category with an atlas

[[!redirects categories with the same collection of objects]]
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### Context
#### Category theory
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[[!include category theory - contents]]
=--
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# Contents
* table of contents
{: toc}

## Idea

### Background comments

Despite the impression one may get from standard texts, the notion of a *[[category]]* (in [[category theory]]) exists in (at least) two distinct versions, which, while closely related, are crucially different in a way that matters when paying attention to fine-print. Namely, categories are traditionally introduced as being [[mathematical structures]] consisting of a [[class]] of [[objects]], etc. ("[[strict categories]]", see below)  but in further discussion these are eventually often regarded as being themselves the objects of the [[2-category]] [[Cat|$CAT$]] of [[categories]], [[functors]] and [[natural transformations]] between these. While this change of perspective may superficially seem to only increase the level of sophistication of the discussion, it actually loses information: By default of [[2-category theory]], the objects of a [[2-category]] like [[CAT]] are identifiable only up to [[equivalence in a 2-category|2-categorical equivalence]] -- which here means: up to [[equivalence of categories]] -- but under this relation the notion of "class of objects" of a category is not fixed (not fixed up to [[isomorphism]], that is) and hence makes no invariant sense: Regarding a category as an object of [[CAT]] (and nothing more), means to *forget* its precise class of objects.

For example, when assuming the relevant [[axiom of choice]], then the [[codiscrete category]] on any [[set]] $S$ (whose class of objects is $S$) is [[equivalence of categories|equivalent]] to the [[terminal category]] (whose class of objects is the [[singleton set]]). From the point of view of the "[[principle of equivalence]]" appropriate within [[CAT]], all these categories are effectively indistinguishable, and yet one finds a strict representative in this equivalence class with any imaginable class of objects.

In order to bring out this distinction, some authors say "[[strict category]]" for the notion of "category" where a class of objects is fixed. While this specific terminology is not common, the notion itself is tacitly ubiquituous: For example, when people say that the [[simplicial nerve]]-construction fully embeds [[small categories]] into [[simplicial sets]] (namely as those which satisfy the [[Segal conditions]]) then they are implicitly referring to such strict categories: Their fixed set of objects constitutes the degree-0 component of their nerve simplicial set.

The point of the following definition of "category with atlas" is to bring out the notion of "strict category" in a way that is itself intrinsic to [[2-category theory]] -- or rather: internal to [[2-topos theory]].

### Idea of the definition

By a *category with an atlas* we shall mean a [[category]] $\mathcal{C}$ which is equipped with an [[essentially surjective functor]]

\[
  \label{AnAtlasOnACategory}
  C_0
    \twoheadrightarrow
  \mathcal{C}
\]

from a [[set]] (or [[class]]) $C_0$ regarded as a ([[large category|large]]) [[discrete category]]. This is also known as a *[[flagged category]]* (in which case one would allow the [[domain]] $C_0$ to be itself a [[groupoid]], but one could agree to do the same here).

This is an "invariant"  or "intrinsic" way of speaking (namely [[internal language|internal]] to the [[2-topos]] [[CAT]]) about categories which are equipped with a notion of a fixed class of objects. Another terminology for this is "[[strict categories]]", but saying "category with atlas" naturally connects to the well-established notion of *[[atlas]]* in the context of [[stacks]], maybe most commonly used in the literature on [[topological stacks]]:

An [[atlas]] for a [[topological stack]] $\mathscr{X}$ is classically taken to be a [[topological space]] $X_0$ and an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] $X_0 \twoheadrightarrow \mathscr{X}$. Given this, its [[homotopy fiber product|homotopy]]-[[Cech nerve]] $X_0 \times_{\mathscr{X}} X_0 \rightrightarrows X_0$ is known as the corresponding *[[topological groupoid]]*, whose space of [[objects]] is nothing but $X_0$.

By the [[Giraud-Rezk-Lurie theorem]], this state of affairs holds in any [[(2,1)-topos]] of [[stacks]]: The ("strict"!) [[groupoid object in an (infinity,1)-category|groupoid objects]] in the underlying [[1-topos]] are equivalently the *[[stacks]] equipped with an [[atlas]]*, namely equivalently objects in the [[(2,1)-topos]] equipped with an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] out of a [[0-truncated]] object.

If here the the underlying [[site]] is the [[terminal category]], then "stacks" are [[groupoids]] regarded up to [[equivalence of groupoids]] (i.e. as objects of the [[(2,1)-category]] [[Grpd]], and "stacks with atlas" are groupoids equipped with a choice of a fixed set of objects, hence "strict groupoids". This is just the notion of "category with an atlas" restricted to categories that happen to be groupoids.

In short, we have the following dictionary between "stack theory" and [[category theory]] (which are unified in [[2-topos theory]]):

| stack theory  | stack theory over the point | category theory |
|---------------|----------------------------|-----------------|
|  stack        |          groupoid          |    groupoid     |
| stack with atlas |  groupoid with atlas    | strict groupoid |

But it is common to understand "[[stacks]]" in the generality of category-valued stacks: [[2-sheaves]]. Now a stack/2-sheaf over the point is just a bare category, and this way the notion of "category with an atlas" blends seamlessly into established terminology in stack theory.

### Consequences of the definition
 {#ConsequencesOfTheDefinition}

The advantage of thinking of [[strict categories]] more intrinsically as *categories equipped with an atlas* (eq:AnAtlasOnACategory) is that it immediately implies natural definitions for a variety of related notions.

For example, the correct category of categories-with-atlas is evidently the full sub 2-category of the $(2,1)$-[[comma category]] $SET/CAT$ on the essential surjections, whose morphisms from $(C_0 \twoheadrightarrow \mathcal{C})$ to $(D_0 \twoheadrightarrow \mathcal{D})$ are squares in $Ho_{(2,1)}(CAT)$ of the form

\begin{tikzcd}
  C_0
  \ar[d, ->>]
  \ar[rr, "{F_0}"]
  &&
  D_0
  \ar[d, ->>, "{\ }"{swap, name=s}]
  \\
  \mathcal{C}
  \ar[rr, "{\ }"{name=t}, "{F}"{swap}]
  &&
  \mathcal{D}
  \ar[from=s, to=t, Rightarrow, "\sim"{sloped}]
\end{tikzcd}

Similarly, for a given [[class]] $C_0$ of objects, the morphisms in the analogous full sub-$(2,1)$-category of the [[co-slice category|co-slice]] of [[CAT]] under $C_0$ are of the form:

\begin{tikzcd}
  &
  C_0
  \ar[dl, ->>]
  \ar[dr, ->>, "{\ }"{swap, name=s}]
  \\
  \mathcal{C}
  \ar[rr, "{\ }"{name=t}, "F"{swap}]
  &&
  \mathcal{C}'
  \ar[from=s, to=t, Rightarrow, "\sim"{sloped}]
\end{tikzcd}

Such morphisms are evidently equivalent (by [[2-morphisms]] in $CAT^{C_0/}$) to
"[[identity-on-objects functors]]".

\linebreak

## See also

* [[atlas]]

* [[identity-on-objects functor]]

* [[Freyd category]]

* [[Freyd multicategory]]

* [[dagger category]]

## References

A generalization of the notion to [[higher category theory]] is considered (under the name "[[flagged categories]]") in

* {#AyalaFrancis18} [[David Ayala]], [[John Francis]], Def. 0.12 of: *Flagged higher categories*, in: *Topology and Quantum Theory in Interaction*,  Contemporary Mathematics **718**, AMS (2018) [[arXiv:1801.08973](https://arxiv.org/abs/1801.08973), [ISBN:978-1-4704-4941-4](https://bookstore.ams.org/conm-718)]

[[!redirects categories with the same type of objects]]
[[!redirects categories with the same class of objects]]
[[!redirects categories with the same set of objects]]

[[!redirects functor between categories with the same collection of objects]]
[[!redirects functor between categories with the same type of objects]]
[[!redirects functor between categories with the same class of objects]]
[[!redirects functor between categories with the same set of objects]]

[[!redirects functors between categories with the same collection of objects]]
[[!redirects functors between categories with the same type of objects]]
[[!redirects functors between categories with the same class of objects]]
[[!redirects functors between categories with the same set of objects]]


category with arities

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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# Categories with arities

* table of contents
{:toc}

## Idea

A *category with arities* is a category equipped with a "small presentation" of its objects.

## Definition

A **category with arities** is a [[locally small category]] $\mathcal{E}$ equipped with a [[small category|small]] [[full subcategory]] $\mathcal{A} \subseteq \mathcal{E}$ which is [[dense subcategory|dense]], in the sense that the [[restricted Yoneda embedding]] $\mathcal{E} \to [\mathcal{A}^{op},Set]$ is [[fully faithful]].

A **functor with arities** or **arity-respecting functor** $F : (\mathcal{E},\mathcal{A}) \to (\mathcal{F},\mathcal{B})$ is a functor $F:\mathcal{E}\to\mathcal{F}$ such that the composite $\mathcal{E} \xrightarrow{F} \mathcal{F} \to [\mathcal{B}^{op},Set]$ preserves the density colimits for $\mathcal{A}$ in $\mathcal{E}$, i.e. such that for any $X\in\mathcal{E}$ and $B\in\mathcal{B}$ the canonical map $\colim^{A\in \mathcal{A} \atop A \to X} \mathcal{F}(B,F A) \to \mathcal{E}(B,F X)$ is an isomorphism of sets.

A **natural transformation with arities** is just an ordinary natural transformation between functors with arities.

This defines the 2-category of categories with arities.  Note that since a natural transformation with arities is determined by its action on the generating objects $\mathcal{A}$, there is only a small set of natural transformations between any two functors with arities.  Thus, even though the 2-category of categories with arities is "very large", its hom-categories are locally small.

