradedCommutativityOfSmashOfSpheres} is the reason why there is no [[symmetric smash product of spectra]] on plain [[sequential spectra]], and in fact no appropriate functorial product operation at all.

To see this, observe, by [this proposition](sequential+spectrum#QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors), that sequential spectra are Quillen equivalent to the model category of [[excisive (infinity,1)-functors]]

$$
  Exc^1(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/}) \hookrightarrow
  Func(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/})
$$

under an equivalence given by restricting from the domain of all [[pointed homotopy types|pointed]] [[finite homotopy types]] $\infty Grpd_{fin}^{\ast/}$ to its non-full subcategory $StdSpheres$ of standard spheres with just the adjuncts of suspension maps between them.

$$
  \iota \colon StdSpheres \hookrightarrow \infty Grpd^{\ast/}_{fin}
  \,.
$$

$$
  Exc^1(\infty Grpd^{\ast/}_{fin}, \infty Grpd^{\ast/})
  \underoverset{\simeq}{\iota^\ast}{\longrightarrow}
  SeqSpectra
$$

Here $StdSpheres$ has objects $S^n_{std} \coloneqq (S^1)^{\wedge n}$ and as hom-spaces it has $StdSpheres(S^{m}, S^{n}) = S^{max(n-m,0)}$, identified as the [[adjunct]] of the canonical isomorphism $S^m \wedge S^{n-m} \to S^n$.

Hence $\iota^\ast$ identifies the $n$th component space of a sequential spectrum with the value $E(S^n)$ of an excisive functor on the $n$-sphere, and it identifies the structure map $S^1 \wedge E_n \to E_{n+1}$ with part of the [[enriched functor|enriched functoriality]] of the excisive functor.

Now in the model category of excisive functors, the correct smash product of spectra is the [[Day convolution]] over the symmetric monoidal category $(\infty Grpd^{\ast/}_{fin},\wedge)$.
The issue then is that the restricted hom-spaces of $StdSpheres$ do not see the non-trivial [[braiding]] of spheres in prop. \ref{GradedCommutativityOfSmashOfSpheres} anymore.

More concretely, the enriched category $StdSpheres$ does not inherit monoidal structure: defining the smash product on hom spaces requires permuting smash copies of spheres, which is not available.  Thus there is no Day convolution product on sequential spectra at all.

One could further restrict along $\mathbb{N} \to StdSpheres$ and use the monoidal structure $(\mathbb{N},+)$ to define at least a smash product on sequences of pointed spaces by [[Day convolution]] over $(\mathbb{N},+)$ as in ([MMSS 00, example 4.1](#MMSS00), [Hovey-Shipley-Smith 00, below prop. 2.3.4](#HoveyShipleySmith00)). But then in addition to the above problem that this does not give a functorial smash product on spectra (it will not respect the structure maps), moreover $(\mathbb{N},+)$ is trivially [[braided monoidal category|braided]] and so, again, under restriction of excisive functors to $\mathbb{N}$ there is no way to recover the information in the smash product of spectra that is encoded in the non-trivial braiding of the smash product of spheres.

=--

## Definitions

* _[smash product of sequential spectra](sequential+spectrum#SmashProduct)_

* _[smash product of excisive functors](model+structure+for+excisive+functors#SmashProduct)_

* _[smash product of symmetric spectra](symmetric+spectrum#SmashProduct)_

* _[smash product of orthogonal spectra](orthogonal+spectrum#SmashProduct)_

## References

The original "handicrafted" constructions of the smash product on the [[stable homotopy category]] are due to

* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onwards

* [[Rainer Vogt]], _Boardman's stable homotopy category_, lectures, spring 1969

* {#Puppe73} [[Dieter Puppe]], _On the stable homotopy category_, Topology and its application (1973)  ([[PuppeStableHomotopyCategory.pdf:file]])

* {#Adams74} [[Frank Adams]], part III, section 4 of _[[Stable homotopy and generalised homology]]_, 1974

The smash product on [[connective spectra]] modeled as [[Gamma spaces]] is discussed in

* {#Lydakis98} Lydakis, _Smash products and $\Gamma$-spaces_, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 ([pdf](http://hopf.math.purdue.edu/Lydakis/smash_gamma.pdf))

The [[symmetric smash product on spectra]] (see there for more) on [[highly structured ring spectra]] is discussed for instance in

for [[S-modules]]:

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland, (1995) pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

for [[symmetric spectra]]:

