strict 2-limit

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## Idea

In general, a _[[2-limit]]_ is the sort of [[limit]] appropriate in a (weak) [[2-category]].  However, when we happen to be in a [[strict 2-category]] we also have another notion at our disposal:

Since strict 2-categories are equivalently categories [[enriched category|enriched]] over [[Cat]] (i.e. over the [[very large category|very large]] [[1-category]] of [[categories]] with [[functors]] between them), we can apply the usual notions of [[weighted limits]] in [[enriched categories]] verbatim to speak of certain 2-limits ([Street 1976](#Street76), [Kelly 1989](#Kelly89)). Here we refer to these [[Cat]]-[[enriched category|enriched]] [[weighted limits]] as *strict 2-limits*.  (Beware that, historically, these were called _2-limits_ while the properly 2-category-theoretic limits were called _bilimits_.)

Because [[enriched category theory]] doesn't know anything about the [[2-category theory|2-category theoretic]] nature of [[Cat]], the resulting *strict 2-limits* can have [[cones]] that [[commuting diagram|commute]] strictly and have [[universal properties]] expressed by [[isomorphisms]] of categories (instead of [[equivalences of categories]]); thus they can violate the [[2-category theory|2-category theoretic]] [[principle of equivalence]].

However, strict 2-limits often turn out to be technically useful even if one is ultimately interested only in weak 2-limits, since in many situations strict 2-limits may serve as stepping stones for the construction of weak [[2-limits]].  This is reminiscent of the use of strict structures in [[homotopy theory]] as a tool to get at weak ones, and in fact a precise comparison can be made (see below).

## Classification

By a **limit** we will mean the fully 2-categorical notion described at [[2-limit]], in which cones [[commuting diagram|commute]] up to invertible [[2-morphisms]] and the [[universal property]] is expressed by an [[equivalence of categories]].

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It just occurred to me that 'strict initial object' conflicts with this.  But unlike 'weak limit', that doesn't generalise very far.

Heh, you're right.  I suppose we could try calling strict initial objects _stable initial objects_, which would make more sense anyway since they are really the 0-ary version of a stable coproduct.  But there's probably not likely to be any real confusion created by the two uses of strict.
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* A **strict 2-limit** (or just _strict limit_) in a strict 2-category is just a [[Cat]]-enriched [[weighted limit]].  This means that its cones must commute strictly (although weakness can be built in via the weighting, see below), and its universal property is expressed by an isomorphism of categories.  Note that a strict limit is not necessarily a limit, because it may violate the [[principle of equivalence]].  (cf. [[red herring principle]].)

* A **pseudo limit** (or _strict pseudo limit_ if it is necessary to emphasize the strictness) is a limit whose cones commute up to coherent 2-cell isomorphism, but whose universal property can still be expressed by an _isomorphism_ of categories.  For any weight $W$, there is another weight $W'$ (a [[cofibrant replacement]] of $W$) such that pseudo $W$-weighted limits are equivalent to strict $W'$-weighted ones.  The idea is that $W'$ includes explicitly all the extra isomorphisms in a pseudo $W$-cone.  Since any isomorphism of categories is _a fortiori_ an equivalence of categories, any pseudo limit is also a limit.

* A **strict lax limit** is a limit whose cones commute only up to a coherent transformation in one direction, but again whose universal property is expressed by an isomorphism.  Likewise we have **strict oplax limits** where the transformation goes in the other direction.  Strict lax and oplax limits can also be rephrased as strict (non-lax) limits for a different weight.  As in the pseudo case, any strict (op)lax limit is also an (op)lax limit.

More generally, any strict limit that respects the [[principle of equivalence]] (one which doesn't demand equality of objects) will also be a limit.  Two formal versions of this statement involve [[flexible limit]]s and the more restrictive [[PIE-limit]]s.  In particular, any strict flexible limit is also a limit.  Since pseudo limits are PIE-limits, it follows that any strict 2-category which admits (strict) PIE-limits also admits all limits, even if it fails to admit some equivalence-violating strict limits.  The category of algebras and pseudo morphisms for any [[2-monad]], such as [[MonCat]], is a good example of a 2-category having strict PIE-limits but not all strict limits.

## Pseudo limits and homotopy limits

If there is a [[model category]] structure on the 1-category underlying the given strict 2-category $C$, then in addition to whatever 2-categorical notions of limit exist in $C$, there is the notion of [[homotopy limit]]s in $C$.  If $C$ is a [[model 2-category]] with the "trivial" or "natural" model structure constructed in (Lack 2006), then these two notions coincide (Gambino 2007).  For example, this is the case in [[Cat]] and [[Grpd]], so the examples listed at [[homotopy limit]] are also examples of pseudo limits.  In general, homotopy limits in a model 2-category give (non-strict) limits in its "homotopy 2-category."

## Examples

Any ordinary 1-limit can be made into a strict 2-limit simply by boosting up its ordinary universal property (a bijection of sets) to an isomorphism of hom-categories.  Thus we have strict [[products]], strict [[pullbacks]], strict [[equalizers]], and so on.  Of these, strict products (including [[terminal objects]]) respect the 2-category theoretic [[principle of equivalence]] (and thus are also 2-limits), while others such as pullbacks and equalizers tend to violate the 2-category theoretic [[principle of equivalence]].

* For example, a strict terminal object is an object 1 such that $K(X,1)$ is _isomorphic_ to the [[terminal category]], for any object $X$.

* Likewise, a strict product of $A$ and $B$ is an object $A\times B$ with projections $p:A\times B\to A$ and $q:A\times B\to B$ such that (1) given any $f:X\to A$ and $g:X\to B$, there exists a unique $h:X\to A\times B$ such thath $p h = f$ and $q h = g$ (equal, not isomorphic) (i.e. $A\times B$ is a [[product]] in the underlying [[1-category]]), and (2) given any $h,k:X\to A\times B$ and $\alpha: p h \to p k$ and $\beta:q h \to q k$, there exists a unique $\gamma:h\to k$ such that $p\gamma = \alpha$ and $q\gamma =\beta$.

As mentioned above, adding **pseudo** in front of an ordinary limit has a precise meaning: it means that all the triangles in the limit cone now commute up to specified isomorphism, and the universal property is still expressed by an isomorphism of categories.  In particular, there is still a specified projection to _each_ object in the diagram.  For example:

* The **pseudo [[pullback]]** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, and $r:P\to C$ and 2-cell isomorphisms $f p \cong r$ and $g q \cong r$.

* The **pseudo [[equalizer]]** of a pair of arrows $f,g:A\rightrightarrows B$ is a universal object $E$ equipped with morphisms $h:E\to A$ and $k:E\to B$ and 2-cell isomorphisms $f h \cong k$ and $g h \cong k$.

These are to be distinguished from:

* The **iso-[[comma object]]** of $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$ and $q:P\to B$ and a 2-cell isomorphism $f p \cong g q$.

* The **iso-[[inserter]]** of $f,g:A\rightrightarrows B$ is a universal object $E$ equipped with a morphism $e:E\to A$ and a 2-cell isomorphism $f e \cong g e$.

The pseudo pullback, pseudo equalizer, iso-comma object, and iso-inserter are all strict Cat-weighted limits; their universal property is expressed by an isomorphism of categories.  Usually the pseudo pullback and iso-comma object are not isomorphic, and likewise the pseudo equalizer and iso-inserter are not isomorphic.  However, both the pseudo pullback and iso-comma object respect the [[principle of equivalence]] and represent a pullback; therefore they are _equivalent_ when they both exist.  Likewise, the pseudo equalizer and iso-inserter both represent an equalizer, and are equivalent when they both exist.

If one is mostly interested in (non-strict) limits, then there is little harm in using "pseudo pullback" to mean "iso-comma object" or "pullback," as is common in the literature.  However, with _lax_ limits the situation is more serious.  Speaking precisely, in the _lax_ version of a limit, the triangles in the limiting cone are made to commute up to a specified transformation in one direction, but there are still specified projections to _each_ object in the diagram.  For example:

* The **strict lax limit of an arrow** $f:A\to B$ is a universal object $L$ equipped with projections $p:L\to A$ and $q:L\to B$ and a 2-cell $f p \to q$.

* The **strict lax pullback** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, $r:P\to C$, and 2-cells $f p \to r$ and $g q \to r$.

In particular, the strict lax pullback in the following list of examples  is quite different from the following more common limit.

* The **comma object** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a generalization of the [[comma category]] in $Cat$; it is a universal object $(f/g)$ equipped with projections $p:(f/g)\to A$ and $q:(f/g)\to B$ and a 2-cell $f p \to g q$.

Even in their non-strict forms, the lax pullback and comma object are distinct.  Usually the comma object is the more important one, but calling it a "lax pullback" should be avoided.

Here are some more important examples of 2-limits, all of which come in strict and weak forms and respect the [[principle of equivalence]].

* The **[[inserter]]** of a pair of parallel arrows $f,g:A \;\rightrightarrows\; B$ is a universal object $I$ equipped with a map $i:I\to A$ and a 2-cell $f i \to g i$.

* The **[[equifier]]** of a pair of parallel 2-cells $\alpha,\beta: f\to g: A\to B$ is a universal object $E$ equipped with a map $e:E\to A$ such that $\alpha e = \beta e$.

* The **[[inverter]]** of a 2-cell $\alpha:f\to g:A\to B$ is a universal object $V$ with a map $v:V\to A$ such that $\alpha v$ is invertible.

* The **[[power]]** of an object $A$ by a category $C$ is a universal object $A^C$ equipped with a functor $C\to K(A^C,A)$.

## References

Original articles on Cat-enriched weighted limits:

* {#Street76} [[Ross Street]], _Limits indexed by category-valued 2-functors_, Journal of Pure and Applied Algebra Volume 8, Issue 2, June 1976, Pages 149-181 (doi:10.1016/0022-4049(76)90013-X)

* {#Kelly89} [[Max Kelly]], *Elementary observations on 2-categorical limits*, Bulletin of the Australian Mathematical Society, 39(2), 301-317 ([doi:10.1017/S0004972700002781](https://doi.org/10.1017/S0004972700002781), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C14F8C3C46D45BCEC748660630EA7535/S0004972700002781a.pdf/div-class-title-elementary-observations-on-2-categorical-limits-div.pdf))

Review and list of examples:

* {#Lack10} [[Steve Lack]], around Sec. 6.8 in: _A 2-categories companion_,  In: Baez J., May J. (eds.) *[[Towards Higher Categories]]*. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 ([arXiv:math.CT/0702535](http://arxiv.org/abs/math.CT/0702535), [doi:10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4))

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 5.3 of: *2-Dimensional Categories*, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

See also:

* [[Steve Lack]], _Homotopy theoretic aspects of 2-monads_ ([arXiv:math.CT/0607646](http://www.arxiv.org/abs/math.CT/0607646)).

* [[Nicola Gambino]], _Homotopy limits for 2-categories_ ([pdf](http://www1.maths.leeds.ac.uk/~pmtng/Publications/homotopy.pdf))

[[!redirects strict 2-limits]]
[[!redirects strict 2-colimit]]
[[!redirects strict 2-colimits]]


strict 2-natural transformation

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## Idea

A **strict 2-natural transformation** (sometimes called simply a "2-natural transformation") is a [[lax natural transformation]] whose [[2-cell]] components are identities.  When the [[2-categories]] and [[2-functors]] involved are also [[strict 2-category|strict]] (as is usually the case), these are the [[enriched natural transformations]] over the monoidal category [[Cat]].