## Related pages

* A monad in the 2-category of categories with arities is a [[monad with arities]].  In [BMW](#BMW) it is claimed that the nerve theorem for monads with arities constructs an [[Eilenberg-Moore object]] in this 2-category.

## References

* {#BMW} [[Clemens Berger]], [[Paul-André Melliès]], [[Mark Weber]], _Monads with Arities and their Associated Theories_ (2011) ([arXiv:1101.3064](http://arxiv.org/abs/1101.3064))

[[!redirects categories with arities]]
[[!redirects functor with arities]]
[[!redirects functors with arities]]
[[!redirects arity-respecting functor]]
[[!redirects arity-respecting functors]]


category with class structure

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

# Contents #
* table of contents
{:toc}

## Idea

Just as the notion of an [[elementary topos]] is an [[axiom|axiomatization]] of basic [[category theory|category-theoretic]] properties of the [[Set|category of sets]], the notion of a "*category with class structure*" or "*category of classes*" is an axiomatization of the basic category theoretic properties of the category of [[classes]].

In contrast to the situation for elementary toposes, however, there is no unique such axiomatization with a privileged status in the literature; the field of [[algebraic set theory]] includes many variations.  Here we describe the first such axiomatization due to [Joyal-Moerdijk](#JM).

## Definition

We work in a category $C$ that is assumed to be a [[Heyting pretopos]]
with a [[natural numbers object]]. Following [Joyal-Moerdijk](#JM), we have the following definition.

\begin{definition}
A __class of open maps__ (with collection) in $C$
is a [[class]] $S$ of morphisms in $C$ that satisfies the following properties:

* $S$ contains all isomorphisms and is closed under compositions;
* $S$ is closed under [[base changes]];
* if $g\in S$ is a [[base change]] of $f$ along an [[epimorphism]], then $f\in S$;
* the canonical maps $0\to 1$ and $1\sqcup 1\to 1$ belong to $S$;
* $S$ is closed under binary [[coproducts]] in the [[category of morphisms]] of $C$;
* if $g=f\circ p$, where $p$ is an [[epimorphism]] and $g\in S$, then also $f\in S$;
* (collection axiom) for any [[epimorphism]] $p\colon Y\to X$ and $f\colon X\to A$ such that $f\in S$ there is an [[epimorphism]] $h\colon B\to A$, a morphism $g\colon Z\to B$ such that $g\in S$, and a morphism $w:Z\to Y$ such that $h g=f p w$ and the induced map $Z\to B\times_A X$ is an [[epimorphism]].

\end{definition}

\begin{definition}
A __class of small maps__ in $C$ is a class of open maps $S$ in $C$
such that every map in $S$ is [[exponential object|exponentiable]]
and there is a universal map $\pi\colon E\to U$ in $S$
with the following property: for any $f\in S$ we can [[base change]] $f$ along some [[epimorphism]] $p$ such that the resulting morphism $f'$ is a
[[base change]] of $\pi$ along some morphism in $C$.
Elements of $S$ are known as __small maps__.
An object $X$ of $C$ is __small__ if the map $X\to 1$ is small.
\end{definition}

Intuitively, small maps are maps $X\to Y$ for which preimages
of any element of $Y$ are sets as opposed to proper classes.

In any [[Heyting pretopos]] the class of [[exponential object|exponentiable maps]]
satisfies all the axioms of a class of open maps
with a possible exception of the collection axiom.

The collection axiom can be reformulated by saying
that the _small powerset_ functor preserves [[epimorphisms]].
One way to define the small powerset of $X$ is as the free $S$-complete
[[suplattice]] generated by $X$.

\begin{definition}
A category with a class of small maps admits __powerclasses__
if for any object $C$ there is an object $P C$
with a small relation $\in\subset C\times P C$
such that any object $X$ and any small relation $R\subset C\times X$
there is a unique morphism $r\colon X\to P C$
such that $R\to C\times X$ is the [[base change]] of $\in\to C\times P C$
along the map $id_C \times r$.
Furthermore, the internal subset relation on $P C$ must be small.
\end{definition}

\begin{definition}
A __universal class__ is an object $U$
such that any object $C$ admits a [[monomorphism]] $U\to C$.
\end{definition}

\begin{definition}
A __category of classes__ or a __class category__
is [[Heyting category]] with a class of small maps
that admits small powerclasses and a universal class.
\end{definition}

The full subcategory of small objects and small maps
of any class category is an [[elementary topos]].

## Related notions

* [[class]]

* [[proper class]]

* [[large category]]

* [[category of classes]]

* [[algebraic set theory]]

## References

* {#JM} [[André Joyal]], [[Ieke Moerdijk]], _Algebraic set theory_.  Cambridge University Press, 1995.  ISBN 0-521-55830-1.

* [[Steve Awodey]], _An outline of algebraic set theory_.

[[!redirects category with class structure]]
[[!redirects categories with class structure]]

[[!redirects regular category with class structure]]
[[!redirects regular categories with class structure]]

[[!redirects coherent category with class structure]]
[[!redirects coherent categories with class structure]]

[[!redirects Heyting category with class structure]]
[[!redirects Heyting categories with class structure]]


category with duals

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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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# Categories with duals
* table of contents
{: toc}

## Idea

A _category with duals_ is a [[category]] where [[objects]] and/or [[morphisms]] have [[dual object|duals]].  This exists in several flavours; this list is mostly taken from a recent [`categories` list post](http://article.gmane.org/gmane.science.mathematics.categories/5807/) from [[Peter Selinger]].

## Categories with duals for objects

*  A __[[left autonomous category]]__ is a [[monoidal category]] in which every [[object]] is [[dualisable object|dualisable]] on the left.

*  A __[[right autonomous category]]__ is a [[monoidal category]] in which every [[object]] is [[dualisable object|dualisable]] on the right.

*  An __[[autonomous category]]__, or __[[rigid category]]__ is a monoidal category that is both left and right autonomous. Note that any braided monoidal category is autonomous on both sides if it is autonomous on either side.

*  A __[[pivotal category]]__ is an autonomous category equipped with a [[monoidal natural transformation|monoidal]] [[natural isomorphism]] from the [[identity functor]] to the [[double dual]] functor. A one-sided autonomous category with such an isomorphism is automatically two-sided autonomous. Although each braided autonomous category has an isomorphism from $A$ to $A^{**}$, such a category is not necessarily pivotal because this isomorphism is not in general monoidal. On the other hand, every [[balanced monoidal category|balanced]] autonomous category is pivotal.

*  A __[[spherical category]]__ is a pivotal category where the left and right [[trace]] operations coincide on all objects.

*  A __[[tortile category]]__, or __[[ribbon category]]__, is a [[balanced monoidal category|balanced]] autonomous (therefore pivotal) category in which the [[twist]] on $A^*$ is the dual of the twist on $A$.

*  A __[[compact closed category]]__ is a [[symmetric monoidal category|symmetric]] tortile category, or equivalently, a symmetric autonomous category.

*  The __$*$-[[star-autonomous category|autonomous categories]]__ do not really belong on this list; being $*$-autonomous is logically independent of being autonomous, and while $*$-autonomous categories have duals, these are not in general duals in the sense of a [[dualisable object]].  However, any compact closed category is $*$-autonomous.

* Likewise, [[closed category|closed categories]] or [[closed monoidal category|closed monoidal categories]] do not really belong on this list, but there is a sense of dual there which should be carefully distinguished from the primary sense here, which is generally stronger. See [[dual object in a closed category]].

## Categories with duals for morphisms

One might write something about these too, or put them on a separate page.  In the meantime, see the table of contents to the right.

There at least two commonspread kinds of categories with duals for morphisms:

* [[dagger categories]] where each morphism $f:X \to Y$ has a $\dagger$-dual $f^\dagger : Y \to X$, without any extra property.
* [[groupoids]], where each morphism $f:X \to Y$ has an inverse $f^{-1} :Y \to X$ defined by the properties $f f^{-1} = 1_Y$, $f^{-1}f = 1_X$.

Moreover, every category enriched in one of the kind of categories listed above will have a notion of 'dual' for its morphisms.

## References

*  Peter Selinger, [`categories` post of 2010-05-15](http://permalink.gmane.org/gmane.science.mathematics.categories/5805);
*  Peter Selinger, _[A survey of graphical languages for monoidal categories](http://arxiv.org/abs/0908.3347)_

[[!redirects category with duals]]
[[!redirects categories with duals]]
[[!redirects monoidal category with duals]]
[[!redirects monoidal categories with duals]]

[[!redirects category with duals for objects]]
[[!redirects categories with duals for objects]]
[[!redirects monoidal category with duals for objects]]
[[!redirects monoidal categories with duals for objects]]


category with path objects

> This is about the notion of *path category* in [[model category theory]] and [[homotopy theory]]. For the notion of path category in ordinary category theory, see [[path category]].

----

+-- {: .rightHandSide}
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

\tableofcontents

## Definition

A **category with path objects** or a **path category** is a [[category]] $\mathcal{C}$ equipped with two [[classes]] of [[morphisms]] called *[[weak equivalences]]* and *[[fibrations]]*, such that:

1. Fibrations are closed under composition

1. The pullback of a fibration along any other map exists and is again a fibration.

1. The pullback of an [[acyclic fibration]] along any other map is again an acyclic fibration.

1. Weak equivalences satisfy the [[two-out-of-six property]]: if $f:A \to B$, $g:B \to C$, $h:C \to D$ are
three composable maps and both $g \circ f$ and $h \circ g$ are weak equivalences, then so are $f$, $g$, $h$ and $h \circ g \circ f$.