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. 13 (2000), 149-208 ([arXiv:math/9801077](http://arxiv.org/abs/math/9801077))

* {#Schwede12} [[Stefan Schwede]], _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf))

Discussion of the smash product as a suitable [[Day convolution]] is, for [[highly structured spectra]], in

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [publisher](http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc))

and for [[excisive functors]] in

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

The uniqueness of the smash product on spectra is discussed in

* {#Shipley01} [[Brooke Shipley]], _Monoidal uniqueness of Stable homotopy theory_, Advances in Mathematics Volume 160, Issue 2, 25 June 2001, Pages 217–240 ([pdf](http://homepages.math.uic.edu/~bshipley/ideal.pdf))

[[!redirects smash products of spectra]]


smash product of spectra > history

< [[smash product of spectra]]


smash product theorem

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

## Statement

### A

For $X$ a [[homotopy type]]/[[spectrum]] and for all $n$, there is a [[homotopy pullback]]

$$
  \array{
    L_{E(n)}X &\longrightarrow& L_{K(n)}X
    \\
    \downarrow && \downarrow
    \\
    L_{E(n-1)}X &\longrightarrow& L_{E(n-1)}L_{K(n-1)}X
  }
  \,,
$$

where $L_{K(n)}$ denotes the [[Bousfield localization of spectra]] at $n$th [[Morava K-theory]] and similarly $L_{E(n)}$ denotes localization at [[Morava E-theory]].

([Lurie 10, lect 23, theorem 4](#LurieLecture23))

This implies that for understanding the [[chromatic tower]] of any [[spectrum]] $X$, it is in principle sufficient to understand all its "chromatic pieces" $L_{K(n)} X$. This is the subject of [[chromatic homotopy theory]].

### B

Let $E$ be a [[ring spectrum]] and $X$ an arbitrary [[spectrum]]. Suppose that there exists an [[integer]] $s \geq 1$ such that, for every [[finite spectrum]] $F$, the $E$-based [[Adams spectral sequence]] for $X \otimes F$ has $E^{p,q}_s$ for $p \geq s$.

If $E$ and $X$ are moreover a [[p-local spectra]] then $L_E X$ is a [[smashing localization]].

([Lurie 10, lect 31](#LurieLect31))

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,
Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf))
 {#LurieLecture23}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 30 _Localizations and the Adams-Novikov spectral sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))
 {#LurieLect30}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,  Lecture 31 _The smash product theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture31.pdf))
 {#LurieLect31}


smash product type

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _smash product type_ is an [[axiom|axiomatization]] of the [[smash product]] in the context of [[homotopy type theory]].

## Definition

The smash product type of two [[pointed type|pointed types]] $A,B$ can be defined as the [[pushout type]] of the span

$$\mathbf 1 \leftarrow A \vee B \rightarrow A \times B$$

where $A \vee B \rightarrow A \times B$ is the inclusion of the [[wedge sum type]] in the [[product type]] both of which are pointed. The resulting pushout is denoted the smash product $A \wedge B$ and is pointed by $\star_{A \wedge B}\equiv\mathrm{inl}(\star_{\mathbf 1})$

It can also be defined as the pushout type of the span

$$\mathbf{2} \leftarrow A + B \rightarrow A \times B$$

## See also

* [[higher inductive type]]

* [[smash product]]

## References ##

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Floris van Doorn]] (2018), _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_, ([arXiv:1808.10690](https://arxiv.org/abs/1808.10690), [web](http://florisvandoorn.com/papers/dissertation.pdf))

* [[Kuen-Bang Hou]], *Higher-Dimensional Types in the Mechanization of Homotopy Theory*, Ph. D. Thesis 2017 ([web](https://favonia.org/files/thesis.pdf))


smash-monoidal diagonals -- section

### Smash-monoidal diagonals
  {#SmashMonoidalDiagonals}

Write

\[
  \label{SmashMonoidalCategoryOfPointedTopologicalSpaces}
  \big( PointedTopologicalSpaces, S^0, \wedge \big)
  \;\;\in\;
  SymmetricMonoidalCategories
\]

* for the [[category]] of [[pointed topological spaces]] (with respect to some [[convenient category of topological spaces]] such as [[compactly generated topological spaces]] or [[D-topological spaces]])

* regarded as a [[symmetric monoidal category]] with tensor product the [[smash product]] and unit the [[0-sphere]] $S^0 \,=\, \ast_+$.

