category of simple graphs

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Introduction

Both [[category theory]] and [[graph theory]] study patterns based on [[diagrams]] consisting of nodes and edges. Despite this surface impression, there is in fact very little interaction between the scientific communities of category theorists and of graph theorists.

This article is a modest bridge, indicating that the _[[category]] of [[graphs]]_ (in the usual graph-theorist's sense -- see for example [Diestel](#Diestel)) has some very nice properties.

## Simple graphs as relations

By a **simple [[graph]]**, we mean a [[set]] $V$ whose elements are called *vertices*, together with a collection of 2-element subsets $\{x, y\}$ of $V$; these are called *edges* of the graph.

There are various ways of forming a category of simple graphs. Perhaps the most straightforward is to define a graph morphism $G \to H$ to be a function $f: V \to W$ between the vertex sets such that $\{f(x), f(y)\}$ is an edge of $H$ whenever $\{x, y\}$ is an edge of $G$.

Another option -- and this is the one chosen for this article -- starts by regarding a simple graph as carrying the same information as a set $V$ equipped with a [[symmetric relation|symmetric]] [[reflexive relation|reflexive]] [[relation]] $E$. Indeed, such a relation determines (and is uniquely determined by) a simple graph $G$ where for given vertices $x, y \in V$, there is an edge $\{x, y\}$ between $x$ and $y$ in $G$ iff both $(x, y) \in E$ and $x \neq y$. We will write $E(x, y)$ to mean $(x, y) \in E$.

Then, under the relational formulation, we define a morphism $(V, E) \to (W, F)$ between simple graphs straightforwardly[^1] as a function $f: V \to W$ that preserves the relevant structure, i.e., that $E(x, y)$ implies $F(f(x), f(y))$. One reason for preferring this notion of morphism is that it allows, for example, consideration of arbitrary edge contractions of a simple graph as [[quotients]] in the category (cf. [[graph minor]]), something that is not possible under the prior notion of morphism.

[^1]: In other words, the usual notion of morphism between structures or models as in [[model theory]].

Thus we will adopt the latter notion of morphism which takes reflexive symmetric relations $E$ as primary. The resulting [[category]] of simple graphs is denoted by $SimpGph$.

Note that graphs and their morphisms can also be understood in functorial language, by regarding simple graphs as special types of [[presheaves]], as follows. Let $C$ be the category of sets $1 \coloneqq \{\ast\}$ and $2 \coloneqq \{s, t\}$ and functions between them. Then a presheaf $X: C^{op} \to Set$ is given by sets $V = X(1)$ and $E = X(2)$ and maps

* $d_0: X(2) \stackrel{X(\ast \mapsto s)}{\to} X(1)$ (source map),

* $d_1: X(2) \stackrel{X(\ast \mapsto t)}{\to} X(1)$ (target map),

* $i: X(1) \stackrel{X(s, t \mapsto \ast)}{\to} X(2)$ (reflexive map),

* $\sigma: X(2) \stackrel{X(s \mapsto t, t \mapsto s)}{\to} X(2)$ (symmetry map).

satisfying appropriate identities imposed by equations in $C$. Simple graphs can then be regarded [[equivalence|equivalently]] as such presheaves where the map

$$\langle d_0, d_1 \rangle: X(2) \to X(1) \times X(1)$$

is a [[monomorphism]]. In that case, a morphism of simple graphs amounts to a [[natural transformation]] between such presheaves.

### An aside on other notions of graph

"Simple graph" as defined in the nLab (see [[graph]]) means that edges are 2-element subsets of $V$, but of course that doesn't preclude consideration of other types of graph. One option is to consider sets $V$ equipped with a collection of subsets of $V$ of cardinality either 1 or 2, i.e., allowing some but not necessarily all loops as edges. We don't call those "simple graphs" (at [[graph]] they are called "loop graphs"), but nevertheless they form a respectable category under the straightforward notion of morphism $f$ (if $\{x, y\}$ is an edge of the domain, possibly with $x = y$, then $\{f(x), f(y)\}$ is an edge of the codomain). Chih and Scull use this category, which they refer to as $Gph$, in their paper [_Homotopy in the Category of Graphs_](#ChihScul).