## Related concepts

* [[strict 2-functor]]

* [[strict adjoint 2-functor]]

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strict 3-category

#Idea

* A strict $3$-category is a [[directed n-graph|directed 3-graph]] equipped with a composition operation on adjacent cells (of all levels) which is strictly unital and associative.

* The concept of a strict $3$-category is the simplest generalization of a [[category]] to a [[3-category]]. It is the one-step [[vertical categorification|categorification]] of the concept of a [[strict 2-category]].

#Definition

A __strict $3$-category__, often called simply a _3-category_, is a [[enriched category|category enriched over]] $\Str2\Cat$, the [[cartesian monoidal category]] of strict $2$-categories.  Similarly, a [[strict 3-groupoid]] is a [[groupoid]] enriched over [[strict 2-groupoid]]s.

These are also called [[geometric shapes for higher structures|globular]] strict $3$-categories and $3$-groupoids, to emphasise the underlying geometry.

#Remarks

* A strict $3$-category is the same as a [[strict omega-category]] which is trivial in degree $j \geq 4$.

* This is to be contrasted with a _weak $3$-category_ called a [[tricategory]] and a _semistrict $3$-category_ called a [[Gray-category]].

* $\Str\Cat$ has the same underlying category as the [[symmetric monoidal category]] [[Gray]].  However, a category enriched over $\Grey$, a [[Gray-category]], is more general than a strict $3$-category.


strict adjoint 2-functor

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## Idea

This entry means to indicate a generalization of the notion of [[adjoint functors]] from [[category theory]] to [[2-category theory]], but specialized to the case of [[strict 2-categories]].

Notice that [[strict 2-categories]] may be understood as [[Cat]]-[[enriched categories]], for [[Cat]] understood as the *[[1-category]]* of [[small categories|small]] [[strict categories]] with [[functors]] between them and equipped with its [[cartesian monoidal category|cartesian monoidal]]-[[structure]] (via forming [[product categories]]).

In this sense the respective notion of adjoint 2-functors between strict 2-categories is that of *[[Cat]]-[[enriched adjoint functors]]*, which (by the discussion there) is equivalently that of [[adjunctions]] in the [[2-category]][[VCat|$\mathcal{V}Cat$]], for $\mathcal{C} = $ [[Cat]].

The following indicates what this means in more explicit detail.

## Definition

Given a [[pair]] of [[strict 2-functors]] between [[strict 2-categories]]

$$
  F
    \,\colon\,
  A \rightleftarrows C
    \,\colon\,
  U
$$

a *strict 2-adjunction* between them is, equivalently:

* for each [[pair]] of [[objects]] $a \in A$, $c\in C$, a [[natural isomorphism]] of [[strict categories|strict]] [[hom-categories]]

  $$
    C\big(F(a), c\big) \,\cong\, A\big(a, U(c)\big)
  $$

  which is [[strict 2-natural transformation|strict 2-natural]] in $a$ and $c$;

* a pair of [[strict 2-natural transformation|strict 2-natural transformations]] of 2-functors:

   1. [[adjunction unit]] $\; \eta \colon Id_A \to U F$,

   1. [[adjunction counit]] $\; \epsilon \colon F U \to Id_B$,

  satisfying the [[triangle identities]] strictly.

## Related concepts

For more general notions of *2-adjunctions*, involving [[weak 2-categories]], [[weak 2-functors]], and/or weak 2-natural transformations; see at *[[2-adjunction]]*.

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strict analytic geometry

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## Idea

Strict analytic geometry is the study of analytic spaces over Banach
rings with building blocs given by strict rational domains (defined, for
example, by relations of the form $\{x,\;|f(x)|\leq |g(x)|\neq 0\}$)
in polydiscs of radius one.

In the $p$-adic setting, strict analytic geometry is
usually called [[rigid analytic geometry]]. In the archimedean setting
(i.e., over $\C$), strict analytic geometry is essentially equivalent to usual
analytic geometry.

There is a natural notion of strict [[global analytic geometry]] that has some
non-trivial relation with the ideas of [[Arakelov geometry]].

## Related concepts

* [[rigid analytic geometry]]
* [[overconvergent global analytic geometry]]


strict and non-strict orders

## Idea

In [[order theory]], there are usually two different versions of an order relation, a [[reflexive relation|reflexive]] version which is usually referred to as a **reflexive order** or **non-strict order** and an [[irreflexive relation|irreflexive]] version which is usually referred to as an **irreflexive order**, **strong order**, or **strict order**. Examples include:

* [[preorder]], [[strict preorder]]

* [[partial order]], [[strict partial order]]

* [[total order]], [[strict total order]]

* [[weak order]], [[strict weak order]]

* [[linear order]], [[strict linear order]]

## References

* Wikipedia, [Preorder](https://en.wikipedia.org/wiki/Preorder)

* Wikipedia, [Partial order](https://en.wikipedia.org/wiki/Partial_order)

* Wikipedia, [Total order](https://en.wikipedia.org/wiki/Total_order)

* Wikipedia, [Weak order](https://en.wikipedia.org/wiki/Weak_order)

category: disambiguation

[[!redirects reflexive order]]
[[!redirects reflexive orders]]
[[!redirects reflexive ordering]]
[[!redirects reflexive orderings]]
[[!redirects reflexively ordered]]
[[!redirects reflexively ordered set]]
[[!redirects reflexively ordered sets]]

[[!redirects irreflexive order]]
[[!redirects irreflexive orders]]
[[!redirects irreflexive ordering]]
[[!redirects irreflexive orderings]]
[[!redirects irreflexively ordered]]
[[!redirects irreflexively ordered set]]
[[!redirects irreflexively ordered sets]]

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[[!redirects strict orders]]
[[!redirects strict ordering]]
[[!redirects strict orderings]]
[[!redirects strictly ordered]]
[[!redirects strictly ordered set]]
[[!redirects strictly ordered sets]]

[[!redirects non-strict order]]
[[!redirects non-strict orders]]
[[!redirects non-strict ordering]]
[[!redirects non-strict orderings]]
[[!redirects non-strictly ordered]]
[[!redirects non-strictly ordered set]]
[[!redirects non-strictly ordered sets]]

[[!redirects strong order]]
[[!redirects strong orders]]
[[!redirects strong ordering]]
[[!redirects strong orderings]]
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strict category

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# Strict categories
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## Idea

A __strict category__ is a [[category]] together with the [[structure]] of a [[set]] (or [[class]]) on its collection of [[objects]]; in particular, the objects can be compared for [[equality]] (not merely [[isomorphism]]).  In contrast, a __weak category__ is a category *without* such structure.  Similarly, a __strict functor__ is one which preserves equality of objects: $F(x) = F(y)$ if $x = y$.  (NB: this is not inherently size-related---both [[large category|large]] and [[small categories]] can be either strict or weak.)

If we use [[set theory|set-theoretic]] [[foundations]] for category theory, then by definition, a category comes with a set (or class) of objects, and therefore admits the structure of a strict category in a canonical way.  Thus, in this case, it is possible to ignore the adjective "strict" in "strict category", while a "strict functor" simply means a "non-[[anafunctor|ana]] functor" (the usual default meaning of 'functor', but not necessarily the best concept).

If we additionally assume the (possibly global) [[axiom of choice]], then not only is any category [[equivalence of categories|equivalent]] to a canonical strict category, but every (ana-)functor is [[naturally isomorphic functors|naturally isomorphic]] to a strict functor, so the notions have little to no mathematical content.  But in other foundations, these facts may not be true.

Moreover, even in set-theoretic foundations with global $AC$, the notion of strict category is not meaningless: calling a category strict signals that we intend to *make use of* the equality relation on its objects.  When taking the [[nPOV]], we often adopt the [[philosophy|philosophical]] position that [[category theory]] is fundamentally about weak categories; see the discussion at [[principle of equivalence]].  However, strict categories still do occasionally arise in mathematical practice; see the examples below.

## Definition

In order to formulate a foundation-independent definition, we make the following assumptions.  Ignoring [[size issues]], suppose that we have a [[bicategory]] $Cat$ consisting of weak [[categories]], [[functors]] (some of which are [[essentially surjective functor|essentially surjective]]), and [[natural transformations]] (some of which are [[natural isomorphisms]]).  Suppose also that we have a [[category]] $Set$ consisting of [[sets]] and [[functions]], and a [[functor]] $Disc\colon Set \to \Cat$ taking each set to the [[discrete category]] on that set.  Note that we do not assume the existence of an "underlying set of objects" functor $Cat\to Set$.

In this context, a __strict category__ $C$ consists of

* a set $Ob(C)$ (to be thought of as the set of [[objects]] of $C$),
* a category $Wk(C)$ (to be thought of as the underlying weak category of $C$), and
* an essentially surjective functor $cl(C)\colon Disc(Ob(C)) \to Wk(C)$ (to be thought of as taking each object of $C$ to its [[clique]], more or less, in $C$).

A __strict functor__ $F$ from $C$ to $D$ (both strict categories) consists of

* a function $Ob(F)\colon Ob(C) \to Ob(D)$,
* a functor $Wk(F)\colon Wk(C) \to Wk(D)$, and
* a natural isomorphism $cl(F)\colon cl(C) ; Wk(F) \to Disc(Ob(F)) ; cl(D)$.

Two strict functors $F, G\colon C \to D$ are __equal__ if

* $Ob(F) = Ob(G)$ and
* there is a (necessarily unique) natural isomorphism from $Wk(F)$ to $Wk(G)$ modulo which $cl(F)$ equals $cl(G)$.

There is an obvious notion of [[composition]] that defines a [[1-category]] $Str Cat$ of strict categories and (equality classes of) strict functors.  Note that this 1-category has a strictly associative and unital composition law, even if $Cat$ is only a [[bicategory]].  (However, $Str Cat$ is not *itself* a strict category, unless the assumed category $Set$ is strict.)  We have a [[forgetful functor]] $Ob\colon Str Cat \to Set$ and a [[pseudofunctor]] $Wk\colon Str Cat \to Cat$.

Finally, a __natural transformation__ between two strict functors $F$ and $G$ is simply

* a natural transformation from $Wk(F)$ to $Wk(G)$.

(That is, a [[natural transformation]] between strict functors is automatically strict.)  With these as 2-cells, we have a [[strict 2-category]] of strict categories, strict functors, and natural transformations.

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###### Remark
An equivalent definition is that a strict category is an $\mathcal{M}$-[[M-category|category]] whose underlying category of tight morphisms is discrete, while a strict functor is an $\mathcal{M}$-functor.  In this case, the tight morphisms (which are necessarily [[isomorphisms]]) are sometimes called **special isomorphisms**.
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## Foundational choices

We now make the above schematic definition explicit in terms of various different foundational systems.

### Set theory with AC

In [[classical mathematics|classical]] [[set theory|set-theoretic]] foundations including the [[axiom of choice]], we take $Cat$ to be the [[strict 2-category]] consisting of (say small) [[categories]], [[functors]], and [[natural transformations]], defined in any of the traditional ways.  The functor $Disc\colon Set \to Cat$ is [[left adjoint]] to the functor $Ob\colon Cat \to Set$, and thus $Str Cat$ is a subcategory of the [[Freyd cover]] of $Cat$.

In this case, the functor $Wk\colon Str Cat \to Cat$ has a [[right adjoint]] which takes a category $C$ to the strict category $Disc(Ob(C)) \to C$.  The adjunction counit is literally the identity, while the adjunction unit consists of [[equivalences]] (but not isomorphisms) in the strict $2$-category $Str Cat$.  Thus, $Str Cat$ is equivalent to $Cat$ (as a bicategory).