This category also has a [[Cartesian product]], given on pointed spaces $X_i = (\mathcal{X}_i, x_i)$ with underlying $\mathcal{X}_i \in TopologicalSpaces$ by

\[
  \label{CartesianProductOfPointedTopologicalSpaces}
  X_1 \times X_2
  \;=\;
  (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2)
  \;\coloneqq\;
  \big(
    \mathcal{X}_1 \times \mathcal{X}_2
    ,
    (x_1, x_2)
  \big)
  \,.
\]

But since this [[smash product]] is a non-trivial [[quotient topological space|quotient]] of the Cartesian product

\[
  \label{SmashProductOfPointedSpacesQuotientDefinition}
  X_1 \wedge X_1
    \,\coloneqq\,
  \frac{X_1 \times X_2}{ X_1 \vee X_2 }
\]

it is not itself [[cartesian monoidal category|cartesian]], but just [[symmetric monoidal category|symmetric monoidal]].

However, via the quotienting (eq:SmashProductOfPointedSpacesQuotientDefinition), it still inherits, from the [[diagonal morphisms]] on underlying topological spaces

\[
  \label{CartesianDiagonalOnTopologicalSpaces}
  \array{
    \mathcal{X}
    &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}&
    \mathcal{X} \times \mathcal{X}
    \\
    x &\mapsto& (x,x)
  }
\]

a suitable notion of [[monoidal category with diagonals|monoidal diagonals]]:

\begin{definition}\label{SmashMonoidalDiagonal}
  [Smash monoidal diagonals]

For $X \,\in\, PointedTopologicalSpaces$, let $D_X \;\colon\; X \longrightarrow X \wedge X$ be the [[composition|composite]]

\begin{xymatrix@C=7pt}
    X
    \ar@/_1.6pc/[rrrrr]|-{ \;D_X\; }
    \ar[rr]^-{\Delta_X}
    &&
    X \times X
    \ar[rr]
    &&
    \frac{X \times X}{ X \vee X}
    \ar@{}[r]|-{ =: }
    &
    X \wedge X
\end{xymatrix}

of the Cartesian [[diagonal morphism]] (eq:CartesianProductOfPointedTopologicalSpaces) with the [[coprojection]] onto the defining [[quotient space]] (eq:SmashProductOfPointedSpacesQuotientDefinition).

\end{definition}

It is immediate that:

\begin{proposition}
  \label{SmashMonoidalDiaginalsAreMonoidalDiagonals}
  The smash monoidal diagonal $D$ (Def. \ref{SmashMonoidalDiagonal})
  makes the [[symmetric monoidal category]] (eq:SmashMonoidalCategoryOfPointedTopologicalSpaces) of [[pointed topological spaces]] with [[smash product]] a [[monoidal category with diagonals]], in that

1. $D$ is a [[natural transformation]];

1. $S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an [[isomorphism]].

\end{proposition}

While elementary in itself, this has the following profound consequence:

\begin{remark}[Suspension spectra have diagonals]

Since the [[suspension spectrum]]-[[functor]]

$$
  \Sigma^\infty
  \;\colon\;
  PointedTopologicalSpaces
   \longrightarrow
  HighlyStructuredSpectra
$$

is a [[strong monoidal functor]] from [[pointed topological spaces]] (eq:SmashMonoidalCategoryOfPointedTopologicalSpaces) to any standard category of [[highly structured spectra]] (by [this Prop.](Introduction+to+Stable+homotopy+theory+--+1-2#SmashProductOfFreeSpectra)) it follows that _[[suspension spectra]] have [[monoidal category with diagonals|monoidal diagonals]]_, in the form of [[natural transformations]]

\[
  \label{SmashMonoidalDiagonalOnSuspensionSpectra}
  \Sigma^\infty X
  \overset{
    \;\;
    \Sigma^\infty(D_X)
    \;\;
  }{\longrightarrow}
  \big(
    \Sigma^\infty X
  \big)
    \wedge
  \big(
    \Sigma^\infty X
  \big)
\]

to their respective [[symmetric smash product of spectra]], which hence makes them into *[[comonoid objects]]*, namely *[[coring spectra]]*.