1. Isomorphisms are acyclic fibrations and every acyclic fibration has a section.

1. For any object $B$ there is a [[path space object]] $P B$ (not necessarily functorial in $B$).

1. $\mathcal{C}$ has a [[terminal object]] $1$ and every map $X \to 1$ to the terminal object is a [[fibration]].

## Examples

* The [[syntactic category]] associated to a [[dependent type theory]] with [[identity types]] is a category with path objects.

* Given a [[Quillen model category]] $\mathcal{C}$ in which every object is cofibrant, the [[full subcategory]] of fibrant objects in $\mathcal{C}$ is a category with path objects.

* The category of [[topological spaces]] is a category with path objects where the [[homotopy equivalences]] are the weak equivalences and the [[Hurewicz fibrations]] are the the fibrations.

* A [[finitely complete category]] is a category with path objects in which every morphism is a fibration and only the [[isomorphisms]] are [[weak equivalences]].

## See also

* [[category of fibrant objects]]

* [[homotopy exact completion]]

## References

* {#vdBergMoerdijk18} [[Benno van den Berg]], [[Ieke Moerdijk]], *Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories*, Journal of Pure and Applied Algebra **222** 10 (2018) 3137-3181 [[doi:10.1016/j.jpaa.2017.11.017](https://doi.org/10.1016/j.jpaa.2017.11.017), [arXiv:1603.02456](https://arxiv.org/abs/1603.02456)]

* {#vdBergBesten21} [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

[[!redirects category with path objects]]
[[!redirects categories with path objects]]


category with star-morphisms

+-- {: .num_remark}
###### Remark
By a _pre-category_ I mean a [[category]] without [[identity morphism|identities]] (a [[semicategory]]).
=--

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

A **pre-category with star-morphisms** consists of
1. a pre-category $C$ (**the base pre-category**);
2. a set $M$ (**star-morphisms**);
3. a function $\operatorname{arity}$ defined on $M$ (how many objects are connected by this multimorphism);
4. a function $\operatorname{Obj}_m : \operatorname{arity} m \rightarrow \operatorname{Obj}
    \left( C \right)$ defined for every $m \in M$;
5. a function (star composition) $\left( m ; f \right)
    \mapsto \operatorname{StarComp} \left( m ; f \right)$ defined for $m \in M$ and
    $f$ being an $(\operatorname{arity} m)$-indexed family of morphisms of $C$ such
    that $\forall i \in \operatorname{arity} m : \operatorname{Src} f_i = \operatorname{Obj}_m i$
    ($\operatorname{Src} f_i$ is the source object of the morphism $f_i$) such that
    $\operatorname{arity} \operatorname{StarComp} \left( m ; f \right) = \operatorname{arity} m$

such that:

1. $\operatorname{StarComp} \left( m ; f \right) \in M$;
2. (*associativiy law*)
    \[ \operatorname{StarComp} \left( \operatorname{StarComp} \left( m ; f \right) ; g \right)
       = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m : g_i \circ
       f_i \right) . \]

(Here $\lambda$ indicates [[function abstraction]].)
=--

The meaning of the set $M$ is an extension of $C$ having as morphisms things
with arbitrary (possibly infinite) indexed set
$\operatorname{Obj}_m$ of objects, not just two objects as
morphims of $C$ have only source and destination.

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

A **category with star-morphisms** is a pre-category with star-morphisms whose base pre-category is a category and the following equality (*the law of composition with identities*) holds for every multimorphism $m$:
  \[ \operatorname{StarComp} \left( m ; \lambda i \in
     \operatorname{arity}m :
     \operatorname{id}_{\operatorname{Obj}_m i}
     \right) = m. \]
=--

See also [[abrupt category|abrupt categories]].

Categories with star-morphisms and [[abrupt category|abrupt categories]] arise in [[Victor Porton]]\'s research on [[cross-composition product]]s.

[[!redirects category with star-morphisms]]
[[!redirects categories with star-morphisms]]


category with translation

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Category with translation#
* table of contents
{:toc}

## Idea

A _category with translations_ is a [[category]] equipped with a rudimentary notion of [[suspension objects]]. Categories with translation underlie [[triangulated categories]] where the "translation" becomes a genuine suspension as in [[homotopy fiber sequences]].

## Definition

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

A **category with translation** is a [[category]] $C$ equipped with an [[equivalence|auto-equivalence]] [[functor]]

$$
  T : C \to C
$$

called the **shift functor** or **translation functor** or **suspension functor**.

=--

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

Frequently $C$ is an [[additive category]] in which case $T$ is also required to be an _[[additive functor]].

=--

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

A **morphism of categories with translation** $F:(C,T)\to (C',T')$ is a [[functor]] $F:C\to C'$ equipped with an [[isomorphism]] $F\circ T\cong T'\circ F$:

$$
  \array{
    C &\stackrel{F}{\to}& C'
    \\
    \downarrow^T &\swArrow^{\simeq}& \downarrow^{T'}
    \\
    C &\stackrel{F}{\to}& C'
  }
  \,.
$$

 If $C$,$C'$ are [[additive category|additive]] and $F$ is additive $F$ is a "morphism of additive categories with translation".

=--

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

In any additive category with translation a **triangle** is a sequence of morphisms of the form
$$
  a\stackrel{f}\to b\stackrel{g}\to c\stackrel{h}\to T a
 \,.
$$

=--

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

In some variants the translation endofunctor $T$ is not required to be an [[equivalence]]. This is the case for instance for the
[[suspended category|presuspended categories]] of Keller and Vossieck.

=--

## Examples

* The "translation" functor models the shift operation in a [[triangulated category]], where one chooses a distinguished collection of triangles satisfying some axioms.

## Related concepts

* [[suspension object]]

* [[triangulated category]]

  * [[exact triangle]], [[distinguished triangle]]

[[!redirects categories with translation]]
[[!redirects category with translations]]
[[!redirects categories with translations]]

[[!redirects shift functor]]
[[!redirects shift functors]]


category with weak equivalences

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
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## Idea

A _category with [[weak equivalence]]s_ is an ordinary [[category]] with a [[class]] of [[morphisms]] singled out -- called 'weak equivalences' -- that include the [[isomorphisms]], but also typically other morphisms.  Such a category can be thought of as a _presentation_ of an [[(∞,1)-category]] that defines explicitly only the 1-morphisms (as opposed to [[n-morphism]]s for all $n$) and the information about which of these morphisms should become _[[equivalence in an (infinity,1)-category|equivalences]]_ in the full [[(∞,1)-category]].

The desired $(\infty,1)$-category in question can be constructed from such a "presentation" by "freely adjoining inverse equivalences" to the [[weak equivalences]], in a suitable $(\infty,1)$-categorical sense.  One way to make this precise is by the process of _[[simplicial localization]]_ .  A single $(\infty,1)$-category can admit many different such presentations. See the section _[Presentations of (∞,1)-categories](#InfCatPres)_ below for more details.

## Definition

A **category with weak equivalences** is a [[category]] $C$ equipped with a [[subcategory]] (in the naïve sense) $W \subset C$

* which contains all [[isomorphisms]] of $C$;

* which satisfies [[two-out-of-three]]: for $f, g$ any two composable morphisms of $C$, if two of $\{f, g, g \circ f\}$ are in $W$, then so is the third.

## Examples and refinements

Often categories with weak equivalences are equipped with further extra [[structure]] that helps with computing the [[simplicial localization]], the [[homotopy category]] and [[derived functor]]s.

* In a [[homotopical category]] the condition on the weak equivalences is slightly stronger; see below.

* In a [[relative category]] the condition is slightly weaker. Relative categories have a good [[homotopy theory]]. See at _[[relative category]]_.

* In a [[category of fibrant objects]] there are additional auxiliary morphisms called fibrations.

* In a [[Waldhausen category]] there are additional auxiliary morphisms called [[cofibrations]].

* In a [[model category]] there are both of these additional auxiliary classes of morphisms with special interrelation between them.

Other variants include

* [[Cartan-Eilenberg category]], ...

## Additional conditions

Three additional conditions which categories with weak equivalences often satisfy are:

* the weak equivalences are closed under [[retracts]], as a subcategory of the [[arrow category]] of $C$.

* the weak equivalences satisfy the [[two-out-of-six property]]; in this case the category is called a [[homotopical category]].

* the weak equivalences are *saturated* in the sense that any morphism which becomes an isomorphism in the [[localization]] $C[W^{-1}]$ is already a weak equivalence.  (This is unrelated to the notion of [[saturated class of maps]] used in the theory of [[weak factorization systems]].)

In fact, these three conditions are closely related.

* Obviously, saturation implies closure under retracts and two-out-of-six, since the isomorphisms in any category satisfy both.

* In any model category, all three conditions hold automatically.

* If the weak equivalences admit a [[calculus of fractions]], or a well-behaved class of cofibrations or fibrations, then the three conditions are equivalent.  See [[two-out-of-six property]] for the proofs, which are from [[Categories and Sheaves]] (for the calculus of fractions) and [Blumberg-Mandell](#BlumbergMandell) (for the case of cofibrations, in the context of a [[Waldhausen category]]).

## Remarks

* If we denote by $Core(C)$ the [[core]] of $C$ -- the maximal sub[[groupoid]] of $C$ -- then we have a chain of inclusions
$
 Core(C) \hookrightarrow W \hookrightarrow C
$.

* Many categories with weak equivalences can be equipped with the further [[structure]] of a [[model category]]. On the other hand, some categories with weak equivalences can _not_ be equipped with a useful structure of a model category.  In particular, categories of [[diagrams]] in a model category do not always inherit a useful model structure (on the other hand often they do, see [[model structure on functors]]).  Several concepts exist that weaken the axioms of a model category in order to still obtain useful results in such a case -- for instance a [[category of fibrant objects]].