The right adjoint $Cat \to Str Cat$ also has a further right adjoint defined as follows: given a strict category $C$, let $C'$ be the category with $Ob(C)$ as its set of objects, and morphisms induced from those in $Wk(C)$.  The unit of this adjunction again consists of identities, while the counit consists of equivalences in $Str Cat$.

This analysis applies not only to [[material set theories]] such as [[ZFC]], but also [[structural set theories]] such as [[ETCS]] and [[SEAR]], as long as they have the axiom of choice.

### Set theory without AC

In this case, it is usually best to define [[Cat]] to be the bicategory of categories, [[anafunctors]], and natural transformations.  Now although every category has an underlying set of objects, we no longer have a functor $Ob\colon Cat \to Set$.  It is still true that every category $C$ admits the structure of a strict category in a canonical way (with $Ob(C)$ its set of objects), but not every strict category necessarily arises in this way, and not every morphism $C \to D$ in $Cat$ (i.e. anafunctor) is a strict functor relative to the canonical strict-category structures of $C$ and $D$.  In fact, an anafunctor is a strict functor precisely when it is naturally isomorphic to an ordinary (non-ana) functor.

If we instead defined $Cat$ to consist only of non-ana functors, then everything would happen exactly as with AC, except that neither adjunction would be an equivalence.

### Type and preset theories with equality

We include under this heading all theories in which the basic objects, generally called [[type theory|types]] or [[presets]] (and sometimes even called 'sets'), come with an [[equality]] [[relation]], but the notion of [[set]] is a defined one.  Generally, the issue is that types (or presets) do not admit [[quotient object|quotients]], and so a **set** is defined to be a type equipped with an [[equivalence relation]] called "equality" (sometimes called a [[setoid]] in this context, especially if the types or presets are themselves confusingly called 'sets').

In this case, it is natural to define a *category* to have only a type/preset of objects, but a set(oid) of morphisms between each pair of objects; see [[type-theoretic definition of category]].  Now every category still has an underlying set of objects, but not every set is the set of objects underlying some category; only the [[completely presented sets]] have this property.

Again, it is true that every category can be made into a strict one in a canonical way, but that not every strict category arises in this way.  This is a different sort of failure than in set theory without $AC$, and can happen even in the presence of full $AC$ (even for setoids).  For instance, consider the [[discrete category]] on the set of [[real numbers]].  Of course this depends on specifics, but typically, there is no type of real numbers per se but only a type of Cauchy (pre)sequences of rational numbers (see [[Cauchy real number]]).  To get the set of real numbers, we must consider this type together with a certain equivalence relation, a setoid.  The discrete category on the set of real numbers is essentially the exact same construction, now with a note that parallel morphisms are always defined to be equal.  The automatic notion of equality on the objects of this category is equality of Cauchy (pre)sequences, but the desired notion of equality for this discrete category as a strict category is isomorphism, which is equality of real numbers.  So while this category may be automatically a strict category, it is a strict category in the wrong way.  (This is not a perfect example, since, given $AC$, the discrete category of real numbers is still equivalent to the discrete category of Cauchy sequences and therefore does arise as a canonical strict category, up to equivalence.)

Note that we still have a functor $Disc\colon Set \to Cat$ which regards an equivalence relation on a type $A$ as a groupoid with $A$ as its type of objects.  The resulting structure of strict categories behaves much like it does in set theory without AC.

### Type and preset theories without equality

If types or presets do not come with a notion of equality, as in some preset theories and [[SEAR+?]], then a category does not in general even *have* an "underlying set of objects", so a category cannot be made into a strict category in any canonical way.  But defining sets as presets with equivalence relations as before, we still have $Disc\colon Set \to Cat$, and we can define strict categories, constructing specific examples as wanted.

### Homotopy type theory

If [[type theory]] is treated as a theory of set-like objects, then it behaves as above, whether [[extensional type theory|extensional]] or [[intensional type theory|intensional]].  However, if we take the point of view of [[homotopy type theory]], then the basic objects of the theory are [[homotopy types]], a.k.a. [[∞-groupoids]].  A *set* is now defined to be an [[h-set]], i.e. a [[truncated object|0-truncated]] type.

Now the natural definition of (1-)category is something like a [[complete Segal space]], where the type of objects is 1-truncated (a [[groupoid]]) and equivalent to the [[core]] of the category.  Now (even though identity types exist in HoTT), not every category has an underlying *set* of objects, so again a category cannot be made into a strict category in any canonical way.  But again, we still have $Disc\colon Set \to Cat$ and we can define strict categories.

### Fibered categories

In [[Jean Bénabou]]'s 1985 paper _[[Fibered categories and the foundations of naive category theory]]_, it is argued that the right notion of a category in/over another category $\mathbf{B}$ is given by a [[Grothendieck fibration|fibered category]] $p\colon \mathbf{F} \to \mathbf{B}$, possibly with some additional structure corresponding to the "definability" (representability of certain sieves of $\mathbf{B}$) of equality of parallel pairs, isomorphisms, etc. (See [[indexed category]] for more motivation.) In this context, equality of objects is available precisely when the fibration is [[split fibration|split]].

In particular, if $\mathbf{B}$ is locally small and has [[pullback|pullbacks]], then an [[internal category]] object $C$ in $\mathbf{B}$ (in the traditional sense) gives rise to an honest functor $\mathbf{B} (-, C)\colon \mathbf{B}^{op} \to Cat$ and so a split fibration $\int_\mathbf{B} \mathbf{B} (-, C) \to \mathbf{B}$ by the [[Grothendieck construction]].

## Examples

There are some at [[M-category]], to put also here.

## Strict higher categories

The notion of strict category is related to the notion of strictness in [[higher category theory]].

For instance, a [[2-category]] must have strict categories as its [[hom-categories]] in order to write down the definition of [[strict 2-category]]; a similar remark holds for [[strict 2-functors]].  As such, strictness in the sense of this page matches the usual sense of strictness in a "local" sense (every strict 2-category is locally a strict 1-category).  However, we now get a notion of __strict $2$-natural transformation__ as a [[2-natural transformation]] that preserves equality (that is, if two objects are equal, then so are the corresponding components of the natural transformation).  (All [[modifications]] between strict transformations are automatically strict ... until we get to $3$-categories, of course!)

On the other hand, the underlying 1-category of a [[strict 2-category]] (its objects and 1-morphisms) need not be a strict 1-category.  For instance, this is the case for the strict 2-category $Str Cat$ defined in a structural set theory, such as [[SEAR]], in which there is no equality predicate on sets.  For a fully consistent terminology, we could say 'locally strict $2$-category' in place of 'strict $2$-category', but this is unlikely to catch on.  Some system of numbered strictness ($1$-strict, $2$-strict, etc) might also work.

[[!redirects strict category]]
[[!redirects strict categories]]
[[!redirects strict functor]]
[[!redirects strict functors]]
[[!redirects strict groupoid]]
[[!redirects strict groupoids]]
[[!redirects StrCat]]
[[!redirects Str Cat]]

[[!redirects weak category]]
[[!redirects weak categories]]


strict epimorphism

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# Strict epimorphisms
* table of contents
{: toc}

## Definition

A **strict epimorphism** in a [[category]] is a [[morphism]] which is the joint [[coequalizer]] of all pairs of [[parallel morphisms]] that it coequalizes.  In other words, $f \colon B\to C$ is a strict epimorphism if it is the [[colimit]] of the (possibly [[large category|large]]) [[diagram]] consisting of all [[parallel morphisms|parallel pairs]] $g,h \colon A \;\rightrightarrows\; B$ such that $f g = f h$.

Although this definition does not include this explicitly, it follows that $f$ is an [[epimorphism]].

A __strict monomorphism__ is a morphism such that its dual is strict epimorphism in the dual category.

## Relation to other epimorphism classes

If $f$ has a [[kernel pair]] $r,s \,\colon\,ker(f) \;\rightrightarrows\; B$ (such as if the ambient category has [[pullbacks]]), then any such pair $g,h$ factor uniquely through the kernel pair, which is itself such a pair (that is, $f r = f s$).  Thus, for any $k \colon B\to D$, we have $k g = k h$ for all $g,h$ with $f g = f h$ if and only if $k r = k s$.  Therefore, $f$ is strict epi if and only if it is the coequalizer of its kernel pair, hence if and only if it is an [[effective epimorphism]] and therefore a [[regular epimorphism]].

For this reason, some sources define "regular epimorphism" in a category without pullbacks to mean what we have called a "strict epimorphism."

It is easy to see that in *any* category, any regular epimorphism is strict.  In a category without pullbacks, it seems that not every strict epimorphism need be regular.  However, every strict epimorphism is [[strong epimorphism|strong]], and hence [[extremal epimorphism|extremal]], for the same reason that any regular epimorphism is.

## Properties
If for an epimorphism $f$, the composition $g \circ f$ is a strict epimorphism then $g$ is a strict epimorphism.

## References

* [[Alexander Grothendieck]], Definition 2.2, p. 304 in:  *Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats* ([[FGA]]), Séminaire Bourbaki, no. 5 (1960) [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/SB_1958-1960__5__299_0)]

See also Definition 2.1 in

* [[Eduardo J. Dubuc]], _From Yoneda to Topoi morphisms_, [arXiv](https://arxiv.org/abs/2312.04716).

Textbook accounts:

* [[Ion Bucur]], [[Aristide Deleanu]], _Introduction to the theory of categories and functors_, Wiley, 1968.

* [[Masaki Kashiwara]], [[Pierre Schapira]], p. 115-116 in: _[[Categories and Sheaves]]_

[[!redirects strict epimorphisms]]
[[!redirects strict epi]]
[[!redirects strict epis]]

[[!redirects strict monomorphism]]
[[!redirects strict monomorphisms]]
[[!redirects strict mono]]
[[!redirects strict monos]]


strict factorization system

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\tableofcontents

## Idea

A **strict factorization system** is like an [[orthogonal factorization system]], but the factorizations are specified uniquely on the nose, rather than merely up to [[isomorphism]].

## Definition

A **strict factorization system** on a category $C$ comprises [[wide subcategories]] $E$ and $M$ of $C$ such that every morphism in $C$ factors *uniquely* (not just uniquely up to unique isomorphism) as $m e$ for $e \in E$ and $m \in M$.

Note that a strict factorization system is not necessarily an [[orthogonal factorization system]], since $E$ and $M$ may not contain the isomorphisms in $C$. However, for every strict factorization system, there is a unique orthogonal factorization system containing it, given by closing $E$ under postcomposition by isomorphisms and closing $M$ under precomposition by isomorphisms (see §2.1 of [Grandis (2000)](#Grandis00)).

## Examples

* Every category has two "trivial" strict factorization systems in which $E$, respectively $M$, consists of the [[identity morphisms]] only.  The corresponding orthogonal factorization systems are those with one class consisting of the isomorphisms.

* Part of the structure of a (non-generalized) [[Reedy category]] is a strict factorization system.

* The category [[Set]], as defined in a [[material set theory]], has a strict factorization system consisting of the surjections and the inclusions of [[subsets]].  Its associated orthogonal factorization system consists of surjections and [[injections]].  Other categories built out of structured sets have similar strict factorization systems.

## As distributive laws in a bicategory

One reason strict factorization systems are of interest is that they can be identified with [[distributive laws]] in the [[bicategory]] of [[spans]], as shown by [Rosebrugh & Wood (2002)](#RosebrughWood02).