For example, given a [[Whitehead-generalized cohomology theory]] $\widetilde E$ [[Brown representability theorem|represented]] by a  [[ring spectrum]]

$$
  \big(E, 1^E, m^E \big)
  \;\;
  \in
  \;
  SymmetricMonoids
  \big(
    Ho(Spectra), \mathbb{S}, \wedge
  \big)
$$

the smash-monoidal diagonal structure (eq:SmashMonoidalDiagonalOnSuspensionSpectra) on suspension spectra serves to define the [[cup product]] $(-)\cup (-)$ in the corresponding [[multiplicative cohomology theory|multiplicative cohomology theory structure]]:

$$
  \begin{aligned}
  &
  \big[
    \Sigma^\infty X
    \overset{c_i}{\longrightarrow}
    \Sigma^{n_i} E
  \big]
  \,\in\, {\widetilde E}{}^{n_i}(X)
  \\
  &
  \Rightarrow
  \;\;
  [c_1] \cup [c_2]
  \,
    \coloneqq
  \,
  \Big[
    \Sigma^\infty X
    \overset{
      \Sigma^\infty(D_X)
    }{\longrightarrow}
    \big(
      \Sigma^\infty X
    \big)
      \wedge
    \big(
      \Sigma^\infty X
    \big)
    \overset{
      ( c_1 \wedge c_2 )
    }{\longrightarrow}
    \big(
      \Sigma^{n_1} E
    \big)
      \wedge
    \big(
      \Sigma^{n_2} E
    \big)
    \overset{
      m^E
    }{\longrightarrow}
    \Sigma^{n_1 + n_2}E
  \Big]
  \;\;
  \in
  \,
  {\widetilde E}{}^{n_1+n_2}(X)
  \,.
  \end{aligned}
$$

\end{remark}


smashing localization

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#Contents#
* table of contents
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## Idea

If a  [[Bousfield localization of spectra]] $L_E$ at a [[spectrum]] $E$ preserves all [[direct sums]], then it is given by [[smash product]] with the $E$-localization of the [[sphere spectrum]]

$$
  L_E X \simeq X \wedge L_E S
$$

and is hence called a _smashing localization_.

Smashing localizations hence in particular

1. preserve [[(∞,1)-colimits]];

1. are [[monoidal (∞,1)-functors]] except possibly for preservation of the [[tensor unit]].

see e.g. ([GGN 13, p. 8](#GGN13)) for discussion.

## Examples

* [[rationalization]] is smashing: $L_{\mathbb{Q}} \simeq (-) \wedge H \mathbb{Q}$ (e.g [Bauer 11, example 1.7 (4)](#Bauer11))

* For $G$ a [[group]] without [[torsion subgroup|torsion]], then localization at the [[Moore spectrum]] $E = S G$ (in particular [[p-localization]]) is smashing (see at _[[Bousfield localization of spectra]]_ [here](Bousfield+localization+of+spectra#pLocalizationIsSmashing)).

* Localization with respect to [[Morava E-theory]] is smashing (Hopkins-Ravenel).

* "finite localizations" are smashing ([Miller 92](#Miller92))

## Counter-examples

* [[p-completion]] is _not_ smashing

## Related concepts

* [[smash product theorem]]

## References

* {#LurieLect22} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture notes, Lecture 22 _Morava E-theory and Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf))

* {#Miller92} Miller, _Finite localizations_, Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390 ([HopfArchive](http://hopf.math.purdue.edu/cgi-bin/generate?/MillerH/finite-localization))

* {#GGN13} [[David Gepner]], Moritz Groth, [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_ ([arXiv:1305.4550](http://arxiv.org/abs/1305.4550))

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ (2011) ([pdf](https://math.mit.edu/events/talbot/2007/tmfproc/Chapter09/Bousfield.pdf), [[Bauer_BousfieldLocalization.pdf:file]]), chapter 6 in:  [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

* {#Nardin12} [[Denis Nardin]], section 3.2 of _Stability and distributivity over orbital ∞-categories_, 2012 ([pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))

[[!redirects smashing localizations]]

[[!redirects smashing localization of spectra]]
[[!redirects smashing localizations of spectra]]


Smith normal form

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#### Algebra
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## Statement

+-- {: .num_prop #MatricesOverPrincipalIdealDomainsHaveSmithNormalForm}
###### Proposition
**([[matrices]] over [[principal ideal domains]] [[matrix equivalence|equivalent]] to [[Smith normal form]])**

For $R$ a [[commutative ring]] which is a [[principal ideal domain]] (for instance $R = \mathbb{Z}$ the [[integers]]), every [[matrix]] $A \in Mat_{n \times m}(R)$ with entries in $R$ is [[matrix equivalence|matrix equivalent]] to a [[diagonal matrix]] filled up with zeros:

There exist [[invertible matrices]] $P \in Mat_{n \times n}(R)$ and $Q \in Mat_{m \times m}(R)$ such that the [[matrix product|product matrix]] $P A Q$ is of the following form:

$$
  P A Q
  \;=\;
$$

such that, moreover, each $a_i$ [[divisor|divides]] $a_{i+1}$.