* Although categories of weak equivalences do not usually have limits and colimits, they are often [[accessible category|accessible]], and can be presented as an [[injectivity class]] or a [[cone-injectivity class]].  This is used in Smith's recognition theorem for [[combinatorial model categories]] and can be "algebraicized" as in [Bourke17](#Bourke17).

## Presentation of $(\infty,1)$-categories {#InfCatPres}

A category $C$ with weak equivalences serves as a presentation of an [[(∞,1)-category]] $\mathbf{C}$ with the same objects and at least the 1-morphisms of $C$, and such that every weak equivalence in $C$ becomes a true equivalence (a [[homotopy equivalence]]) in $\mathbf{C}$.

The procedure (or one of its equivalent variants) that constructs the [[(∞,1)-category]] $\mathbf{C}$ from the category with weak equivalences $C$ is called Dwyer-Kan [[simplicial localization]].

In fact, every [[(∞,1)-category]] may be presented this way (and indeed [[posets]] equipped with [[wide subcategory|wide subcategories]] of morphisms called weak equivalences are sufficient). This is discussed at

* [[model structure on categories with weak equivalences]].

Alternatively, we may further project to the 1-category in which all weak equivalences become true [[isomorphism]]s: this is the [[homotopy category]] of $C$ with respect to $W$. Equivalently this is the [[homotopy category of an (∞,1)-category]] of $\mathbf{C}$.

Note that the category with weak equivalences which presents a given $(\infty,1)$-category can not, in general, be taken to be the [[homotopy category of an (∞,1)-category|homotopy category]] of that $(\infty,1)$-category; more "flab" must be built into it.

It also cannot, in general, be the underlying 1-category of a simplicially enriched presentation of that $(\infty,1)$-category.  For instance, every $\infty$-groupoid can be realized as a simplicially enriched groupoid, but the underlying 1-category of a simplicially enriched groupoid is a 1-groupoid, which cannot be localized any further to produce a non-1-truncated $\infty$-groupoid.

## Related concepts

* [[localizer]], [[modelizer]]

* [[localizing subcategory]]

* [[relative category]]

[[!include algebraic model structures - table]]

## References
 {#References}

The terminology and specific definition of "category with weak equivalences" appears in:

* [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], and [[Jeffrey H. Smith]], §2.3 of: _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_, Mathematical Surveys and Monographs **113**, American Mathematical Society (2004) [[doi:10.1090/surv/113](http://dx.doi.org/10.1090/surv/113), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf)]

though the basic idea goes back, at least, to:

* {#DwyerKan1980Localizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** (1980) 267-284. [[doi:10.1016/0022-4049(80)90049-3](https://doi.org/10.1016/0022-4049%2880%2990049-3)]

in the context of [[simplicial localization]] (cf. the [references](relative+category#References) at *[[relative category]]*).

See also:

* {#Bourke17} [[John Bourke]], _Equipping weak equivalences with algebraic structure_, Mathematische Zeitschrift **294** (2020) 995–1019 [[arXiv:1712.02523](https://arxiv.org/abs/1712.02523), [doi:10.1007/s00209-019-02305-w](https://doi.org/10.1007/s00209-019-02305-w)]

[[!redirects categories with weak equivalences]]


category-theoretic approaches to probability theory

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

There are a number of approaches to apply [[category theory]] to [[probability]] and related fields, such as [[statistics]], [[information theory]] and [[dynamical systems]].

On one hand, one can study the existing structures in traditional probability theory (such as [[probability spaces]], [[integration]], and so on) using a categorical lens. For instance, the [[Giry monad]] models the formation of spaces of [[probability measures]] and its iterations, used for example in the context of [[de Finetti's theorem]].

On the other hand, one can try to express certain aspects of [[probability]] and [[statistics]] [[analytic versus synthetic|synthetically]]. One looks for [[structures]] and [[axioms]] which can be thought of as "fundamental" in probability and statistics, and which one can use to prove theorems, without having to use [[measure theory]] directly. One then proves that the usual measure-theoretic treatment is a [[model]] (or [[semantics]]) of this theory. This approach is often called **synthetic probability theory**, in analogy for example with [[synthetic differential geometry]].
One of the most recent approaches to synthetic probability theory is given by [[Markov categories]].

The main end goals of categorical probability are

* To generalize existing results in probability theory to more general settings, for example with less stringent conditions on countability, separability, etc.;
* To find new results, which with the traditional methods would have been too complex to prove;
* To make probability and related fields more accessible to practitioners, thanks to the fact that the formalism incorporates measure theory without requiring any deep knowledge of it.

## Main structures of interest

[[category theory|Category theory]] was first developed to model particular structures in [[algebraic topology]], and subsequently [[algebraic geometry]], [[algebra]], [[logic]] and [[computer science]]. Each one of these intended applications shaped a piece of the theory, adding to category theory the relevant structures of interest for each application.

The applications of category theory to probability are among the most recent, and are both bringing new categorical structures into the theory (such as [[Markov categories]]), as well as repurposing and reinterpreting existing ideas (such as [[monads]]).

### Markov categories

[[Markov categories]] are a recent framework that models categories whose morphisms can be thought of as having randomness, such as [[stochastic maps]] and [[Markov kernels]].
It has a graphical formalism which keeps track of the stochastic dependencies, and which can be used to prove theorems in probability purely graphically.

For more details, see [[Markov category]].

### Probability monads

[[probability monads|Probability monads]] can be thought of as a way of adding a notion of "randomness" to an existing theory.
A [[monad]] often models the idea of "forming spaces of particular structures", and in probability theory, one is interested in forming spaces of [[probability measures]].
Monads are particularly useful when this construction needs to be iterated, for example, when in [[de Finetti theorem|de Finetti]] situations one needs to form _probability measures over probability measures_.

For more details, see [[probability monad]].

### Dagger categories

[[dagger categories|Dagger categories]] can be thought of as "undirected" categories, where morphisms can be seen as going either way as in an undirected graph.
In [[probability theory]], joint distributions, or [[transport plans]] exhibit such a behavior, sometimes called [[Bayesian inversion]].
Several probabilistic ideas can be modelled in terms of dagger-categorical concepts, for example, [[conditional expectation]].

For more details, see [[category of couplings]].

## Main results

Using category-theoretic methods, several results have been obtained in the past few years.

Firstly, some known concepts and results of probability theory have been given a category-theoretic description.
(For example: expressing [[Kolmogorov's extension theorem]] as a [[cofiltered limit]] condition.) This allows to incorporate the existing theory of probability into the categorical framework, and is the basic starting point for further results.
(For example, every time in traditional probability Kolmogorov's extension theorem is invoked, one now knows that a certain universal property is being used.)

Secondly, some classical results of probability theory have been restated and reproven using category theory.
Often this adds new insight into the problem, and allows, for example, to drop further unnecessary assumptions (see the next point).
In addition, the category-theoretic formalism often trades higher complexity for higher abstraction. This way, while more abstract, the categorical proofs tend to be simpler than their measure-theoretic counterparts. (And so, they also allow to prove more difficult results more easily.)

Thirdly, and most importantly, *new theorems* have been proven, as well as generalizations and extensions of old theorems, especially from the discrete to the continuous case.

Here is a partial list, roughly in alphabetical order, roughly divided by subject.

(Work in progress, for more material see also the references below.)

### Probability and measure theory

* Approximations of Markov chains: [CDPP'09](#approx_markov), [DDGS'18](#krn).

* [[Carathéodory's extension theorem]]: proven using [[probability monads]] (as [[codensity monads]]) in [Van Belle'22](#vb_codensity).

* [[conditional expectation|Conditional expectations]]: first expressed categorically in [Panangaden](#panangaden_condexp)  and [CDPP'09](#approx_markov), then in terms of [[probability monads]] (as [[codensity monads]]) in [Van Belle'23](#vb_martingales) then using [[partial evaluations]] in [Perrone'18](#Perrone_thesis), [FP'20](#pev) and [FGPR'23](#fritz_representable), and finally in [[categories of couplings]] in [Ensarguet-Perrone'23](#ergodic_dagger).

* [[De Finetti's theorem]]: stated, interpreted and proven in terms of [[Markov categories]] with [[probability monads]]. Main results in [Fritz-Gonda-Perrone'21](#fritz_definetti), see also [Moss-Perrone'22](#det_submonad) for further context. An analogous result was proven in the [[category of couplings]], [Ensarguet-Perrone'23](#ergodic_dagger). An additional, independent categorical approach is given in [Jacobs-Staton'20](#definetti_limit).

* [[ergodic decomposition|Ergodic decomposition theorem]]: proven for deterministic [[dynamical systems]] using [[Markov categories]] in [Moss-Perrone'23](#markov_ergodic), and extended to the stochastic case using [[Markov categories]] and [[categories of couplings]] in [Ensarguet-Perrone'23](#ergodic_dagger).

* Stochastic versions of [[Gelfand duality]]: [Furber-Jacobs'15](#gelfand_furber) and [Parzygnat'17](#stoch_gelfand_parzygnat).

* Probabilistic graphical models: a categorical study of Bayesian networks [Fong'12](#fong12), a d-separation criterion for Markov categories [Fritz-Klingler'22](#d-sep).

* [[idempotent completion|Idempotent completion]] of [[BorelStoch]]: a new measure-theoretic result, with several structural consequences, proven in [FGLPS'23](#markov_supports).