## Properties of $SimpGph$

The category $SimpGph$ has very good properties. For example,

+-- {: .num_theorem}
###### Theorem
$SimpGph$ is a [[Grothendieck quasitopos]]. In particular, it is a [[regular category]] and even a [[logos]], and $SimpGph^{op}$ is also regular. It is also $\infty$-[[extensive category|extensive]].
=--

+-- {: .proof}
###### Proof
(See also [Adamek and Herrlich](#AdamHerr).) As above, let $C$ be the category of sets $1$ and $2$ and functions between them, and regard the category of simple graphs as a full subcategory of the presheaf [[topos]] $Set^{C^{op}}$. For this presheaf topos, there is just one nontrivial $\neg\neg$-dense sieve, namely the inclusion

$$(s, t): C(-, 1) + C(-, 1) \to C(-, 2)$$

(where $s$ is shorthand for $C(-, \ast \mapsto s)$ and similarly for $t$) and so the category of $\neg\neg$-[[separated presheaves]] is equivalent to the category of presheaves $X$ such that the induced map

$$X(2) \cong Set^{C^{op}}(C(-, 2), X) \stackrel{(s, t)^\ast}{\to} Set^{C^{op}}(C(-, 1) + C(-, 1), X) \cong X(1) \times X(1),$$

which is the source-target pairing $\langle d_0, d_1 \rangle: X(2) \to X(1) \times X(1)$, is monic. In other words, a simple graph in this language is exactly a separated presheaf. On the other hand, a Grothendieck quasitopos is, essentially by definition, the category of separated presheaves for a topology on a presheaf topos, in this case the $\neg\neg$-topology.

Being a quasitopos with small coproducts, it is $\infty$-extensive provided that coproducts are disjoint. However, this is trivial to check (it even suffices to check, according to [[Elephant]] 2.6.5, that $0 \to 1$ is a [[regular monomorphism]], or that $1 + 1$ is a disjoint coproduct, which it obviously is).
=--

+-- {: .num_remark #quasitopos}
###### Remark
Implicit in this proof is the way in which (for example) limits of simple graphs are calculated. Since $SimpGph$ is realized as a category of separated presheaves which is a [[reflective subcategory]] of all presheaves, limits in $SimpGph$ are calculated just as they are in the presheaf category $Set^{C^{op}}$, which is to say: they are computed objectwise, at the objects $1, 2$ of $C$.

For example, consider how to construct the [[equalizer]] of a pair of graph maps $f, g: G \rightrightarrows H$. For the vertex set of the equalizer (computing the equalizer at the object $1$ of $C$), one just takes the equalizer of the functions between the vertex sets of $G$ and $H$. At the edge set level where we have maps $E = G(2) \rightrightarrows F = H(2)$: since there is at most one edge between two vertices, the maps $f(2), g(2)$ must agree on $e \in E$ provided $f(d_0 e) = g(d_0 e)$ and $f(d_1 e) = g(d_1 e)$, so the equalizer graph is just the full or [[graph|induced subgraph]] of $G$ given by the equalizer vertex set.

Moreover, every induced subgraph $H \hookrightarrow G$ arises as an equalizer (consider the equalizer of the two embeddings of $G$ into the amalgamation $G +_H G$, which at the vertex level agrees with $H \hookrightarrow G$).

Of course this is elementary, but we get much more more: the quasitopos $SimpGph$ is also a [[locally cartesian closed category]], and [[dependent products]] can also be read off directly from how they work for presheaves.
=--

It is easy to describe [[monomorphism|monos]] and [[epimorphism|epis]] in $SimpGph$. For, let $\Gamma = \hom(1, -): SimpGph \to Set$ be the underlying vertex-set [[forgetful functor]].