=--

## Related concepts

* [[Hermite normal form]]

* [[Gram-Schmidt process]]

## References

The results is named after

* H.J.S. Smith, _On  systems  of  linear  indeterminate  equations  and congruences_ Philos. Trans. Royal Soc. London cli, 293-326.  Reprinted in _The Collected Mathematical Papers of Henry John Stephen Smith_, Volume 1. New York: Chelsea (1965)

Lecture notes include

* [[Patrick Morandi]], _The Smith Normal Form of a Matrix_, 2005 ([pdf](http://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf))

* Sam Evans, _Smith normal form over the integers_ ([pdf](https://www3.nd.edu/~sevens/smithform.pdf))

* Bill Casselman, _Hermite and Smith forms_, 2011 ([pdf](http://www.math.ubc.ca/~cass/siegel/Smith.pdf))

* George Havas, Leon Sterling, _Integer matrices and abelian groups_ ([pdf](https://pdfs.semanticscholar.org/1f4c/9ae8000e1e9fb6f3131b7812be84d6a9a0ad.pdf), [doi:10.1007/3-540-09519-5_94](https://link.springer.com/chapter/10.1007/3-540-09519-5_94))

* George Havas, Bohdan Majewski, _Integer matrices and diagonalization_, J. Symbolic Computation (1997) 24, 399-408 ([pdf](http://staff.itee.uq.edu.au/havas/1997hm.pdf))

See also

* Wikipedia, _[Smith normal form](https://en.wikipedia.org/wiki/Smith_normal_form)_

[[!redirects Smith normal forms]]


Smith space (functional analysis)

> For the [[generalized smooth space]] of the same name see at [[Smith space (generalized smooth space)]].

#Contents#
* table of contents
{:toc}

## Idea

## Definition

In the context of [[functional analysis]], a **Smith space** is a [[complete topological vector space|complete]] [[locally convex topological vector space|locally convex]] topological $\mathbb{R}$-vector space $V$ that admits a [[compact object|compact]] [[absolutely convex]] subset $K \subset V$ such  that $V = \bigcup_{c \gt 0} c K$ with the induced [[compactly generated topological space|compactly generated topology]] on $V$.

## Examples

* For a compact absolutely convex set, $S$, the space of [[measures]], $M(S)$, with the compactly generated topology is a Smith space.

## Properties

The category of Smith spaces is [[equivalence of categories|equivalent]] to the opposite of the category of [[Banach spaces]]. More precisely, if $V$ is a Banach space, then $Hom(V,\mathbb{R})$ is a Smith space; and if $W$ is a Smith space, then $Hom(W,\mathbb{R})$ is a Banach space, where in both cases we endow the dual space with the compact-open topology. The corresponding biduality maps are isomorphisms.

Smith spaces are more natural than Banach spaces from the perspective of [[condensed mathematics]] because they are controlled by a nice compact subset instead of a nice open subset (the unit ball) as in the Banach setting. Also, Banach spaces are [[filtered colimits]] of Smith spaces while Smith spaces are [[filtered limits]] of Banach spaces. Filtered colimits have better algebraic and homological properties.

Smith spaces embed fully faithfully in condensed R-vector spaces, but Smith spaces do not form an [[abelian category]]. There is an induced functor from Waelbrock’s enlarged (abelian) category of quotients of Smith spaces, but that functor is not fully faithful.

Smith spaces are the same as Waelbrock dual spaces.

A Banach space is a Smith space if and only if it is finite-dimensional.

##References

The original treatment is in

* [[Marianne Smith]], _The Pontrjagin duality theorem in linear spaces_, Annals of Mathematics. 56(2): 248–253, ([doi:10.2307/1969798](https://doi.org/10.2307%2F1969798)).

In the context of [[condensed mathematics]], see

* [[Peter Scholze]], _Lectures on Analytic Geometry_, ([pdf](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf))


Smith space (generalized smooth space)

> For the concept of the same name in [[functional analysis]] see at [[Smith space (functional analysis)]].