* [[Kantorovich duality]] in terms of [[probability monads]]: [Perrone'18](#Perrone_thesis), [Fritz-Perrone'19](#FPKant), [Fritz-Perrone'20](#orderedkantorovich).

* [[Kolmogorov extension theorem]]: described in terms of [[Markov categories]] in [Fritz-Rischel'20](#fritzrischel), and in terms of [[probability monads]] (as [[codensity monads]]) in [Van Belle'23](#vb_martingales).

* [[martingales|Martingales]] Described in terms of [[probability monads]] (as [[codensity monads]]) in [Van Belle'23](#vb_martingales) and using [[partial evaluations]] in [Perrone'18](#Perrone_thesis).

* Multinomial and hypergeometric distributions: described in terms of [[Markov categories]] in [Jacobs'21](#multinomial).

* [[Radon-Nikodym theorem]]: proven using [[probability monads]] (as [[codensity monads]]) in [Van Belle'23](#vb_martingales).

* [[support|Supports]] of [[probability measures]]: studied in terms of [[probability monads]] in [Fritz-Perrone-Rezagholi'21](#monads_supports) and in terms of [[Markov categories]] in [FGLPS'23](#markov_supports)

* Urn models, multinomials, and probability on sets, using category theory, including Markov categories: [Jacobs'21a](#multisets) [Jacobs'21b](#multinomial) [Jacobs'22](#urns_tubes), [Jacobs-Stein'23](#overdrawing), [Jacobs'24](#urn_isometric).

* Kolmogorov and Hewitt-savage's [[zero-one laws]]: proven in terms of [[Markov categories]] in [Fritz-Rischel'20](#fritzrischel), and in [[categories of couplings]] in [Ensarguet-Perrone'23](#ergodic_dagger).

(...)

### Statistics

* Bahadur's theorem on ancillary statistics: proven in terms of Markov categories in [Fritz'20](#fritzmarkov)

* Basu's theorem on minimal [[sufficient statistics]]: proven in terms of Markov categories in [Fritz'20](#fritzmarkov).

* [[Bayesian inverses]]: expressed categorically, with proofs about their core properties and their existence, in [DDGS'18](#krn), [Cho-Jacobs'19](#cd_categories), [Fritz'20](#fritzmarkov), [Parzygnat'24](#retrodiction), and others.

* [[Bayesian updating]]: categorically [Jacobs'23](#pearl_jeffrey), [Jacobs-Zanasi'18](#essential_bayes), and including using Markov categories, [Di Lavore-Román'23](#partial_markov).

* Blackwell-Sherman-Stein theorem on statistical experiments: proven and generalized beyond the discrete case using [[Markov categories]] in [FGPR'23](#fritz_representable).

* Causal inference: using string diagrams [JKZ'18](#causal_surgery), and using Markov categories [Yin'22](#markov_causal).

* Filtering and smoothing for hidden Markov models: described in terms of [[Markov categories]] in [Virgo'23](#filters_virgo), as well as in [FKMSW'24](#hidden_markov).

* Fisher-Neyman factorization theorem for [[sufficient statistics]]: proven in terms of Markov categories in [Fritz'20](#fritzmarkov).

* General results on [[sufficient statistics]]: in terms of Markov categories [Fritz'20](#fritzmarkov), and in terms of [[idempotents]] [Jacobs'23](#idemp_discrete).

* Improper priors for Gaussians, using [[Markov categories]]: [Stein-Samuelson'23](#gauss_nondet).

(...)

### Information theory

* Characterizations of [[relative entropy]] in the discrete case: [Baez-Fritz-Leinster'11](#entropy_infloss), [Baez-Fritz'14](#entropy_bayes), [Leinster'19](#entropy_short), [Fullwood-Parzygnat'21](#information_loss_stoch). (Quantum case in [Parzygnat'22](#von_neumann), see below.)

* Characterizations of [[relative entropy]] on [[standard Borel spaces]]: [Gagne-Panangaden'18](#entropy_borel).

* [[entropy|Entropy]] in terms of [[operads]]: [Bradley'21](#entropy_operads).

* [[entropy|Entropy]] from the point of view of [[topos theory]]: [Constantin-Döring'20](#entropy_topos).

* [[entropy|Entropy]], [[relative entropy]] and [[data processing inequalities]]: expressed in terms of [[enriched category|enriched]] [[Markov categories]] in [Perrone'24](#markov_entropy).

* A book on the relationship between [[metrics]], [[entropy]] and diversity in ecology, with a category-theoretic mindset: [Leinster'21](#entropy_diversity).

(...)

### Quantum probability and information theory

* Quantum versions of Bayesian inverses: [Coecke-Spekkens'12](#coecke_spekkens12), [Parzygnat-Russo'22](#noncommutative_bayes), [Parzygnat-Buscemi'23](#retrodiction), [Fullwood-Parzygnat'23](#quantum_bayes), [Parzygnat'24](#retrodiction).

* Quantum versions of conditioning and disintegrations:  [Parzygnat'21](#conditional_quantum), [Parzygnat-Russo'23](#noncomm_disintegrations).

* Quantum versions of [[de Finetti's theorem]]: [Staton-Summers'22](#definetti_ned) and [Fritz-Lorenzin'23](#quantum_definetti).

* Categorical descriptions of dilations: [Parzygnat'19](#stinespring) and [GHL'21](#gauguin_dilations).

* Categorical characterization of von Neumann entropy: [Parzygnat'22](#von_neumann).

* Quantum versions of Markov categories: [Parzygnat'20](#quantum_markov) and [Fritz-Lorenzin'23](#quantum_definetti).

(...)

### Probability in computer science

* Bayesian updating in probabilistic programming: [Jacobs'23](#pearl_jeffrey).

* A [[cartesian closed category]] with a [[probability monad]]: [HKSY'17](#QBS), [SSSW'21](#name_gen).

* Stochastic $\lambda$-calculus: [AKM'21](#stoch_lambda).

* Probability monads for probabilistic computation: [Jones-Plotkin'89](#jones_plotkin89), [Jacobs'18](#jacobs_pmonads), [DKPS'2023](#affine_lazy).

* Privacy of name generation: [HKSY'17](#QBS), [SSSW'21](#name_gen), [FGGPS'23](#dilations).

* [[probabilistic programming|Probabilistic programming]] and [[random graphs]]: [AFKKMRSY'24](#graphons).

* Stochastic memoization via monads: [Kaddar-Staton'23](#stoch_memo).

(...)

### Structural results of categorical probability

* Foundational results on [[Markov categories]]: [Golubtsov'99](#golubtsov99), [Cho-Jacobs'19](#cd_categories), [Fritz'20](#fritzmarkov).

* Constructions and analysis of [[categories]] of [[Markov kernels]] and of [[transport plans]]: [Lawvere'62](#Lawvere62), [Chentsov'65](#Chentsov65), [Giry'82](#giry'82), [Panangaden'99](#panangaden_cat_markov), [DDGS'18](#krn), [Perrone'21](#lifting), [Kozen-Silva-Voogd'23](#krn_ext).

* Constructions of free Markov and CD categories: [Fritz-Liang'23](#free_gs).

* Study of joint and marginal distributions for [[probability monads]]: [Fritz-Perrone'18](#bimonoidal_monads).

* Markov categories for [[partial maps]]: [Di Lavore-Román'23](#partial_markov).

* Connections between [[Markov categories]] and [[categories of couplings]]: [Fritz'20](#fritzmarkov) and [Ensarguet-Perrone'23](#ergodic_dagger).

* Connections between [[Markov categories]] and [[probability monads]]: [Golubtsov'02](#golubtsov02), [FGPR'23](#fritz_representable), [Moss-Perrone'22](#det_submonad), [FGGPS'23](#dilations).

* Connections between [[Markov category#positivity_and_causality|positive Markov categories]] and [[semicartesian monoidal categories]]: [FGGPS'23](#dilations).

* Construction of [[locale|point-free]] [[probability monads]] (on [[locales]]): [Vickers'11](#vmonad).

* Results expressing [[probability monads]] as [[codensity monads]]: [Avery'16](#avery_giry) and more recently [Van Belle'22](#vb_codensity) with concrete applications to measure theory.

* Constructions of [[probability monads]] on [[metric spaces]], and connections with [[metric space|metric geometry]] and [[Banach spaces]]: [AJK'04](#jung), [Fritz-Perrone'19](#FPKant), [Fritz-Perrone'20](#orderedkantorovich).

* Constructions of [[probability monads]] on [[topological spaces]], and connections with [[functional analysis]]: [Swirszcz'74](#swirszcz), [Giry'82](#giry'82), [Jones-Plotkin'89](#jones_plotkin89), [Heckmann'96](#Heckmann96), [Keimel'08](#radonkeimel), [GLJ'19](#algebras), [Fritz-Perrone-Rezagholi'21](#monads_supports).

* A monad encoding [[probabilistic point processes]] combining probability and nondeterminism: [Dash-Staton'20](#point_processes).

* [[random variables|Random variables]] in terms of [[sheaves]]: [Simpson'17](#probability_shaves).

(...)

## See also

* [[Markov category]]

* [[probability monad]]

* [[probability theory]]

## References

(in alphabetical order by author)

* {#jung} Mauricio Alvarez-Manilla, Achim Jung, [[Klaus Keimel]], _The probabilistic powerdomain for stably compact spaces_, Theoretical Computer Science 328, 2004. [Link here](https://www.sciencedirect.com/science/article/pii/S0304397504004074).

* {#graphons} Nathanael L. Ackerman, Cameron E. Freer, Younesse Kaddar, Jacek Karwowski, Sean K. Moss, Daniel M. Roy, Sam Staton, Hongseok Yang, _Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets_, Proceedings of POPL, 2024. ([arXiv](https://arxiv.org/abs/2312.17127))

* {#avery_giry} Tom Avery, _Codensity and the Giry monad_, Journal of Pure and Applied Algebra, 220(3), 2016.