Ordinary orthogonal factorization systems can be similarly characterized by:

* using a type of relaxed [[distributive law]], as in [Rosebrugh & Wood (2002)](#RosebrughWood02);

* using [[wreaths]] as in [Lack & Street (2002)](#LackStreet02);

* by working in the [[bicategory]] of [[profunctors]] instead, as in [Lack (2004)](#Lack04) (see also [[factorization system over a subcategory]]); or by

* using [[weak distributive laws]], as in [Böhm (2012)](#Böhm12).

## References

Strict factorization systems were defined in:

* {#Grandis00} [[Marco Grandis]]. _Weak subobjects and the epi-monic completion of a category_. Journal of Pure and Applied Algebra 154.1-3 (2000): 193-212.

See also:

* {#RosebrughWood02} [[Robert Rosebrugh]], [[Richard J. Wood]], *Distributive Laws and Factorization*, Journal of Pure and Applied Algebra **175** 1–3 (2002) 327-353 [doi:10.1016/S0022-4049(02)00140-8[pdf](http://www.mta.ca/~rrosebru/articles/fac.pdf)]

* {#LackStreet02} [[Stephen Lack]], [[Ross Street]], *The formal theory of monads II*, J. Pure Appl. Alg. **175** 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

* {#Lack04} [[Stephen Lack]], *Composing PROPs*, Theory and Applications of Categories, **13** 9 (2004) 147-163 [[tac:13-09](http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html)]

* {#Böhm12} [[Gabriella Böhm]], *Factorization systems induced by weak distributive laws*, Appl. Categ. Structures **20** 3 (2012) 275-302 [[doi:10.1007/s10485-010-9243-y](https://doi.org/10.1007/s10485-010-9243-y), [arXiv:1009.0732](https://arxiv.org/abs/1009.0732)]

[[!redirects strict factorization systems]]
[[!redirects strict factorisation system]]
[[!redirects strict factorisation systems]]


strict ind-object

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#Contents#
* table of contents
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## Definition

If $C$ is a [[category]] then an [[ind-object]] $x\in Ind(C)$ is a __strict ind-object__ (alias __essentially monomorphic ind-object__) if it can be represented in $Ind(C)$ as (the apex of) a [[colimit]] of a small [[filtered diagram]] (whose [[objects]] are in $C$ and) whose [[morphisms]] are specifically [[monomorphisms]] in $C$.

([Grothendieck-Verdier 71, Exposé I.§8.12.1](#GrothendieckVerdier71)). See also [Blanc 96, def. 4.1](#Blanc96).

[[abstract duality|Dually]], __strict [[pro-objects]]__ (alias __essentially epimorphic pro-objects__) are [[limits]] of small [[cofiltered diagrams]] involving only [[epimorphisms]].

## References

* {#GrothendieckVerdier71} [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], _Prefaisceaux_, in _Theorie de Topos et Cohomologie Etale de Schemas_ 1971

* {#Blanc96} [[David Blanc]], _Colimits for the pro-category of towers of simplicial sets_, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1996) Volume: 37, Issue: 4, page 258-278 ([numdam](http://www.numdam.org/item?id=CTGDC_1996__37_4_258_0))

[[!redirects strict ind-objects]]

[[!redirects strict pro-object]]
[[!redirects strict pro-objects]]
[[!redirects essentially epimorphic pro-object]]
[[!redirects essentially monomorphic ind-object]]


strict initial object

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* table of contents
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## Idea

The [[empty set]], among all sets, has two characteristic properties:

1. there is a *unique* [[function]] *out* of the empty set, to any other set;

1. there is *no* [[function]] *to* the empty set, except from itself.

The first property generalizes to arbitrary [[categories]] as the property of an *[[initial object]]*.

The corresponding generalization including also the second property is that of a *strict initial object*:

## Definition

An [[initial object]] $\varnothing$ is called a **strict initial object** if every [[morphism]] [[codomain|to]] $\varnothing$ is an [[isomorphism]]:

\[
  \label{MorphismToStrictInitialObjectIsIsomorphism}
  X \overset{f}{\longrightarrow} \varnothing
  \;\;\;\;\;\;\;\;\;
    \Rightarrow
  \;\;\;\;\;\;\;\;\;
  X \underoverset{\simeq}{f}{\longrightarrow} \emptyset
  \,.
\]

## Properties
 {#Properties}

* The [[Cartesian product]] of any [[object]] $X$ with a strict initial object is isomorphic to the strict initial object, $X \times \varnothing \simeq \varnothing$, because the [[projection]] $pr_1 \colon \varnothing \times X \to \varnothing$ exists by definition of Cartesian products, whence (eq:MorphismToStrictInitialObjectIsIsomorphism) implies that it is an isomorphism

  $$
    \varnothing \times X
      \underoverset
        {\;\;\;\simeq\;\;\;}
        {pr_1}
        {\longrightarrow}
    \varnothing
    \,.
  $$

* Strict initial objects may be understood as [[van Kampen colimits]], see e.g. [Sobocinski & Heindel (2011), Exp. 4.5 (i)](#SobocinskiHeindel11). Indeed, the van Kampen-property in this case requires that the [[slice category]] $C/\varnothing$ be [[equivalence of categories|equivalent]] to the [[terminal category]] $\mathbf{1}$.

## Examples
 {#Examples}

* The [[empty set]] is a strict initial object in [[Sets]].

* The [[empty topological space]] is a strict initial object in [[TopologicalSpaces]].

* The [[empty groupoid]] is a strict initial object in [[Groupoids]].

* The [[empty simplicial set]] is a strict initial object in [[SimplicialSets]].

* The initial objects of any of the following types of categories are strict:

  * in [[posets]],

  * in [[toposes]],

  * in [[extensive categories]],

  * in [[distributive categories]].

* Specifically the initial objects of [[Set]], [[Cat]], [[Top]] are all strict.

* [[John Baez]] showed on the [Category Theory Zulip](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Integers.20and.20rational.20numbers.20as.20strict.20initial.20objects.3F/near/422742440) that the [[rational numbers]] are a strict initial object in the category of [[characteristic zero]] [[fields]] and [[ring homomorphisms]]. This implies that the rational numbers are a strict initial object in the category of [[ordered fields]] and [[ring homomorphisms]].

* In the [same discussion thread on the Category Theory Zulip](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Integers.20and.20rational.20numbers.20as.20strict.20initial.20objects.3F/near/422749731) [[Madeleine Birchfield]] showed that the [[integers]] are a strict initial object in the category of [[ordered integral domains]] and [[monotonic function|strictly monotonic]] [[ring homomorphisms]].

At the other extreme, a [[zero object]] is a strict initial object only if the category is trivial (i.e. [[equivalence of categories|equivalent]] to the [[terminal category]]).

## Related concepts

* An [[empty bundle]] is a [[morphism]] out of a strict initial object (an [[empty morphism]]), regarded as a [[bundle]].

* [[formal dual]] concept: [[strict terminal object]]

* [[weakly initial object]]

## References

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], Def. 2.7 in: _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

Discussion as [[van Kampen colimits]]:

* {#SobocinskiHeindel11} [[Pawel Sobocinski]], [[Tobias Heindel]], Exp. 4.5 (i) in: *Being Van Kampen is a universal property*, Logical Methods in Computer Science, **7** 1 (2011) [[arXiv:1101.4594](https://arxiv.org/abs/1101.4594), doi:10.2168/LMCS-7(1:14)2011]

[[!redirects strict initial objects]]


strict inverse function

\tableofcontents

## Idea

In [[dependent type theory]], there are two notions of [[equality]],

1. *strict* or *[[judgmental equality]]*,

1. *weak* or *[[typal equality]]*.

Accordingly there are two corresponding notions of the [[inverse function]] of a function $f:A \to B$:

1. There are the various notions of *weak* [[inverse functions]], such as [[quasi-inverse functions]] and [[coherent inverse functions]], which is a function $f^{-1}:B \to A$ with [[homotopies]] $h:\prod_{x:A} f^{-1}(f(x)) =_A x$ and $k:\prod_{y:B} f(f^{-1}(y)) =_B y$ and possibly other conditions and structure.

1. There is the notion of *strict* inverse function, which is a function $f^{-1}:B \to A$ with [[judgmental equalities]] $x:A \vdash f^{-1}(f(x)) \equiv x:A$ and $y:B \vdash f(f^{-1}(y)) \equiv y:B$. The other structures are unimportant since non-trivial paths for weak inverse functions reduce to [[reflexivity]] for $x:A$ or $y:B$ for strict inverse functions.

## Examples

* If [[copy types]] are defined using judgmental [[computation rules]] and [[uniqueness rules]], then the constructor and eliminator functions are strict inverse functions of each other.

* In [[Mike Shulman]]'s [[higher observational type theory]] the $\mathrm{idtoequiv}$ function for the [[univalent universes]] has a strict inverse function.

## See also

* [[judgmental equality]]

* [[inverse function]]

[[!redirects strict inverse functions]]
[[!redirects strict inverse function]]


strict Lie 2-algebra

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#Contents#
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## Idea

A strict Lie 2-algebra is the [[infinitesimal object|infinitesimal]] approximation to a smooth [[strict 2-group]] in generalization of how an ordinary [[Lie algebra]] is the infinitesimal approximation to a [[Lie group]].

## Definition

The notion of _strict Lie 2-algebra_ is the special case of a general notion [[Lie 2-algebra]] for which the [[Jacobi identity]] does hold (and not just up to nontrivial isomorphism).

More precisely: a strict Lie 2-algebra is an [[∞-Lie algebra]] with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.

Equivalently, this is a [[dg-Lie algebra]] with generators in the lowest two degrees.

In direct analogy to how [[strict 2-group]]s are equivalently encoded in a smooth [[crossed module]] of [[group]]s, a strict Lie 2-algebra is equivalently encoded in a [[differential crossed module]] of ordinary [[Lie algebra]]s.

## Examples

### Derivation Lie 2-algebra

The Lie version of a smooth [[automorphism 2-group]] is the derivation Lie 2-algebra $Der(\mathfrak{g})$ of an ordinary [[Lie algebra]] $\mathfrak{g}$. This is the one coming from the [[differential crossed module]] $(\mathfrak{g} \stackrel{ad}{\to} der(\mathfrak{g}))$.

## Related concepts

* [[Lie algebra]], [[Lie group]]

* [[Lie 2-algebra]], [[Lie 2-group]]

* [[L-infinity algebra]], [[smooth ∞-group]]

* [[Lie algebroid]], [[Lie groupoid]]

* [[L-∞ algebroid]], [[smooth ∞-groupoid]]

[[!redirects strict Lie 2-algebras]]


strict local ring

[[!redirects strict local rings]]

## Idea

For $Spec \mathbb{Z}$ an [[affine scheme]], the objects which the [[étale topos]] $Sh(X_{et})$ classifies are **strict local rings**. The [[point of a topos|points of this topos]] are _[[strict Henselian rings]]_ ([Hakim, III.2-4](#Hakim)) and ([Wraith](#Wraith)).

## Definition

We need to knead the definition of strict Henselian ring into being geometric.

Let $p = \sum_{i=0}^{n} a_i x^i \in R[x]$ be a monic polynomial of degree $n$ (so $a_n = 1$).

If $R$ is a field, a simple root of $p$ is an $r \in R$ with $p(r) = 0$ and $(p/(x-r))(r) \neq 0$. We generalise this as:

+-- {: .num_defn}
###### Definition
More generally if $R$ is a ring, a _simple root_ of $p$ is an $r \in R$ with $p(r) = 0$ and $p'(r)$ invertible.
=--

If $R$ is a field, $p$ is _unramifiable_ if in any field extension $R \subseteq Q$ in which we can write $p = \prod_i (x-s_i)$, some $s_i$ is a simple root.