+-- {: .num_prop #EmbeddingOfSimpGphIntoSet}
###### Proposition
The forgetful functor $\Gamma = \hom(1, -): SimpGph \to Set$ is faithful, in fact exhibits $SimpGph$ as a [[topological concrete category]].
=--

We omit the easy proof. Next we have two easy results:

+-- {: .num_prop}
###### Proposition
$\Gamma$ [[reflected limit|reflects]] monos and epis (because $\Gamma$ is faithful).
=--

+-- {: .num_prop}
###### Proposition
$\Gamma$ [[preserved limit|preserves]] [[limits]] and [[colimits]] (because it has a [[left adjoint]] $\Delta$ and a [[right adjoint]] $\nabla$).
=--

It follows that $\Gamma: SimpGph \to Set$ both preserves and reflects monos and epis. As a result, we can prove various simple exactness results in $SimpGph$. For instance:

+-- {: .num_lemma}
###### Lemma
In $SimpGph$, the [[pushout]] of any mono is a mono.
=--

+-- {: .proof}
###### Proof
Suppose we have a pushout diagram in $SimpGph$:

$$\array{
G & \to & H \\
 ^\mathllap{mono} \downarrow & & \downarrow^\mathrlap{k} \\
G' & \to & H'
}$$

Since $\Gamma$ preserves both pushouts and monos, and since the pushout of a mono in $Set$ is a mono, we have that $\Gamma(k)$ is monic in $Set$. Since $\Gamma$ reflects monos, this means $k$ is monic in $SimpGph$.
=--

As already observed, there is a chain of adjoint functors $\Delta \dashv \Gamma \dashv \nabla: Set \to SimpGph$. But in fact there is a fourth functor in the chain: $\Delta$ has a left adjoint $\Pi: SimpGph \to Set$, the [[connected object|connected components]] functor. If $E$ is the simple graph consisting of two vertices $a, b$ with an edge between them, then there is a reflexive fork

$$a, b: 1 \rightrightarrows E \stackrel{!}{\to} 1$$

and $\Pi$ is formed as a [[reflexive coequalizer]] of the induced diagram:

$$SimpGph(1, -) \to SimpGph(E, -) \rightrightarrows SimpGph(1, -) \to \Pi$$

+-- {: .num_prop}
###### Proposition
$\Pi \dashv \Delta$, and $\Pi$ preserves products.
=--

+-- {: .proof}
###### Proof
$\Pi$ preserves products because being a reflexive coequalizer, it is a [[sifted colimit]] of product-preserving functors. For graphs $G$, there is a natural map $u G: G \to \Delta \Pi G$ which at the underlying vertex-set level sends a vertex to its connected component; if $S$ is any set, then a graph map $f: G \to \Delta S$ must send any two vertices with an edge between them to the same point, and so $f$ factors as $G \stackrel{u G}{\to} \Delta \Pi G \stackrel{\Delta g}{\to} \Delta S$ for a unique map $g: \Pi G \to S$, and this establishes the adjunction $\Pi \dashv \Delta$.
=--

### Equalizers and coequalizers

If $f, g \colon G \stackrel{\to}{\to} H$ are maps in $SimpGph$, then their [[equalizer]] $Eq(f, g) = i \colon K \to G$ is given at the vertex level by

$$\Gamma(i) = Eq(\Gamma(f), \Gamma(g))$$

and at the edge level by $K(x, y) \Leftrightarrow G(i(x), i(y))$; in other words, if $x, y$ belong to $K$ and there is an edge between them in $G$, then that edge lies in $K$. To express this last condition, we say the subgraph $i$ is _full_ (at the edge level).

Thus, $i$ is a regular mono in $SimpGph$ iff both $\Gamma(i)$ is monic in $Set$ and $i$ is full at the edge level (see Remark \ref{quasitopos}).

The [[coequalizer]] of $f$ and $g$, say $Coeq(f, g) = q \colon H \to Q$, is given at the vertex level by

$$\Gamma(q) = Coeq(\Gamma(f), \Gamma(g))$$

and at the edge level by taking the image of the composite $E_H \hookrightarrow Vert(H)^2 \stackrel{q^2}{\to} Vert(Q)^2$, so that

$$Q(x, y) \Leftrightarrow \exists_{u, v} H(u, v) \wedge q(u) = x \wedge q(v) = y.$$

Thus, $q$ is a [[regular epimorphism]] if it is surjective at both the vertex and edge levels.