* {#stoch_lambda} Pedro H. Azevedo de Amorim, Dexter Kozen, Radu Mardare, [[Prakash Panangaden]], Michael Roberts, _Universal semantics for the stochastic $\lambda$-calculus_, Proceedings of LICS, 2021. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/lics-2021-1.pdf))

* {#algebraic_markov} Giorgio Bacci, Radu Mardare, [[Prakash Panangaden]], [[Gordon Plotkin]], _An algebraic theory of Markov processes_, Proceedings of LICS, 2018. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/Markov-LICS.pdf))

* {#entropy_bayes} [[John Baez]] and [[Tobias Fritz]], _A Bayesian characterization of relative entropy_, Theory and Application of Categories, 29(16), 2014. ([arXiv](https://arxiv.org/abs/1402.3067))

* {#entropy_infloss} [[John Baez]], [[Tobias Fritz]] and [[Tom Leinster]], _A characterization of entropy in terms of information loss_, Entropy 13(2), 2011. ([arXiv](https://arxiv.org/abs/1106.1791))

* {#entropy_operads} Tai-Danae Bradley, _Entropy as a topological operad derivation_, Entropy 23(9), 2021. ([arXiv](https://arxiv.org/abs/2107.09581))

* {#vanbreugel} Franck van Breugel, _The metric monad for probabilistic nondeterminism_, unpublished, 2005. ([pdf](http://www.cse.yorku.ca/~franck/research/drafts/monad.pdf))

* {#approx_markov} Philippe Chaput, Vincent Danos, [[Prakash Panangaden]] and [[Gordon Plotkin]], _Approximating Markov processes by averaging_, Proceedings of ICALP 2009. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/new_plain.pdf))

* {#Chentsov65} N. N. Chentsov, _The categories of mathematical statistics_, Dokl. Akad. SSSR 164, 1965.

* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* {#structuralism} [[Bob Coecke]], Eric Oliver Paquette, Dusko Pavlovic, _Classical and quantum structuralism_, ([arXiv:0904.1997](https://arxiv.org/abs/0904.1997))

* {#coecke_spekkens12} [[Bob Coecke]] and Robert W. Spekkens, _Picturing classical and quantum Bayesian inference_, Synthese, 186(3), 2012. ([arXiv](https://arxiv.org/abs/1102.2368))

* {#entropy_topos} Carmen Constantin and Andreas Döring, _A topos-theoretic notion of entropy_, 2020. ([arXiv](https://arxiv.org/abs/2006.03139))

* {#krn} Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Alexandra Silva, _Borel kernels and
their approximation, categorically_. In MFPS 34: Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, volume 341, 91–119, 2018.  [arXiv](https://arxiv.org/abs/1803.02651).

* {#affine_lazy} Swaraj Dash, Younesse Kaddar, Hugo Paquet and Sam Staton, _Affine monads and lazy structures for Bayesian programming_, Proceedings of POPL, 2023. ([pdf](https://www.cs.ox.ac.uk/people/hugo.paquet/lazyppl.pdf))

* {#point_processes} Swaraj Dash and Sam Staton, _A monad for probabilistic point processes_, Proceedings of Applied Category Theory, 2020. ([pdf](https://cgi.cse.unsw.edu.au/~eptcs/paper.cgi?ACT2020:61.pdf))

* {#partial_markov} Elena Di Lavore and Mario Román, _Evidential Decision Theory via Partial Markov Categories_, Proceedings of LICS, 2023. ([arXiv](https://arxiv.org/abs/2301.12989))

* {#ergodic_dagger} Noé Ensarguet, [[Paolo Perrone]], Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, [arXiv:2310.04267](https://arxiv.org/abs/2310.04267)

* {#fong12} [[Brendan Fong]], _Causal theories: A categorical prespective on Bayesian networks_, master thesis, University of Oxford, 2012. ([arXiv](https://arxiv.org/abs/1609.05382))

* {#2-entropy} James Fullwood, _On a 2-relative entropy_, Entropy 24(1), 2022. ([arXiv](https://arxiv.org/abs/2112.03582))

* {#retrodiction} [[Arthur J. Parzygnat]], _Reversing information flow: retrodiction in semicartesian categories_, 2024. ([arXiv](https://arxiv.org/abs/2401.17447))

* {#quantum_bayes} James Fullwood and [[Arthur J. Parzygnat]], _From time-reversal symmetry to quantum Bayes rules_, PRX Quantum 4, 2023. ([arXiv](https://arxiv.org/abs/2212.08088))

* {#quantum_states_time} James Fullwood and [[Arthur J. Parzygnat]], _On quantum states over time_, Proceedings of the Royal Society A 478, 2022. ([arXiv](https://arxiv.org/abs/2202.03607))

* {#information_loss_stoch} James Fullwood and [[Arthur J. Parzygnat]], _The information loss of a stochastic map_, Entropy 2021, 23(8), 2021. ([arXiv](https://arxiv.org/abs/2107.01975))

* {#gelfand_furber} Robert W. J. Furber and [[Bart Jacobs]]: _From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality_, LMCS 11(2), 2015. ([arXiv](https://arxiv.org/abs/1303.1115))

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

* {#weaklymarkov} [[Tobias Fritz]], [[Fabio Gadducci]], [[Paolo Perrone]] and Davide Trotta, _Weakly Markov categories and weakly affine monads_, Proceedings of CALCO, LIPIcs 10, 2023. ([arXiv](https://arxiv.org/abs/2303.14049))

* {#oplax_cartesian} [[Tobias Fritz]], [[Fabio Gadducci]], Davide Trotta, Andrea Corradini, _From gs-monoidal to oplax-cartesian categories: constructions and functorial completeness_, Applied Categorical Structures 31, 42, 2023. ([arXiv](https://arxiv.org/abs/2205.06892))

* {#dilations} [[Tobias Fritz]], Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, [[Paolo Perrone]] and Dario Stein, _Dilations and information flow axioms in categorical probability_. Mathematical Structures in Computer Science, 2023. ([arXiv:2211.02507](https://arxiv.org/abs/2211.02507)).

* {#markov_supports} [[Tobias Fritz]], Tomáš Gonda, Antonio Lorenzin, [[Paolo Perrone]], Dario Stein, _Absolute continuity, supports and idempotent splitting in categorical probability_, ([arXiv:2308.00651](https://arxiv.org/abs/2308.00651))

* {#fritz_definetti} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], _De Finetti's theorem in categorical probability_. Journal of Stochastic Analysis, 2021. ([arXiv:2105.02639](https://arxiv.org/abs/2105.02639))

* {#fritz_representable} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], Eigil Fjeldgren Rischel, _Representable Markov categories and comparison of statistical experiments in categorical probability_, Theoretical Computer Science 961, 2023. ([arXiv:2010.07416](https://arxiv.org/abs/2010.07416))

* {#d-sep} [[Tobias Fritz]] and Andreas Klingler, _The d-separation criterion in Categorical Probability_, Journal of Machine Learning Research 24(46), 2023. ([arXiv](https://arxiv.org/abs/2207.05740))

* {#hidden_markov} [[Tobias Fritz]], Andreas Klingler, Drew McNeely, Areeb Shah-Mohammed and Yuwen Wang, _Hidden Markov models and the Bayes filter in categorical probability_, 2024. ([arXiv](https://arxiv.org/abs/2401.14669))

* {#free_gs} [[Tobias Fritz]], [[Wendong Liang]], _Free gs-monoidal categories and free Markov categories_, Applied Categorical Structures 31, 21, 2023. ([arXiv:2204.02284](https://arxiv.org/abs/2204.02284))

* {#quantum_definetti} [[Tobias Fritz]] and Antonio Lorenzin, _Involutive Markov categories and the quantum de Finetti theorem_, 2023. ([arXiv](https://arxiv.org/abs/2312.09666))

* {#orderedkantorovich} [[Tobias Fritz]] and [[Paolo Perrone]], _Stochastic order on metric spaces and the ordered Kantorovich monad_, Advances in Mathematics 366, 2020. ([arXiv:1808.09898](https://arxiv.org/abs/1808.09898))

* {#pev} [[Tobias Fritz]] and [[Paolo Perrone]], _Monads, partial evaluations, and rewriting_, Proceedings of MFPS, 2020. ([arXiv](https://arxiv.org/abs/1810.06037))

* {#FPKant} [[Tobias Fritz]] and [[Paolo Perrone]], _A Probability Monad as the Colimit of Spaces of Finite Samples_, Theory and Applications of Categories 34, 2019. ([arXiv:1712.05363](https://arxiv.org/abs/1712.05363))

* {#bimonoidal_monads} [[Tobias Fritz]], [[Paolo Perrone]], _Bimonoidal Structure of Probability Monads_, Proceedings of MFPS, 2018, ([arXiv:1804.03527](https://arxiv.org/abs/1804.03527))

* {#monads_supports} [[Tobias Fritz]], [[Paolo Perrone]], Sharwin Rezagholi, _Probability, valuations, hyperspace: Three monads on Top and the support as a morphism_, Mathematical Structures in Computer Science 31(8), 2021. ([arXiv:1910.03752](https://arxiv.org/abs/1910.03752))

* {#fritzrischel} [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _Infinite products and zero-one laws in categorical probability_, Compositionality 2(3) 2020. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))

* {#entropy_borel} Nicolas Gagne and [[Prakash Panangaden]], _A categorical characterization of relative entropy on standard Borel spaces_, ENTCS 336, 2018. ([arXiv](https://arxiv.org/abs/1703.08853))

* {#gauguin_dilations} Nicholas Gauguin Houghton-Larsen, _A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing_, PhD thesis, University of Copenhagen, 2021. ([arXiv](https://arxiv.org/abs/2103.02302))

* {#giry82} Michèle Giry, _A categorical approach to probability theory_, Categorical aspects of topology and analysis, Lecture notes in Mathematics 915, Springer, 1982.