To make this definition work for more general rings, some work must be done. Let $R'$ be the ring $R[s_1, \cdots, s_n]/\langle a_k - \sigma_k(\vec s) \mid k \in \{0,\cdots,n\} \rangle$, where $\sigma_k$ is the [[elementary symmetric polynomial]]. Then the _hyperdiscriminant polynomial_ is the polynomial $\Delta(p) = \prod_{i=1}^n (1 + p'(s_i) x) \in R'[x]$. As this polynomial is symmetric in the $s_i$, the theory of symmetric polynomials tells us that $\Delta(p) \in R[x]$.

+-- {: .num_defn}
###### Definition
More generally if $R$ is a ring, $p$ _unramifiable_ if $\Delta(p)-1$ is nonconstant i.e. $1 \in \langle \Delta_1(p), \cdots, \Delta_n(p) \rangle$ where $\Delta_i(p)$ is the $x_i$ coefficient of $\Delta(p)$
=--

This is roughly justified by the following:

+-- {: .num_lemma}
###### Proposition
If $R$ is a local ring and $t_1, \cdots, t_n \in R$, then at least one of $t_1, \cdots, t_n$ is invertible iff $1 \in \langle a_1, \cdots, a_n \rangle$ where $a_i$ are defined so that $\prod_{i=1}^n (1 + t_i x) = 1 + \sum_{i=1}^n a_i x^i$
=--
+-- {: .proof}
###### Proof
Each $a_i$ is a finite sum of a nonempty product of some $t_j$, so if $1$ is in the ideal some $t_j$ is invertible.

Conversely, if $t_j$ is invertible, substituting $x = -1/t_j$ gives $1 = \sum_{i=1}^n a_i (-1)\cdot(-1/t_j)^i$
=--

This lets us arrive at our definition:

+-- {: .num_defn}
###### Definition
A ring $R$ is _separably closed_ iff whenever $p$ is a monic polynomial of positive degree, if $p$ is unramifiable, $p$ has a simple root.

A _strict local ring_ is a separably closed local ring.
=--

## References

III.2-4 of

* [[Monique Hakim]], _Topos annelés et schémas relatifs_
 {#Hakim}

and

* [[Gavin Wraith]], _Generic Galois theory of local rings_
 {#Wraith}

Section 21 of

* [[Ingo Blechschmidt]], _Using the internal language of toposes in algebraic geometry_

includes discussion of what some similar topoi classify, including the étale topology over $Sh(S)$ partially via the internal language.

See also [this MO discussion](http://mathoverflow.net/questions/48690/what-does-an-etale-topos-classify)

See also at _[classifying topos -- For strict local rings](http://ncatlab.org/nlab/show/classifying+topos#StrictLocalRings)_


strict localization

Given a class $W$ of arrows in a category, the [[localization]] $Q_W: C\to C[W^{-1}]$ (if it exists) inverts all the arrows in $W$, i.e. $Q_W(w)$ is an isomorphism in $C[W^{-1}]$ and is universal with respect to all functors with such a property. The universality makes it a 2-categorical colimit called a coinverter in the meta-2-category of large 2-categories. The universality says that every $W$-inverting functor factors through $Q_W$ up to an iso 2-cell; i.e. for every $F:C\to D$ which is inverting all arrows in $W$, there exist a functor $G:C[W^{-1}]\to D$ such that $F \cong G\circ Q_W$; in addition, for all categories $D$, the functor $Fun(Q_W,D)$ of precomposing with $Q_W$ is fully faithful functor from $Fun(C[W^{-1}],D)\to Fun(C,D)$ (it follows that $G$ is unique up to unique iso). It follows that the image of $Fun(Q_W,D)$ is precisely the subcategory of $Fun(C,D)$ consisting of $W$-inverting functors.

We say that $Q_W$ is a __strict localization__ (functor) if for all $F$, the corresponding $G$ can be chosen in a unique way such that $F = G\circ Q_W$. It follows that every two strict localizations are not only equivalent but in fact isomorphic. For example, when $W$ admits the calculus of (say, left) fractions then both the large construction via the path category and the construction via one sided fractions give the isomorphic categories. The additional requirement that for each $D$, $Fun(Q_W, D)$ is fully faithful automatically follows essentially from the uniqueness of $G$ satisfying strict equality $F = G\circ Q_W$ (or, alternatively, for small categories, using lemma 1.2 in Gabriel-Zisman which claims this for the construction of the localized category via the path category, and the fact that every two strict localizations are isomorphic).

[[!redirects strict localizations]]
[[!redirects strict localization functor]]


strict monoidal category

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## Definition

A [[monoidal category]]-[[structure]] is **strict** if its [[associator]] and left/right [[unitors]] are [[identity natural transformations]]. By the [[coherence theorem for monoidal categories]], every monoidal category is monoidally [[equivalent]] to a strict one.

Explicitly, this means that:

\begin{definition} A strict monoidal category is a category $\mathcal{C}$ equipped with an object $1 \in \mathcal{C}$ and a bifunctor $\otimes:\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ such that for every objects $A,B,C$ and morphisms $f,g,h$, we have:

* $(A \otimes B) \otimes C = A \otimes (B \otimes C)$
* $1 \otimes A = A$
* $A \otimes 1 = A$
* $(f \otimes g) \otimes h = f \otimes (g \otimes h)$
* $Id_{1} \otimes f = f$
* $f \otimes Id_{1} = f$

\end{definition}

## Examples

\begin{example}
  The [[skeleton|skeletal]] version of the [[symmetric groupoid]] is the [[free construction|free]] strict [[symmetric monoidal category]] on a single object.
\end{example}

## Related concepts

* [[coherence theorem for monoidal categories]]

* [[permutative category]]

* [[commutative monoidal category]]

## References

* [[Saunders MacLane]], §XI.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Schauenburg01} [[Peter Schauenburg]], *Turning Monoidal Categories into Strict Ones*, New York Journal of Mathematics **7** (2001) 257-265 [[nyjm:j/2001/7-16](http://nyjm.albany.edu/j/2001/7-16.html), [eudml:121925](https://eudml.org/doc/121925)]

[[!redirects strict monoidal categories]]


strict morphism

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## Idea

A _strict morphism_ is a [[morphism]] for which the notion of [[regular image]] and [[regular coimage]] coincide.

Compare with [[strict epimorphism]].

## Definition

### In a category with limits and colimits

Let $C$ be a category with finite [[limits]] and [[colimits]]. Let $f : c \to d$ be a [[morphism]] in $C$.

Recall that the [[regular image]] of $f$ is the [[limit]]

$$
  Im f \simeq lim( d \rightrightarrows d \sqcup_c d )
  \,,
$$

i.e. the [[equalizer]] of $d \rightrightarrows d \sqcup_c d$,

while the [[regular coimage]] is the [[colimit]]

$$
  Coim f \simeq colim( c \times_d c \rightrightarrows c)
  \,.
$$

By the various universal properties, there is a unique morphism

$$
  u : Coim f \to Im f
$$

such that

$$
  \array{
      c &\stackrel{f}{\to}& d
      \\
      \downarrow && \uparrow
      \\
      Coim f &\stackrel{u}{\to}& Im f
  }
  \,.
$$

The morphism $f$ is called a **strict morphism** if $u$ is an [[isomorphism]].

## Examples

Examples of categories in which _every_ morphism is strict include

* [[Set]];
* the category $Mod(R)$ of [[modules]] over a [[ring]] $R$;
* the category $PSh(C) = [C^{op},Set]$ of [[presheaf|presheaves]] on any [[small category]] $C$;
* any [[abelian category]];
* any [[topos]].

[[!redirects strict morphisms]]


strict n-category

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## Definition

A _strict $n$-category_ is a [[strict omega-category]] all whose [[k-morphisms]] for $k \gt n$ are identities.

The [[category]] $n Cat$ of strict $n$-categories and [[n-functors]] between them can also be defined [[induction|inductively]] by

* starting by setting $0 Cat :=$ [[Set]];

* noticing that [[Set]] is canonically a (symmetric, in fact cartesian) [[closed monoidal category]] such that one can consider [[enriched category|categories enriched]] over it;

* noticing that for $V$ any [[complete category|complete]] and [[cocomplete category|cocomplete]] closed monoidal category, also $V Cat$ (the category of $V$-[[enriched categories]]) has these same properties;

* finally setting, recursively,

  $$
    (n+1)Cat := (n Cat) Cat
    \,.
  $$

The category $Str\omega Cat$ of strict $\omega$-categories can then in turn be defined as a suitable [[colimit]] of the categories $n Cat$.

## Properties

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

Write $Str n Cat$ for the 1-[[category]] of [[strict n-categories]].

Write

$$
  Str n Cat_{gaunt} \hookrightarrow  Str n Cat
$$

for the [[full subcategory]] on the _gaunt $n$-categories_, those $n$-categories whose only invertible [[k-morphisms]] are the identities.

=--

This subcategory was considered in ([Rezk](#Rezk)). The term "gaunt" is due to ([Barwick, Schommer-Pries](#BarwickSchommerPries)). See prop. \ref{GauntIs0Truncated} below for a characterization intrinsic to $(\infty,n)$-categories.

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

For $k \leq n $ the $k$-[[globe]] is gaunt, $G_k \in Str n Cat_{gaunt} \hookrightarrow \in Str n Cat$.

Write

$$
  \mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}
$$

for the [[full subcategory]] of the [[globe category]] on the $k$-globes for $k \leq n$.

Being a [[subobject]] of a gaunt $n$-category, also the [[boundary]] of a globe $\partial G_k \hookrightarrow G_k$ is gaunt, i.e. the $(k-1)$-[[skeleton]] of $G_k$.

=--

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

Write

$$
  \sigma_k : Str (k) Cat \to Str (k+1) Cat
$$

for the "categorical suspension" functor which sends a strict $k$-category to the object $\sigma(X) \in Str (k+1) Cat \simeq (Str k Cat)Cat$ which has precisely two objects $a$ and $b$, has $\sigma(C)(a,a) = \{id_a\}$, $\sigma(C)(b,b) = \{id_b\}$, $\sigma(C)(b,a) = \emptyset$ and

$$
  \sigma(C)(a,b) = C
  \,.
$$

=--

We usually suppress the subscript $k$ and write $\sigma^i = \sigma_{k+i} \circ \cdots \circ \sigma_{k+1} \circ \sigma_k$, etc.

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

The $k$-[[globe]] $G_k$ is the $k$-fold suspension of the 0-globe (the point)

$$
  G_k = \sigma^k(G_0)
  \,.
$$

The [[boundary]] $\partial G_k$ of the $k$-globe is the $k$-fold suspension of the
empty category

$$
  \partial G_k = \sigma^k(\emptyset)
  \,.
$$

Accordingly, the boundary inclusion $\partial G_k \hookrightarrow G_k$ is the $k$-fold suspension of the initial morphism $\emptyset \to G_0$

$$
  (\partial G_k \hookrightarrow G_k) = \sigma^k(\emptyset \to G_0)
  \,.
$$

=--

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

The category $Str n Cat_{gaunt}$ is a [[locally presentable category]] and in fact a [[locally finitely presentable category]].

=--

([B-PS, lemma 3.5](#BarwickSchommerPries))

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

For $A,B$ two categories, a [[profunctor]] $A^{op} \times B \to Set$
is equivalently a functor $K \to G_1$ equipped with an identification
$A \simeq K_0$ and $B \simeq K_1$.