#Contents#
* table of contents
{:toc}

{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

In 1966, J. Wolfgang Smith studied the following [[generalized smooth space|extension of the notion of a smooth manifold]].

+-- {: mynumdef #SmithSp}
###### Definition
A _differentiable structure_ on an arbitrary topological space $X$ is a family, $\mathcal{F}$, of real-valued functions on $X$ satisfying a certain closure condition.

To express the closure condition, we need an auxiliary notion.
A _plot_ of $(X, \mathcal{F})$ is a continuous map $\phi : U \to X$ with domain an open subset of some Euclidean space with the property that $f \circ \phi \in C^\infty(U)$ for all $f \in \mathcal{F}$.

The closure condition is that if a continuous map $f: X \to \mathbb{R}$ has the property that whenever $\phi: U \to X$ is a plot for $(X, \mathcal{F})$ then $f \circ \phi \in C^\infty(U)$ then $f \in \mathcal{F}$.
=--

## References ##
* [[J. Wolfgang Smith]], _The de Rham theorem for general spaces_, Tohoku Math. J. (2), Volume 18, Number 2 (1966), 115-137, ([project euclid](https://projecteuclid.org/euclid.tmj/1178243443))


Smith-Minkowski-Siegel mass formula

## References

* Wikipedia, _[Smith–Minkowski–Siegel mass formula](http://en.wikipedia.org/wiki/Smith–Minkowski–Siegel_mass_formula)_

[[!redirects Minkowski-Siegel mass formula]]


smooth

in [[differential geometry]]:

* [[smooth structure]], [[smooth manifold]], [[smooth space]], [[smooth groupoid]], [[smooth infinity-groupoid]]

in [[algebraic geometry]]

* [[smooth algebra]], [[smooth scheme]]


smooth (infinity,1)-algebra

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## Idea

The notion of smooth $(\infty,1)$-algebra is the analog in [[higher category theory]] of [[smooth algebra]]. This is the basis for the [[derived geometry]] version of [[differential geometry]]/[[synthetic differential geometry]].

## Definition

A **smooth $\infty$-algebra** is an [[∞-algebra over an (∞,1)-algebraic theory]] $T$ for $T$ the ordinary [[Lawvere theory]] of [[smooth algebra]]s.

## Properties

The [[model structure on simplicial algebras]] on [[simplicial C-∞-ring]] is a [[presentable (infinity,1)-category|presentation]] for smooth $(\infty,1)$-algebras.

## Applications

Smooth $(\infty,1)$-algebras appear as the algebras of functions in [[derived differential geometry]], for instance on [[derived smooth manifolds]].

[[!redirects smooth (∞,1)-algebra]]
[[!redirects smooth (∞,1)-algebras]]
[[!redirects smooth (infinity,1)-algebras]]


smooth 2-group

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#### $\infty$-Lie theory
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#Contents#
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{:toc}

## Definition

A [[smooth infinity-group]] that is [[1-truncated]], hence that is just a [[2-group]].

## Examples

Every [[Lie 2-group]] presents a smooth 2-group.

[[!redirects smooth 2-groups]]


smooth algebra in algebraic sense

In $n$lab, the entry [[smooth algebra]] is dedicated to the notion of $C^\infty$-rings, hence it is coming from an extension of the theory of [[differentiable manifold]]s. On the other hand in commutative algebra and in the study of associative algebras there are many notions of smoothness, regularity and formal smoothness which correspond to nonsingularity in an algebraic category, while the algebras are quite different from the $C^\infty$-setup. In [[algebraic geometry]], Grothendieck introduced many notions of [[smooth scheme|smoothness for commutative algebraic schemes]] (they say 19 different notions). See also [[formally smooth morphism]].

* W.F. Schelter, _Smooth algebras_, J. Algebra 103 (1986), 677–685.


smooth E-infinity-groupoid

> under construction

#Contents#
* table of contents
{:toc}

## Idea

For $\mathbf{B} = Sh_\infty(CRing_\infty^{op}, et)$ the [[(∞,1)-topos]] of [[E-∞ geometry]], let

$$
  \mathbf{H} \coloneqq Sh_\infty(SmoothMfd, \mathbf{B})
$$

be the [[(∞,1)-category of (∞,1)-sheaves]] on the [[site]] of [[smooth manifolds]] with values in $\mathbf{B}$.