* {#golubtsov99} Peter V. Golubtsov, _Axiomatic description of categories of information converters_, Problemy Peredachi Informatsii (Problems in Information Transmission) 35(3), 1999.

* {#golubtsov02} Peter V. Golubtsov, _Monoidal Kleisli category as a background for information transformers theory_. Information Theory and Information Processing 2, 2002.

* {#golubtsov04} Peter V. Golubtsov, _Information transformers: category-theoretical structure, informativeness, decision-making problems_, Hadronic Journal Supplements 19(3), 2004.

* {#golubtsov_mosaliuk02} Peter V. Golubtsov and Stepan S. Moskaliuk, _Method of additional structures on the objects of a monoidal Kleisli category as a background for information transformers theory_, Hadronic Journal 25(2), 2002.

* {#algebras} [[Jean Goubault-Larrecq]] and Xiaodong Jia, _Algebras of the extended probabilistic powerdomain monad_, ENTCS 345, 2019
([doi:10.1016/j.entcs.2019.07.015](https://doi.org/10.1016/j.entcs.2019.07.015))

* {#Heckmann96} Reinhold Heckmann, _Spaces of valuations_, Papers on General Topology and Ap-plications, 1996 ([doi:10.1111/j.1749-6632.1996.tb49168.x](https://doi.org/10.1111/j.1749-6632.1996.tb49168.x),[pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf))

* {#stoch_memo} Younesse Kaddar and Sam Staton, _A model of stochastic memoization and name generation in probabilistic programming: categorical semantics via monads on presheaf categories_, Proceedings of MFPS, 2023. ([arXiv](https://arxiv.org/abs/2309.09467))

* {#QBS} [[Chris Heunen]], Ohad Kammar, [[Sam Staton]] and Hongseok Yang, _A convenient category for higher-order probability theory_, Proceedings of LICS 2017. ([arXiv](https://arxiv.org/abs/1701.02547))

* {#urn_isometric} [[Bart Jacobs]], _Drawing from an Urn is Isometric_, Proceedings of FoSSaCS, 2024. ([pdf](http://www.cs.ru.nl/B.Jacobs/PAPERS/draw-isometries.pdf))

* {#pearl_jeffrey} [[Bart Jacobs]], _Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming_ ([arXiv:2309.07053](https://arxiv.org/abs/2309.07053))

* {#urns_tubes} [[Bart Jacobs]], _Urns & Tubes_, Compositionality 4(4), 2022. ([arXiv](https://arxiv.org/abs/2110.02155))

* {#idemp_discrete} [[Bart Jacobs]], _Sufficient statistics and split idempotents in discrete probability theory_, Proceedings of MFPS, 2023. ([arXiv](https://arxiv.org/abs/2212.09191))

* {#multinomial} [[Bart Jacobs]], _Multinomial and Hypergeometric Distributions in Markov Categories_, Proceedings of MFPS 2021. ([arXiv](https://arxiv.org/abs/2112.14044))

* {#multisets} [[Bart Jacobs]], _From Multisets over Distributions to Distributions over Multisets_, Proceedings of LICS, 2021. ([arXiv](https://arxiv.org/abs/2105.06908))

* {#jacobs_pmonads} [[Bart Jacobs]], _From probability monads to commutative effectuses_, Journal of Logical and Algebraic Methods in Programming 94, 2018.
([doi:10.1016/j.jlamp.2016.11.006](https://doi.org/10.1016/j.jlamp.2016.11.006))

* {#parameter_learning} [[Bart Jacobs]], _Categorical aspects of parameter learning_, 2018. ([arXiv](https://arxiv.org/abs/1810.05814))

* {#exact_inference} [[Bart Jacobs]], _A Channel-based Exact Inference Algorithm for Bayesian Networks_, 2018. ([arXiv](https://arxiv.org/abs/1804.08032))

* {#conj_priors} [[Bart Jacobs]], _A Channel-Based Perspective on Conjugate Priors_, 2017. ([arXiv:1707.00269](https://arxiv.org/abs/1707.00269))

* {#hypernormalization} [[Bart Jacobs]], _Hyper Normalisation and Conditioning for Discrete Probability Distributions_, LMCS 13, 2017. ([arXiv](https://arxiv.org/abs/1607.02790))

* {#causal_surgery} [[Bart Jacobs]], Aleks Kissinger, Fabio Zanasi, _Causal Inference by String Diagram Surgery_, ([arXiv:1811.08338](https://arxiv.org/abs/1811.08338))

* {#definetti_limit} [[Bart Jacobs]], [[Sam Staton]], _De Finetti's construction as a categorical limit_, Proceedings of CMCS, 2021. ([arXiv:2003.01964](https://arxiv.org/abs/2003.01964))

* {#overdrawing} [[Bart Jacobs]] and [[Dario Stein]], _Overdrawing Urns using Categories of Signed Probabilities_, Proceedings of Applied Category Theory, EPTCS 397, 2023. ([arXiv](https://arxiv.org/abs/2312.12453))

* {#essential_bayes} [[Bart Jacobs]], Fabio Zanasi, _The Logical Essentials of Bayesian Reasoning_, ([arXiv:1804.01193](https://arxiv.org/abs/1804.01193))

* {#bayes_adjoint} [[Bart Jacobs]], Fabio Zanasi, and Octavio Zapata, _Bayesian Factorisation as Adjoint, [abstract](http://wadt18.cs.rhul.ac.uk/submissions/WADT18A33.pdf)

* {#jones_plotkin89} Claire Jones and [[Gordon Plotkin]], _A probabilistic powerdomain of evaluations_, Procedings of LICS, 1989. ([doi](https://doi.org/10.1109/LICS.1989.39173))

* {#radonkeimel} [[Klaus Keimel]], _The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras_, Topology and its Applications, 2008 ([doi:10.1016/j.topol.2008.07.002](https://doi.org/10.1016/j.topol.2008.07.002))

* {#stone_markov} Dexter Kozen, Kim G. Larsen, Radu Mardare, [[Prakash Panangaden]], _Stone duality for Markov processes_, Proceedings of LICS, 2013. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/mfcs2013.pdf))

* {#markov_logid} Dexter Kozen, Radu Mardare, [[Prakash Panangaden]], _Stone duality for Markov processes_, Proceedings of MFCS, 2013. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/lics2013.pdf))

* {#krn_ext} Dexter Kozen, Alexandra Silva, Erik Voogd, _Joint Distributions in Probabilistic Semantics_, MFPS 2023. ([arXiv](https://arxiv.org/abs/2309.06913))

* {#Lawvere62} [[Bill Lawvere]], _The category of probabilistic mappings_, ms. 12 pages, 1962
([[lawvereprobability1962.pdf:file]])

* {#entropy_short} Tom Leinster, _A short characterization of relative entropy_, Journal of Mathematical Physics 60, 2019. ([arXiv](https://arxiv.org/abs/1712.04903))

* {#entropy_diversity} Tom Leinster, _Entropy and Diversity_, Cambridge University Press, 2021.

* {#wasserstein_algebras} Radu Mardare, [[Prakash Panangaden]], [[Gordon Plotkin]], _Free complete Wasserstein algebras_, LMCS 14, 2018. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/fcwa.pdf))

* {#quantitative_reasoning} Radu Mardare, [[Prakash Panangaden]], [[Gordon Plotkin]], _Quantitative algebraic reasoning_, Proceedings of LICS, 2016. ([pdf](https://www.cs.mcgill.ca/~prakash/Pubs/lics2016.pdf))

* Cristina Matache, Sean Moss, Sam Staton, Ariadne Si Suo, _Denotational semantics for languages for inference: semirings, monads, and tensors_, Proceedings of LAFI, 2023. ([arXiv](https://arxiv.org/abs/2312.16694))

* {#moggi89} [[Eugenio Moggi]], *Computational lambda-calculus and monads*, Proceedings of LICS, 1989. ([doi](https://doi.org/10.1109/LICS.1989.39155))

* {#markov_ergodic} Sean Moss and [[Paolo Perrone]], _A category-theoretic proof of the ergodic decomposition theorem_, Ergodic Theory and Dynamical Systems, 2023. ([arXiv:2207.07353](https://arxiv.org/abs/2207.07353))

* {#det_submonad} Sean Moss, [[Paolo Perrone]], _Probability monads with submonads of deterministic states_, LICS 2022. ([arXiv:2204.07003](https://arxiv.org/abs/2204.07003))

* {#panangaden_cat_markov} [[Prakash Panangaden]], _The category of Markov kernels_, ENTCS, 1999. ([full text](https://www.sciencedirect.com/science/article/pii/S1571066105806024))

* {#panangaden_markov} [[Prakash Panangaden]], _Labelled Markov processes_, Imperial College Press, 2009.