=--

This motivates the following definition.

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

A _$k$-profunctor_ / $k$-correspondence of strict $n$-categories is a morphism
$K \to G_k$ in $Str n Cat$. The category of $k$-correspondences is the [[slice category]] $Str n Cat/ G_k$.

=--

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

The categories  $Str n Cat_{gaunt}/G_k$ of $k$-correspondences between gaunt $n$-categories are [[cartesian closed category]].

=--

([B-SP, cor. 5.4](#BarwickSchommerPries))

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

By standard facts, in a [[locally presentable category]] $\mathcal{C}$ with [[finite limits]], a [[slice category|slice]] $\mathcal{C}/X$ is cartesian closed precisely if [[pullback]] along all morphisms $f : Y \to X$ with codomain $X$ preserves [[colimits]] (see at _[[locally cartesian closed category]]_ the section _[Cartesian closure in terms of base change and dependent product](locally%20cartesian%20closed%20category#EquivalentCharacterizations)_).

=--

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

Without the restriction that the codomain of $f$ in the above is a [[globe]], the pullback $f^*$ in $Str n Cat$ will in general fail to preserves colimits. For a simple example of this, consider the [[pushout]] diagram in [[Cat]] $\hookrightarrow Cat_{(\infty,1)}$ given by

$$
  \array{
     \Delta[0] &\stackrel{\delta_1}{\to}& \Delta[1]
     \\
     {}^{\mathllap{\delta_0}}\downarrow && \downarrow^{\mathrlap{\delta_0}}
     \\
     \Delta[1] &\stackrel{\delta_2}{\to}& \Delta[2]
  }
  \,.
$$

Notice that this is indeed also a [[homotopy pushout]]/[[(∞,1)-pushout]] since, by remark \ref{GauntIs0Truncted}, all objects involved are 0-truncated.

Regard this canonically as a pushout diagram in the [[slice category]] $Cat_{/\Delta[2]}$ and consider then the pullback $\delta_1^* : Cat_{/\Delta[1]} \to Cat_{/\Delta[1]}$ along the remaining face $\delta_1 : \Delta[1] \to \Delta[2]$. This yields the diagram

$$
  \array{
     \emptyset &\stackrel{}{\to}& \emptyset
     \\
     {}^{}\downarrow && \downarrow^{}
     \\
     \emptyset &\stackrel{}{\to}& \Delta[1]
  }
  \,,
$$

which evidently no longer is a pushout.

=--

(See also the discussion [here](http://golem.ph.utexas.edu/category/2011/11/the_1category_of_ncategories.html#c040335)).

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

Write

$$
  Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}
$$

for the smallest [[full subcategory]] that

1. contains the [[globe category]] $\mathbb{G}_{\leq n}$, example \ref{Globes};
1. is closed under [[retracts]] in $Str n Cat_{gaunt}$;
1. has all [[fiber products]] over [[globes]] (equivalently: such that all [[slice categories]] over globes have [[products]]).

=--

([B-SP, def. 5.6](#BarwickSchommerPries))

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

The following categories are naturally [[full subcategories]] of $Str n Cat_{gen}$

* the $n$-fold [[simplex category]] $\Delta^{\times n}$;

* the $n$th [[Theta-category]].

=--

This is discussed in more detail in [[(infinity,n)-category]]  in _[Presentation by Theta-spaces and by n-fold Segal spaces](%28infinity,n%29-category#PresentationByThetaSpaces)_.

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

The following [[pushouts]] in $Str n Cat$ we call the **fundamental pushouts**

1. Gluing two $k$-[[globes]] along their [[boundary]] gives the boundary of the $(k+1)$-globle

   $$G_k \coprod_{\partial C_{k-1}} G_k \simeq \partial G_{k+1}$$

1. Gluing two $k$-globes along an $i$-face gives a [[pasting]] composition of the two globles

   $$G_k \coprod_{G_i} G_k $$

1. The [[fiber product]] of globes along non-degenerate morphisms $G_{i+j} \to G_i$ and $G_{i+k} \to G_i$ is built from gluing of globes by

   $$
     G_{i+j} \times_{G_i} G_{i+k}
     \simeq
     (G_{i+j} \coprod_{G_i} G_{i+k}) \coprod_{\sigma^{i+1}(G_{j-1} \times G_{k-1})} (G_{i+k} \coprod_{G_i} G_{i+j})
   $$

1. The [[interval groupoid]] $(a \stackrel{\simeq}{\to} b)$ is obtained by forcing in $\Delta[3]$ the morphisms $(0\to 2)$ and $(1 \to 3)$ to be identities and it is equivalent, as an $n$-category, to the 0-globe

   $ \Delta[3] \coprod_{\{0,2\} \coprod \{1,3\}} (\Delta[0] \coprod \Delta[0])
   \stackrel{\sim}{\to} G_0$

   and the analog is true for all suspensions of this relation

   $$
     \sigma^k(\Delta[3]) \coprod_{\sigma^k\{0,2\} \coprod \sigma^k\{1,3\}} (G_k\coprod G_k)
     \stackrel{\sim}{\to} G_k
     \,.
   $$

We say a functor $i$ on $Str n Cat$ _preserves_ the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism
$i(\sigma^k(\Delta[3])) \coprod_{i(\sigma^k\{0,2\}) \coprod i(\sigma^k\{1,3\})} (i(G_k \coprod G_k)) \to i(G_k)$ is an equivalence.

=--

## Examples

A strict 1-category is just a [[category]].

[[strict 2-category|Strict 2-categories]] are important, because the [[coherence theorem for bicategories]] states that every ("weak") [[2-category]] is [[equivalence of categories|equivalent]] to a strict one, and also because many 2-categories, such as [[Cat]], are naturally strict. However, for $n\ge 3$, these two properties fail, so that strict $n$-categories become less useful (though not useless).  Instead, one needs to use (at least) [[semistrict categories]].

## Related concepts

* [[(∞,n)-category]]

* [[strict category]]

## References

With an eye towards the generalization to [[(∞,n)-categories]], strict $n$-categories are discussed in

* [[Charles Rezk]], _A cartesian presentation of weak n-categories_ ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))
 {#Rezk}

and in section 2 of

* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([pdf](http://arxiv.org/abs/1112.0040), [unusual slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))
 {#BarwickSchommerPries}

[[!redirects strict n-categories]]
[[!redirects strict infinity-category]]
[[!redirects strict infinity-categories]]


strict n-groupoid

A **strict $n$-groupoid** is an [[∞-groupoid]] that is both a [[strict ∞-groupoid]] as well as an [[n-groupoid]].


strict omega-category

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## Idea

A _strict $\omega$-category_ is a [[globular set|globular]] [[∞-category]] in which all operations obey their respective laws strictly.

This was the original notion of [[∞-category]], and the original meaning of the term [[∞-category]]. Even today, most authors who use that term still mean this notion.

This means that

 1. all composition operations are strictly associative;

 2. all composition operations strictly commute with all others (strict [[exchange laws]]);

 3. all identity $k$-morphisms are strict identities under all compositions.

## Definition

An **$\omega$-category** $C$ [[internalization|internal to]]  $Sets$ is

* a [[globular set]]
$$
  C :=
  (\cdots
  C_3 \stackrel{\to}{\to}
  C_2 \stackrel{\to}{\to}
  C_1 \stackrel{\to}{\to}
  C_0
 )
$$

* together with the structure of a [[category]] on
all
$(  C_{k} \stackrel{\to}{\to} C_l )$ for all $k \gt l$;

* such that
$(  C_{k} \stackrel{\to}{\to}
  C_{l} \stackrel{\to}{\to} C_m )$ for all $k \gt l \gt m$;
is a [[strict 2-category]].

Similarly for an $\omega$-category [[internalization|internal to]]  another ambient category $A$.

The category $\omega Cat(A)$ of $\omega$-categories [[internalization|internal to]] $A$ has $\omega$-categories
as its objects and morphism of the underlying globular objects respecting all the above extra structure as morphisms.

## Remarks

* The last condition in the above definition says that all pairs of composition operations satisfy the [[exchange law]].

* $\omega$-Categories can also be understood as the directed limit of the sequence of iterated [[enriched category theory|enrichments]]
$$
  (0 Cat = Set)
  \hookrightarrow
  (1 Cat = Set-Cat)
  \hookrightarrow
  (2 Cat = Cat-Cat)
  \hookrightarrow
  \left(3 Cat = (2Cat)-Cat = (Cat-Cat)-Cat\right)
  \hookrightarrow
  \cdots
  \,.
$$

* The category of strict $\omega$-categories admits a [[closed monoidal category|biclosed monoidal structure]] called the [[Crans-Gray tensor product]].

* The category of strict $\omega$-categories also admits a [[canonical model structure]].

* Terminology on $\omega$-categories varies. We here follow section 2.2 of [[Sjoerd Crans]]: [Pasting presentations for $\omega$-categories](http://home.tiscali.nl/secrans/papers/thpp.html), where it says

  * _[[Ross Street|Street]] allowed $\omega$-categories to have infinite dimensional cells. Steiner has the extra condition that every cell has to be finite dimensional, and called them $\infty$-categories, following [[Ronnie Brown|Brown]] and Higgins. I will use Steiner's approach here since that's the one that reflects the notion of higher dimensional homotopies closest, but I will nonethless call them $\omega$-categories, and I agree with [[Dominic Verity|Verity]]'s suggestion to call the other ones $\omega^+$-categories._

* [[Simpson's conjecture]], a statement about [[semi-strict infinity-category|semi-strictness]], states that every weak $\infty$-category should be equivalent to an $\infty$-category in which strictness conditions 1. and 2. hold, but not 3.

## As simplicial sets

Under the [[oriental|∞-nerve]]

$$
  N : Str \omega Cat \to SSet
$$

strict $\omega$-categories yield simplicial sets that are called [[complicial sets]].

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

The categories of $\omega$-categories and complicial sets are [[equivalence of categories|equivalent]].
=--

This is sometimes called the _Street-Roberts conjecture_. It was completely proven in

* Dominic Verity, _Complicial sets_ ([arXiv](http://arxiv.org/abs/math/0410412))

which also presents the history of the conjecture.

Based on this fact, there are attempts to weaken the condition on a [[simplicial set]] to be a [[complicial set]] so as to obtain a notion of [[simplicial weak ∞-category]].

## Related concepts

* [[strict ∞-groupoid]]

* [[parity complex]], [[oriental]]

* [[complicial set]]

## References

Strict $\omega$-categories have probably been independently invented by several people.

According to [Street 09, p. 10](#Street09) the concept was first brought up in

* {#Roberts79} [[John E. Roberts]], *Mathematical Aspects of Local Cohomology*, in: *Algèbres d’opérateurs et leurs applications en physique mathématique*, Colloques Internationaux du Centre National de la Recherche Scientifique (C.N.R.S) **274**, Paris (1979) 321–332 [ISBN:2-222-02441-2, [pdf](https://dmitripavlov.org/scans/roberts-mathematical-aspects-of-local-cohomology.pdf), [[Roberts-LocalCohomology.pdf:file]]]

in an attempt to define [[non-abelian cohomology]] (of [[local nets of observables]] in [[algebraic quantum field theory]]).