### Ternary factorization

The category of simple graphs has a [[ternary factorization system]] as follows: each morphism $f \colon G \to H$ factors as

$$G \stackrel{q}{\to} G' \stackrel{a}{\to} H' \stackrel{i}{\to} H$$

where

* $q$ is a surjection between vertex-sets and at the edge level, i.e., is a [[regular epi]];

* $a$ induces an identity between vertex-sets (hence is [[bimorphism|jointly monic and epic]]), but without necessarily being full at the edge level;

* $i$ is given by an injection between vertex-sets and is full at the edge level, and (thus) is a [[regular mono]].

The factorization of $f$ into $q$ followed by $i \circ a$ is the (regular epi)-mono factorization, while the factorization of $f$ into $a \circ q$ followed by $i$ is the epi-(regular mono) factorization. (The fact that regular monos are stable under pushout, used to ensure that the epi-(regular mono) factorization is an [[orthogonal factorization system]], is true because $SimpGph$, being a [[quasitopos]], is a *coregular category*, meaning $SimpGph^{op}$ is [[regular category|regular]].) Both factorizations coincide at the level of underlying functions between vertex sets, both being the usual epi-mono factorization.

In particular, we have the easy facts that

* $f$ is a mono iff $q$ is a graph isomorphism: monos in $SimpGph$ are simple graph maps that are injective on vertices and edges,

* $f$ is an epi iff $i$ is a graph isomorphism: epis in $SimpGph$ are simple graph maps that are surjective on vertices.

### Subcategories

Graph theory suggests both full and wide subcategories of $SimpGph$. See [[the category of simple graphs from a graph-theoretic perspective]] for more details.

### Natural numbers object

The category $SimpGph$ has a [[natural numbers object]]; on abstract grounds this is formed by applying the reflection functor

$$L: Set^{C^{op}} \to SimpGph$$

to the natural numbers object in $Set^{C^{op}}$.

## References

* [[Jiří Adámek]] and Horst Herrlich, _Cartesian closed categories, quasitopoi, and topological universes_. Comm. Math. Univ. Carol., Vol. 27, No. 2 (1986), 235-257. ([web](http://dml.cz/handle/10338.dmlcz/106447))
{#AdamHerr}

* [[Tien Chih]], [[Laura Scull]], _Homotopy in the Category of Graphs_, ([arXiv:1901.01619](https://arxiv.org/abs/1901.01619))
{#ChihScul}

* Reinhard Diestel, Graph Theory (Second Edition), Graduate Texts in Mathematics 173, Springer (2000).
{#Diestel}

category: category

[[!redirects category of simple graphs]]
[[!redirects Simp Gph]]
[[!redirects SimpGph]]

[[!redirects simple graph]]
[[!redirects simple graphs]]


category of simple graphs and some graph theory

[[!redirects category of simple graphs and graph theory]]
[[!redirects category of simple graphs and some graph-theoretical questions]]
[[!redirects the category of simple graphs from a graph-theoretic perspective]]
Graph theory studies both full and wide subcategories of the category SimpGph defined in [[category of simple graphs]].
(Needless to say, other categories of graphs have and will be used, but this page focuses on SimpGph.)

#### Wide subcategories

Definition \ref{threeWideSubcategories} gives three examples of wide subcategories, the definitons being graph-theoretic, the names immediately chosen to be category-theoretic, for systematic reasons.