* {#panangaden_condexp} [[Prakash Panangaden]], _A categorical view of conditional expectation_, ([slides](http://math.ucr.edu/home/baez/mathematical/ACTUCR/Panagaden_Conditional_Expectation.pdf))

* {#von_neumann} [[Arthur J. Parzygnat]], _A functorial characterization of von Neumann entropy_, Cahiers de Topologie et de Géométrie Différentielle Catégorique LXIII(1), 2022. ([arXiv](https://arxiv.org/abs/2009.07125))

* {#conditional_quantum} [[Arthur J. Parzygnat]], _Conditional distributions for quantum systems_, EPTCS 343, 2021. ([arXiv](https://arxiv.org/abs/2102.01529))

* {#quantum_markov} [[Arthur J. Parzygnat]], _Inverses, disintegrations, and Bayesian inversion in quantum Markov categories_, 2020. ([arXiv](https://arxiv.org/abs/2001.08375))

* {#stinespring} [[Arthur J. Parzygnat]], _Stinespring's construction as an adjunction_, Compositionality 1(2), 2019. ([arXiv](https://arxiv.org/abs/1807.02533))

* {#stoch_gelfand_parzygnat} [[Arthur J. Parzygnat]], _Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem_, 2017. ([arXiv](https://arxiv.org/abs/1708.00091))

* {#retrodiction} [[Arthur J. Parzygnat]], Francesco Buscemi, _Axioms for retrodiction: achieving time-reversal symmetry with a prior_, Quantum 7(1013), 2023. [arXiv](https://arxiv.org/abs/2210.13531)

* {#noncomm_disintegrations} [[Arthur J. Parzygnat]], Benjamin P. Russo, _Non-commutative disintegrations: existence and uniqueness in finite dimensions_, Journal of Noncommutative Geometry 17(3), 2023. ([arXiv](https://arxiv.org/abs/1907.09689))

* {#noncommutative_bayes} [[Arthur J. Parzygnat]] and Benjamin P. Russo, _A noncommutative Bayes theorem_, Linear Algebra Applications 644, 2022. ([arXiv](https://arxiv.org/abs/2005.03886))

* {#markov_entropy} [[Paolo Perrone]], _Markov Categories and Entropy_, IEEE Transactions on Information Theory 70(3), 2024. ([arXiv](https://arxiv.org/abs/2212.11719))

* {#lifting} [[Paolo Perrone]], _Lifting couplings in Wasserstein spaces_, 2021. ([arXiv](https://arxiv.org/abs/2110.06591))

* {#Perrone_thesis} [[Paolo Perrone]], _Categorical Probability and Stochastic Dominance in Metric Spaces_, ([thesis](http://www.paoloperrone.org/phdthesis.pdf))

* {#name_gen} Marcin Sabok, Sam Staton, Dario Stein, Michael Wolman, _Probabilistic Programming Semantics for Name Generation_, Proceedings of POPL, 2021. ([arXiv](https://arxiv.org/abs/2007.08638))

* {#probability_sheaves} [[Alex Simpson]], _Probability Sheaves and the Giry Monad_, ([pdf](https://coalg.org/mfps-calco2017/calco-papers/calco2017-1.pdf))

* {#definetti_ned} Sam Staton and Ned Summers, _Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras_, Proceedings of Quantum Physics and Logic, 2022. ([arXiv](https://arxiv.org/abs/2207.05832))

* {#gauss_nondet} Dario Stein and Richard Samuelson, _A Category for unifying Gaussian Probability and Nondeterminism_, Proceedings of CALCO, LIPIcs 10, 2023. ([arXiv](https://arxiv.org/abs/2204.14024))

* {#exact_cond} Dario Stein and [[Sam Staton]], _Probabilistic Programming with Exact Conditions_, JACM, 2023. ([arXiv](https://arxiv.org/abs/2312.17141))

* {#swirszcz} T. Swirszcz, _Monadic functors and convexity_, Bulletin de l'Academie Polonais des Sciences 22, 1974 ([pdf](https://www.fuw.edu.pl/~kostecki/scans/swirszcz1974.pdf))

* {#vb_martingales} Ruben Van Belle, _A categorical treatment of the Radon-Nikodym theorem and martingales_, 2023. ([arXiv](https://arxiv.org/abs/2305.03421))

* {#vb_caratheodory} Ruben Van Belle, _A categorical proof of the Carathéodory extension theorem_, 2022. ([arXiv](https://arxiv.org/abs/2210.01720))

* {#vb_codensity} Ruben Van Belle, _Probability monads as codensity monads_, Theory and Applications of Categories, 38(21), 2022. ([arXiv](https://arxiv.org/abs/2111.01250))

* {#vmonad} [[Steve Vickers]], _A monad of valuation locales_, 2011. [Link here](https://www.cs.bham.ac.uk/~sjv/Riesz.pdf).

* {#filters_virgo} Nathaniel Virgo, _Unifilar Machines and the Adjoint Structure of Bayesian Filtering_, Proceedings of ACT, 2023. ([arXiv](https://arxiv.org/abs/2305.02826))

* {#markov_causal} Yimu Yin, Jiji Zhang ,_Markov categories, causal theories, and the do-calculus_, 2022. ([arXiv](https://arxiv.org/abs/2204.04821))

(...more to add...)

category: probability

[[!redirects category-theoretic probability]]
[[!redirects category-theoretic approach to probability theory]]
[[!redirects categorical approach to probability theory]]
[[!redirects categorical approaches to probability theory]]
[[!redirects categorical probability]]
[[!redirects categorical probability theory]]
[[!redirects categorical probability and statistics]]
[[!redirects synthetic probability]]
[[!redirects synthetic probability theory]]


categorytheorycontents

**[[category theory|Category Theory]]**

##Contents

* Item 1

   * Sub-Item

* Item 2

[Edit this sidebar](/nlab/edit/categorytheorycontents)


Caterina Consani

* [webpage](http://www.math.jhu.edu/~kc/)

## related $n$Lab entries

* [[F1]]

category: people


Catharina Stroppel

* [website](http://www.math.uni-bonn.de/people/stroppel)

category: people
[[!redirects C. Stroppel]]


Catherine Meusburger

[website](http://www.algeo.math.fau.de/people/meusburger-catherine/prof-dr-catherine-meusburger)

category: people


Catherine Ray

Catherine Ray is a graduate student at Northwestern University studying the interplay between number theory and homotopy theory.

* [blog](http://rin.io)

She graduated with a B.S. in Computational Physics from George Mason University at 16, and accepted the Thiel Fellowship in 2014 to study mathematical physics full-time.


Cathy Kriloff

* [personal page](https://www2.cose.isu.edu/~krilcath/)

## Selected writings

* [[Briana Foster-Greenwood]], [[Cathy Kriloff]], *Spectra of Cayley graphs of complex reflection groups*, J. Algebraic Combin., 44(1):33–57, 2016 ([arXiv:1502.07392](https://arxiv.org/abs/1502.07392), [doi:10.1007/s10801-015-0652-8](https://doi.org/10.1007/s10801-015-0652-8))

category: people


Catégories Tannakiennes

This entry is about the article

* [[Pierre Deligne]]

  __Catégories Tannakiennes__,

  [[Grothendieck Festschrift]], vol. II,

  Birkhäuser Progress in Math. 87 (1990) pp.111-195.

  ([pdf](https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf))

on [[Tannakian categories]].

The main theorem is a [[Tannaka duality|Tannakian reconstruction theorem]], which repairs a problem in the work of Grothendieck's student Saavedra Rivano:

* Neantro Saavedra Rivano, _Catégories Tannakiennes_, Bulletin de la Société Mathématique de France **100** (1972)  417-430 [[eudml:87193](https://eudml.org/doc/87193)]

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

which had a substantial gap in the proof of the main theorem.

At the end of the article there is an application to the [[differential Galois theory]] (which has been earlier much studied by A. R. Magid) which is the Galois theory for differential rather than algebraic equations.

The article had a huge influence not only in Tannakian theory but also in creating more interactions between modern [[algebraic geometry]], [[category theory]], and the theory of [[bialgebra|bialgebras]].

In the followup

* {#Deligne02} [[Pierre Deligne]], _Catégorie Tensorielle_, Moscow Math. Journal 2 (2002) no. 2, 227-248. ([pdf](https://www.math.ias.edu/files/deligne/Tensorielles.pdf))

the conditions for Tannakian reconstruction are relaxed further, see at _[[Deligne's theorem on tensor categories]]_.

A generalization to "non-neutral" Tannakian categories, and representations of [[gerbes]] is in

* {#DeligneMilne12} [[Pierre Deligne]] and [[James Milne]], _Tannakian categories_, 2012. ([pdf](http://www.jmilne.org/math/xnotes/tc.pdf))

See also

* [wikipedia](http://en.wikipedia.org/wiki/Tannakian_category)
* [[Tannakian category]]

category: reference


Cauchy approximation

[[!redirects Cauchy approximations]]

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

#Contents#
* table of contents
{:toc}

## Definition ##

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$$

be the set of positive rational numbers. Let $(S, \sim)$ be a [[Booij premetric space]]. A __Cauchy approximation__ is a function $x: \mathbb{Q}_{+} \to S$ such that for all positive rational numbers $\delta$ and $\eta$, $x(\delta) \sim_{\delta + \eta} x(\eta)$.

The set of all Cauchy approximations is defined as

$$C(S) \coloneqq \{x \in S^{\mathbb{Q}_{+}} \vert \forall \delta \in \mathbb{Q}_{+}.\forall \eta \in \mathbb{Q}_{+}.x(\delta) \sim_{\delta + \eta} x(\eta)\}$$

## Properties ##

Every Cauchy approximation is a [[Cauchy net]] indexed by the positive rational numbers $\mathbb{Q}_{+}$.

A Cauchy approximation is the composition $x \circ M$ of a [[net]] $x$ and an [[modulus of convergence]] $M$.

## See also ##

* [[Booij premetric space]]
* [[Cauchy structure]]
* [[modulus of convergence]]
* [[Cauchy real numbers]]
* [[HoTT book real numbers]]

## References ##

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

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