Possibly the earliest published definition is due to

* [[Ronnie Brown]] and P. J. Higgins, _The equivalence of $\infty$-groupoids and crossed complexes_, Cah. Top. Géom. Diff. 22 (1981) no. 4, 371-386 [web](http://www.numdam.org/item?id=CTGDC_1981__22_4_371_0).

which also contains the definitions of [[n-fold category]] and of what was later called [[globular set]]. There these strict, globular higher categories are called "$\infty$-categories" while "$\omega$-groupoid" is used  to mean a cubical set with connections and compositions, each a groupoid, as in

* R. Brown and P.J. Higgins,  _On the algebra of cubes_,  J. Pure Appl.  Algebra 21 (1981) 233-260.

Applications to homotopy theory were given in

* R. Brown and P.J. Higgins,  _Colimit theorems for relative homotopy
groups_, J. Pure Appl. Algebra 22 (1981) 11-41.

* R. Brown, _Non-abelian cohomology and the homotopy classification of maps_,  in Homotopie algébrique et algebre locale, Conf. Marseille-Luminy 1982,  ed. J.-M. Lemaire et J.-C. Thomas,
Astérisques 113-114 (1984), 167-172.

Related monoidal closed structures were developed in:

* R. Brown and P.J. Higgins, _Tensor products and homotopies for
$\omega$-groupoids and crossed complexes_,  J. Pure Appl.
Alg. 47 (1987) 1-33.

Another 1980s reference is

* [[Ross Street]], _The algebra of oriented simplices_, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 ([pdf](http://www.math.mq.edu.au/~street/aos.pdf), doi:10.1016/0022-4049(87)90137-X),

in which strict $\omega$-categories are called "$\omega$-categories."  This paper was also the first to define [[orientals]].

A review of some of the theory in the context of some of the history is given in

* {#Street09} [[Ross Street]], _An Australian conspectus of higher categories_, in: *[[Towards Higher Categories]]*, The IMA Volumes in Mathematics and its Applications **152**, Springer (2010) 237-264 [[pdf](http://www.math.uchicago.edu/~may/IMA/Street.pdf), [[Street-Conspectus.pdf:file]], [doi:10.1007/978-1-4419-1524-5](https://link.springer.com/book/10.1007/978-1-4419-1524-5)]

and also in

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([arXiv](http://arxiv.org/abs/math.CT/0303175))

The theory of $\omega$-categories was further developed by Sjoerd Crans in parts 2 and 3 of his [thesis](http://home.tiscali.nl/secrans/papers/comb.html)

* [[Sjoerd Crans]], _Pasting presentations for $\omega$-categories_ ([link](http://home.tiscali.nl/secrans/papers/thpp.html))

* Sjoerd Crans, _Pasting schemes for the monoidal biclosed structure on $\omega$-Cat_ ([link](http://home.tiscali.nl/secrans/papers/thten.html))

See also the

* [Introduction](http://home.tiscali.nl/secrans/papers/thintro.ps.gz)

to his thesis, in particular section I.3 "$\omega$-categories".

The relationship between strict $\omega$-categories and cubical $\omega$-categories was considered in

* F.A. Al-Agl, R. Brown, R. Steiner _Multiple categories: the equivalence of a globular and a cubical approach_,  Adv. Math.  170  (2002),  no. 1, 71--118

where they prove that strict globular $\omega$-categories are equivalent to $\omega$-fold categories (aka "cubical $\omega$-categories") equipped with [[connections]]. This paper also develops the monoidal closed structures.

* R. Steiner, _Omega-categories and chain complexes_, Homology, Homotopy and Applications __6__(1), 2004, pp.175–200, [pdf](http://www.intlpress.com/HHA/v6/n1/a12/v6n1a12.pdf)

A direct proof that the category of strict $\omega$-categories is [[monadic]] over the category of [[polygraphs]] is given in:

* [[François Métayer]], _Strict $\omega$-categories are monadic over polygraphs_, Theory and Applications of Categories, Vol. 31, No. 27, 2016, pp. 799-806. [[TAC](http://www.tac.mta.ca/tac/volumes/31/27/31-27abs.html)]

[[!redirects strict ∞-category]]
[[!redirects strict ∞-categories]]
[[!redirects strict omega-categories]]


strict omega-groupoid

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{:toc}

## Idea

A _strict $\omega$-groupoid_ is an [[algebraic definition of higher categories|algebraic model]] for certain simple [[homotopy types]]/[[∞-groupoids]] based on [[globular sets]]. It is almost like a [[chain complex]] of [[abelian groups]] (under [[Dold-Kan correspondence]]) except that the [[fundamental group]] is allowed, more generally, to be non-abelian and to act on all the other [[homotopy groups]]. In fact, strict $\omega$-groupoids are [[equivalence of categories|equivalent]] to [[crossed complexes]].

The strict $\omega$-groupoids form an (∞,1)-category [[StrωGrpd]].

## Definition

A **strict $\omega$-groupoid** or **strict $\infty$-groupoid** is a [[strict ∞-category]] in which all [[k-morphisms]] have a strict [[inverse]] for all $k \in \mathbb{N}$

Equivalently, it is a [[globular set]] $X_\bullet$ equipped with a [[unit law|unital]] and [[associativity|associative]] [[composition]] in each degree such that for all pairs of degrees $(k_1 \lt k_2)$ it induces on the 2-graph $X_{k_2} \stackrel{\to}{\to} X_{k_1} \stackrel{\to}{\to} X_0$ the structure of a [[strict 2-groupoid]].

## Properties

### Relation to crossed complexes
 {#RelationToCrossedComplexes}

Following work of [[J. H. C. Whitehead]], in  ([Brown-Higgins](#BrownHiggins)) it is shown that the [[1-category]] of strict $\omega$-groupoids is equivalent to that of [[crossed complexes]]. This equivalence is a generalization of the [[Dold-Kan correspondence]] to which it reduces when restricted to crossed complexes whose fundamental group is abelian and acts trivially. More details in this are at _[[Nonabelian Algebraic Topology]]_.

Strict $\infty$-groupoids form one of the vertices of the [[cosmic cube]] of [[higher category theory]].

### Model structure

There is a [[model structure on strict ∞-groupoids]].

This should [[presentable (∞,1)-category|present]] the full [[sub-(∞,1)-category]] of [[∞Grpd]] of strict $\infty$-groupoids.

## Related concepts

* [[weak ∞-groupoid]], [[weak ∞-category]]

* [[hypercrossed complex]]

## References

A textbook reference is

* [[Ronnie Brown]],  [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_
 {#BrownHigginSivera}

The equivalence of strict $\omega$-groupoids and [[crossed complex]]es is discussed in

* [[Ronnie Brown]], [[Philip Higgins]], _The equivalence of $\infty$-groupoids and crossed complexes_ , Cah. Top. Géom. Diff. 22, 371-386, 1981. ([pdf](http://www.bangor.ac.uk/~mas010/pdffiles/x-comp.pdf))
 {#BrownHiggins}

Notice that this article says "$\infty$-groupoid" for _strict globular $\infty$-groupoid_ and "$\omega$-groupoid" for _strict cubical $\infty$-groupoid_, and also contains  definitions  of $n$-fold categories, and of what are now called  [[globular sets]].

[[!redirects strict omega-groupoids]]
[[!redirects strict ∞-groupoid]]
[[!redirects strict ∞-groupoids]]
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strict partial order

## Disambiguation

There are multiple notions of **strict partial order** used in mathematics:

* an [[asymmetric relation|asymmetric]] [[strict preorder]], which are just [[strict preorders]]

* a [[weakly linear relation|weakly linear]] [[strict preorder]], which are [[strict weak orders]]

A **strict poset**, is a [[set]] equipped with a strict partial order.

## References

For strict partial orders as strict preorders, which are always asymmetric, see:

* Wikipedia, *[Strict partial order](https://en.wikipedia.org/wiki/Strict_partial_order)*

For strict partial orders as strict weak orders, see section 4.1 of:

* [[Auke Booij]], *Analysis in univalent type theory* (2020) $[$[etheses:10411]( 	http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$

category: disambiguation

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strict preorder

\tableofcontents

## Definitions

A **strict preorder** or **strict quasiorder** on a set $S$ is a (binary) [[relation]] $\lt$ on $S$ that is both [[irreflexive relation|irreflexive]] and [[transitive relation|transitive]].  That is:

* $x \nless x$ always;
* If $x \lt y \lt z$, then $x \lt z$.

A **strictly preordered set**, or **strict proset**, is a [[set]] equipped with a strict preorder.

## Properties

\begin{theorem}
Strict preorders orders are asymmetric.
\end{theorem}

\begin{proof}
Transitivity of $\lt$ says that for all $x \in S$ and $y \in S$, $x \lt y$ and $y \lt x$ implies $x \lt x$. However, irreflexivity says that for all $x$, $x \leq x$ is always false. This implies that for all $x$ and $y$, $x \lt y$ and $y \lt x$ is always false, which is precisely the condition of asymmetry.
\end{proof}

As a result, sometimes the term *[[strict partial order]]* is used for strict preorders, since they are always asymmetric. However, the term "strict partial order" is also used for other order relations.

## In constructive mathematics

Unlike with other notions of [[order]], a set equipped with a strict preorder cannot be [[constructive mathematics|constructively]] understood as a kind of [[enriched category]] (at least, not as far as I know ...).  Using [[excluded middle]], however, a strict preorder is the same as a [[partial order]]; interpret $x \leq y$ literally to mean that $x \lt y$ or $x = y$, while $x \lt y$ conversely means that $x \leq y$ but $x \ne y$.

Instead, the relation $\lt$ should be defined as an [[irreflexive comparison]] when generalising mathematics to other categories and to constructive mathematics.

If a strict preorder satisfies comparison (if $x \lt z$, then $x \lt y$ or $y \lt z$), then it is a [[strict weak order]], and additionally, if it is a [[connected relation]], it is a [[strict total order]].

There are also certainly examples of strictly preordered sets that are also partially ordered, where $\lt$ and $\leq$ (while related and so denoted with similar symbols) don\'t correspond as above. For example, if $A$ is any [[inhabited set]] and $B$ is any [[linearly ordered set]], then the [[function set]] $B^A$ is partially ordered with $f \leq g$ meaning that $f(x) \leq g(x)$ always and strictly preordered with $f \lt g$ meaning that $f(x) \lt g(x)$ always. Except in degenerate cases, it\'s quite possible to have $f \ne g$, $f \nless g$, and $f \leq g$ simultaneously.

## Related concepts

* [[preorder]]
* [[strict total order]]

## References

* Wikipedia, *[Strict preorder](https://en.wikipedia.org/wiki/Strict_preorder)*

* Wikipedia, *[Strict partial order](https://en.wikipedia.org/wiki/Strict_partial_order)*

[[!redirects strict preorder]]
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strict proposition

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## Idea

In [[dependent type theory]], there are two notions of [[equality]],

1. *strict* or *[[judgmental equality]]*,

1. *weak* or *[[typal equality]]*.

Accordingly there are two corresponding notions of [[propositions]] or [[subsingletons]]:

1. The usual notion of proposition, [[h-propositions]], are the types $A$ where every two elements $a:A$ and $b:A$ are typally equal to each other $p(a, b):a =_A b$.

1. *Strict propositions*, which are the types $A$ where every two elements $a:A$ and $b:A$ are [[judgemental equality|judgmentally equal]] to each other $a \equiv b:A$.

Strict propositions are used to formalize [[logic over type theory]] in [[dependent type theory]] without having to postulate a separate proposition judgment, since strict propositions are judgmentally [[proof relevance|proof irrelevant]], and thus behave as propositions in traditional [[propositional logic]].

## Examples

* The [[empty type]] is a strict proposition.