An [[object]] in this $\mathbf{H}$ combines the properties of a [[smooth ∞-groupoid]] and an object in [[E-∞ geometry]], hence might be called a "smooth $E_\infty$-groupoid".

It is useful to regard this as a [[cohesive (∞,1)-topos]] over $\mathbf{B}$

$$
  \mathbf{H}
  \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  \mathbf{B}
  \,.
$$

As such this appears for instance in the discussion at

* [[differential algebraic K-theory]]

* [[equivariant elliptic cohomology]].

## Constructions

### Multiplicative group

Write

$$
  \mathbb{G}_m \in \mathbf{B}
$$

for the sheaf which sends each ring to its [[∞-group of units]]

$$
  \mathbb{G}_m \;\colon\; R \mapsto R^\times
  \,.
$$

This is the canonical group object in $\mathbf{B}$. The [[mapping stacks]] into it are the [[Picard ∞-stacks]].

(...)

For [[E-∞ rings]] over the [[complex numbers]], hence [[E-∞ algebras]] over $\mathbb{C}$, the [[multiplicative group]]

$$
  \mathbb{G}_m = \mathbb{C}^\times
$$

naturally carries both the structure of an object in [[smooth ∞-groupoids]] and in [[E-∞ geometry]], which may be combined to the structure of a smooth $E_\infty$-groupoid.

For $U \in SmthMfd$ and $A \in CRing_\infty(\mathbb{C})$ let $\mathbb{G}_m \in \mathbf{H}$ be given by

$$
  \mathbb{G}_m \;\colon\; (U,A) \mapsto GL_1(A)\otimes C^\infty(U,\mathbb{C}^\times)
  \,,
$$

where on the right we have the [[∞-groupoid]] underlying the [[abelian ∞-group]] which is the [[tensor product]] of the [[∞-group of units]] of $A$ with the [[abelian group]] of non-vanishing complex-valued [[smooth functions]] on $X$.

[[!redirects smooth E-∞-groupoid]]

[[!redirects smooth E-infinity-groupoids]]
[[!redirects smooth E-∞-groupoids]]

[[!redirects smooth E-∞ groupoid]]
[[!redirects smooth E-∞ groupoids]]

[[!redirects smooth E-infinity groupoid]]
[[!redirects smooth E-infinity groupoids]]

[[!redirects smooth E? groupoid]]
[[!redirects smooth E? groupoids]]

[[!redirects smooth E∞-groupoid]]
[[!redirects smooth E∞-groupoids]]


smooth functor

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#### Differential geometry
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=--
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#Contents#
* table of contents
{:toc}

## Idea

### General

By a *smooth functor* one may generally understand an [[internal functor]] [[internalization|in]] the [[category]] of [[SmoothManifolds]], hence a [[homomorphism]] between [[internal categories]] or [[internal groupoids]] ([[Lie groupoids]]) in [[SmoothManifolds]].

Slightly alternatively, a smooth functor may be understood as an [[enriched functor]] between [[enriched categories]] over [[SmoothManifolds]].

### In global analysis

Much more specifically, "smooth functor" is used by [Kriegl & Michor 97, Sec 29.5](KrieglandMichor1997) to refer to an [[endofunctor]] of [[FinDimVect]], when $FinDimVect$ is viewed as a [[enriched category|category enriched]] in the category of [[smooth manifolds]].

That is, a smooth functor is a functor $F \colon FinDimVect \to FinDimVect$ such that the map $FinDimVect(X,Y) \to FinDimVect(F(X),F(Y))$ is smooth for every $X$, $Y$.

Currently, the remainder of this entry focuses on this specific notion.

## Examples

The iterated tensor product $X \mapsto X^{\otimes n}$ is a smooth functor.

The iterated wedge product $X \mapsto \bigwedge_{i=1}^n X$ is a smooth functor

## Extensions

The concept of smooth functor can be extended to multivariate functors $FinDimVect^n \to FinDimVect$, and also to [[contravariant]] functors $FinDimVect^{op} \to FinDimVect$, such as the dual $V \mapsto V^*$.

## Related concepts

* [[linear functor]]

## References

* {#KrieglandMichor1997} [[Andreas Kriegl]] and [[Peter Michor]], *The Convenient Setting of Global Analysis*, Mathematical Surveys and Monographs 53, American Mathematical Society, 1997 (ISBN: 978-0-8218-0780-4)

[[!redirects smooth functors]]