* The strict [[unit type]] with judgmental [[computation rules]] is a strict proposition.

* The [[squash type]] of any type is a strict proposition.

## Related concepts

* [[judgmental equality]]

* [[mere proposition]]

* [[squash type]]

* [[type of strict propositions]]

* [[strict singleton]]

## References

* [[Gaëtan Gilbert]], [[Jesper Cockx]], [[Matthieu Sozeau]], [[Nicolas Tabareau]], *Definitional Proof-Irrelevance without K*. Proceedings of the ACM on Programming Languages, Volume 3, Issue POPL, Article 3, pp 1–28, January 2019. ([doi:10.1145/3290316](https://doi.org/10.1145/3290316))

[[!redirects strict proposition]]
[[!redirects strict propositions]]


strict singleton

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## Idea

In [[dependent type theory]], a **strict singleton** or a **strict contractible type** is a pointed [[strict proposition]], or equivalently a strict proposition which is also a [[singleton]] or [[contractible type]]. In dependent type theories where [[strict propositions]] instead of [[h-propositions]] are used to represent the [[predicate logic]] of the type theory, strict singletons represent the notion of *[[truth]]* in the theory.

## Examples

* The strict [[unit type]] with judgmental [[computation rules]] is a strict singleton.

* Given a type $A$, the strict [[function type]] $\emptyset \to A$ from the [[empty type]] to $A$, with judgmental [[computation rules]], is a strict singleton.

* The [[squash type]] of any [[pointed type]] is a strict singleton.

## Related concepts

* [[judgmental equality]]

* [[strict proposition]]

[[!redirects strict singleton]]
[[!redirects strict singletons]]

[[!redirects strict contractible type]]
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strict terminal object

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## Idea

The [[trivial ring]], among all unital rings, has two characteristic properties:

1. there is a *unique* [[function]] *into* the trivial ring, from any other unital ring;

1. there is *no* [[function]] *from* the trivial ring, except to itself.

The first property generalizes to arbitrary [[categories]] as the property of a *[[terminal object]]*.

The corresponding generalization including also the second property is that of a *strict terminal object*:

## Definition

A [[terminal object]] $\mathbf{1}$ is called a **strict terminal object** if every [[morphism]] [[domain|from]] $\mathbf{1}$ is an [[isomorphism]]:

$$
  \mathbf{1}
    \overset{\simeq}{\longrightarrow}
  \X
  \,.
$$

In other words, a strict terminal object is a [[maximal object|maximal terminal object]].

## Examples
 {#Examples}

* Trivial unital ring

* Trivial [[Boolean algebra]]

* Trivial [[absorption monoid]]

* In general, for any algebraic theory with two constants $0$ and $1$ and a binary operation for which $0$ is a (left) absorbing element and $1$ is a (left) unit, the trivial model is strictly terminal.

## Related concepts

* the [[formal dual]] concept: [[strict initial object]]
* [[maximal object]]

##References

* [[Zhen Lin Low]], _What do algebraic theories with strictly terminal trivial models look like?_, [MO question:159234](https://mathoverflow.net/q/159234/)

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[[!redirects strict terminal object]]
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strict weak order

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## Idea

A [[strict total order]] which is not a [[connected relation]].

## Definition

A **strict weak order** is a [[strict preorder]] which is also a [[cotransitive relation]].

These are sometimes called *[[strict partial orders]]* in the literature, but the term "strict partial order" is also used for other order relations in mathematics.

Strict weak orders are used in models of the real numbers which have [[infinitesimals]], such as in the smooth real numbers in [[synthetic differential geometry]], and thus where not every real number not greater than or less than zero is equal to zero.

## Properties

\begin{theorem}
Strict weak orders are [[asymmetric relations]].
\end{theorem}

\begin{proof}
Transitivity of $\lt$ says that for all $x \in S$ and $y \in S$, $x \lt y$ and $y \lt x$ implies $x \lt x$. However, irreflexivity says that for all $x$, $x \leq x$ is always false. This implies that for all $x$ and $y$, $x \lt y$ and $y \lt x$ is always false, which is precisely the condition of asymmetry.
\end{proof}

\begin{theorem}
Connected strict weak orders are [[pseudo-orders]].
\end{theorem}

\begin{proof}
The proof is the same as above, except both relevant orders are [[connected relations]].
\end{proof}

### Preorder of a strict weak order

An important part of a strict weak order is that it is a preorder.

\begin{theorem}
The negation of a strict weak order is transitive.
\end{theorem}

\begin{proof}
The [[contrapositive]] of cotransitivity says that

> for all $x$, $y$, and $z$, if $x \lt y$ or $y \lt z$ is false, then $x \lt z$ is false.

By one of [[de Morgan's laws]], that $x \lt y$ or $y \lt z$ is false is logically to equivalent to that $\neg(x \lt y)$ and $\neg(y \lt z)$, and substituting this into the original statement results in

> if $\neg(x \lt y)$ and $\neg(y \lt z)$, then $\neg(x \lt z)$

which is precisely transitivity for the negation of the strict weak order.
\end{proof}

\begin{theorem}
The negation of a strict weak order is reflexive.
\end{theorem}

\begin{proof}
Irreflexivity states that $\neg (x \lt x)$ is true, which is precisely reflexivity for the negation of the strict weak order.
\end{proof}

\begin{theorem}
The negation of a strict weak order is a preorder.
\end{theorem}

\begin{corollary}
The incomparability relation of a strict weak order, $\neg (x \lt y) \wedge \neg (y \lt x)$, is an [[equivalence relation]]
\end{corollary}

\begin{proof}
For every preorder, $(x \leq y) \wedge (y \leq x)$ is an [[equivalence relation]]. Since $\neg (x \lt y)$ is a [[preorder]], $\neg (x \lt y) \wedge \neg (y \lt x)$ is an equivalence relation.
\end{proof}

## See also

* [[total preorder]]

* [[weak order]]

* [[pseudo-order]]

* [[strictly weakly ordered ring]]

* [[ordered local ring]]

* [[ordered Kock field]]

* [[apartness relation]]

* [[irreflexive comparison]]

## References

* Wikipedia, [Weak order](https://en.wikipedia.org/wiki/Weak_order)

Strict weak orders are called [[strict partial orders]] in section 4.1 of:

* [[Auke Booij]], *Analysis in univalent type theory* (2020) $[$[etheses:10411]( 	http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$

Strict weak orders are used in defining the smooth real numbers in:

* [[David Jaz Myers]], *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* ([arXiv:2205.15887](https://arxiv.org/abs/2205.15887))

[[!redirects strict weak order]]
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strict ω-category > history

< [[strict ∞-category]]
[[!redirects strict ∞-category/history]]
[[!redirects strict ∞-category -- history]]


strict ω-groupoid > history

< [[strict ∞-groupoid]]
[[!redirects strict ∞-groupoid/history]]
[[!redirects strict ∞-groupoid -- history]]


strictly full subcategory

A [[subcategory]] is **strictly full** if it is [[full subcategory|full]] and closed under isomorphism (i.e. [[replete subcategory|replete]]).

Note that, as a condition on subcategories, being strictly full is a property that does not respect the [[principle of equivalence]] of [[category theory]]; however, seeing the full subcategory as a condition on objects, that condition is in accord with the [[principle of equivalence]] precisely when the full subcategory is strictly full!

* The Stacks Project, Definition 4.2.10 [tag/001D](https://stacks.math.columbia.edu/tag/001D)

[[!redirects strictly full subcategories]]


strictly weakly ordered ring

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## Idea

A notion of [[ordered ring]] for [[strict weak orders]].

## Definition

### With an order relation

A strictly weakly ordered ring is an [[ring]] $R$ with a [[strict weak order]] $\lt$ such that

* $0 \lt 1$

* for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

* for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

We define the predicate $\mathrm{isPositive}$ as

$$\mathrm{isPositive}(a) \coloneqq 0 \lt a$$

### With a positivity predicate

Let $R$ be a [[commutative ring]]. $R$ is an **pseudo-ordered ring** if there is a [[predicate]] $\mathrm{isPositive}$ stating an element $a$ is positive, such that

* 0 is not positive
* given element $a \in R$, if $a$ is positive, then $-a$ is not positive
* given elements $a \in R$ and $b \in R$; if $a$ is positive, then either $b$ is positive or $a - b$ is positive
* 1 is positive
* given elements $a \in R$ and $b \in R$, if $a$ is positive and $b$ is positive, $a + b$ is positive; alternatively, $a + b$ being positive implies that $a$ is positive or $b$ is positive.
* given elements $a \in R$ and $b \in R$, if $a$ is positive and $b$ is positive, $a \cdot b$ is positive

We define the order relation $\lt$ as

$$a \lt b \coloneqq \mathrm{isPositive}(b - a)$$

## Properties

Every strictly weakly ordered ring is a [[preordered ring]] given by the [[negation]] of the strict weak order. In the presence of [[excluded middle]], every strictly weakly ordered ring is a [[total order|totally preordered]] ring.

## Examples

* Every [[pseudo-ordered ring]] is a strictly weakly ordered ring.

* Every [[ordered local ring]] is a strictly weakly ordered ring where every element greater than zero or less than zero is invertible.

* In particular, the set of [[Cauchy sequences]] of [[rational numbers]] is a strictly weakly ordered ring.

* Every [[ordered Kock field]] is a example of a strictly weakly ordered ring which in the presence of [[excluded middle]] is a [[linearly ordered ring]].

## See also

* [[ordered local ring]]

* [[ordered Kock field]]

[[!redirects strictly weakly ordered ring]]
[[!redirects strictly weakly ordered rings]]


string

> This entry is about the notion in [[particle physics]]/[[quantum gravity]]. For the notion in [[computer science]] see at *[[string (computer science)]]*.

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## Idea

A _string_ or *1-brane* is a [[brane]] of dimension one higher than an ordinary [[particle]]:

where a 1-[[dimension|dimensional]] [[sigma-model]] may be thought of a describing the dynamics of [[particle]]s propagating of a [[target space]] $X$, a 2-dimensional [[sigma-model]] is said to described the dynamics of a string on some target space.

Much of traditional [[quantum field theory]] on $X$ may be understood in terms of [[second quantization]] of 1-dimensional [[sigma-models]] with [[target space]] $X$. What is called _[[string theory]]_ is the corresponding study of what happens to this situation as the 1-dimensional $\sigma$-model is replaced by a 2-dimensional one (for more on this see at *[[worldline formalism]]*).

A key motivation/application for strings is their identification with [[flux tubes]] in [[confinement|confined]] [[Yang-Mills theory]]/[[QCD]] [[Polyakov (1998)](#Polyakov98), [Polyakov (1999)](#Polyakov98)], for more on this see at *[[AdS-CFT correspondence]]* and *[[AdS-QCD correspondence]]*.

There is

* the [[bosonic string]] of [[bosonic string theory]]

* the [[superstring]] of [[superstring theory]]

(also called the _[[F1-brane]]_)

* there is also the [[D1-brane]].

## Related entries

* [[relativistic particle]]

* [[relativistic string]]

* [[relativistic membrane]]

* [[vortex string]]

* [[string theory]]

  * [[string scattering amplitudes]]

  * [[string field theory]]

  * [[string theory FAQ]]

$\,$

[[!include table of branes]]

## References

### General

See references at *[[string theory]]*, *[[bosonic string]]* and *[[superstring]]*.

[[!include Polyakov gauge-string duality -- references]]

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[[!redirects fundamental string]]
[[!redirects fundamental strings]]

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