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

Let MonoSimpGph, RegMonoSimpGph, and IsomeRegMonoSimpGph be the wide subcategories of SimpGph with the following hom-sets:

MonoSimpGph($G_0,G_1$) $:=$  $\{$ $f$ $\in$ SimpGph$(G_0,G_1)$: the function $f$ is injective $\}$,

RegMonoSimpGph($G_0,G_1$) $:=$  $\{$ $f$ $\in$ SimpGph$(G_0,G_1)$: $f$ is a graph-isomorphism onto its image $\}$,

IsomeRegMonoSimpGph($G_0,G_1$) $:=$ $\{$ $f$ $\in$ SimpGph$(G_0,G_1)$: $f$ is a graph-isomorphism onto its image, and, for any $x,y\in V$ we have $\mathrm{dist}_{\mathrm{im}(f)}(f(x),f(y))=\mathrm{dist}_{G_1}(f(x),f(y))\}$.

(Here $\mathrm{dist}_G\colon V\times V\rightarrow\omega$ denotes the function which takes any pair of vertices to the length of a shortest graph-theoretic path with these endpoints.)

 =--

Closedness under composition of each of the three subclasses of $\mathrm{Mor}(SimpGph)$ defined by Definition \ref{threeWideSubcategories} is immediate from the respective definitions.

Evident remarks:

* Each of MonoSimpGph, RegMonoSimpGph, IsomeRegMonoSimpGph are [[concrete categories]]. (because of Proposition \ref{EmbeddingOfSimpGphIntoSet})

*  There are the class inclusions (of which the first would be false for multigraphs),

Mor(IsomeRegMonoSimpGph) $\subset$ Mor(RegMonoSimpGph) $\subset$ Mor(MonoSimpGph) $=$ Monos(SimpGph).

* Because of the latter equality, each of MonoSimpGph, RegMonoSimpGph, IsomeRegMonoSimpGph is a [[left cancellative category]]. (And none is a groupoid or a preorder.)

* None of
MonoSimpGph, RegMonoSimpGph, IsomeRegMonoSimpGph has any [[terminal object]].

* The relation $(\emptyset,\emptyset)$, which is symmetric and reflexive, is the (only) initial object of each of MonoSimpGph, RegMonoSimpGph, IsomeRegMonoSimpGph, because this is the only initial object of SimpGph.

We now explain the names of the three subcategories.

* It is evident (using the functor $\Gamma$ from the Proposition [here](/nlab/show/category+of+simple+graphs#EmbeddingOfSimpGphIntoSet) that the morphisms of Simp.Gph.Monos are indeed precisely the [[monos]] of SimpGph.

* How to get from the _definition_ of RegMonoSimpGph given in Definition \ref{threeWideSubcategories} to [[regular monomorphism]]s? Answer: directly by definition of _graph-isomorphism_, Definition \ref{threeWideSubcategories} says that the morphisms $f$ of RegMonoSimpGph are those for which the vertex-set $\mathrm{im}(\Gamma(f))$, with $\Gamma$ from Proposition \ref{EmbeddingOfSimpGphIntoSet}, induces a graph isomorphic to the graph $\mathrm{dom}(f)$. Therefore, these morphisms are those injective graph homomorphisms whose image is an _induced_ subgraph of the target. Further, the term used in the nLab instead of _induced_ is _full_, so the question reduces to why the morphisms $f$ of $SimpGph$ with the two properties that (1) $\Gamma(f)$ is injective and (2) the graph $im(f)$ is a full subgraph of $\mathrm{cod}(f)$, are the regular monos; but sufficient explanations for this _have already been given_ in Section "Equalizers and coequalizers" of [[category of simple graphs]].

* By definition of IsomeRegMonoSimpGph, for each morphism of that subcategory, the graph im($f$) is what in graph theory usually is called an _isometric subgraph_.

Isometric subgraphs have several uses in graph-theoretical considerations, which to describe here would take us too far afield.

## References

* [[Hans-Jürgen Bandelt, Victor Chepoi]], Metric graph theory and geometry: a survey. Contemporary Mathematics, Volume 453, 2008,
{#BandeltChepoi2008}


category of simplices

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Categories of simplices
* table of contents
{: toc}

## Idea

For $X$ a [[simplicial set]] its _category of simplices_ is the [[category]] whose [[objects]] are the [[simplices]] _in_ $X$ and whose [[morphisms]] are maps between these, as simplices in $X$.

In particular the [[subcategory]] on the non-degenerate simplices has a useful interpretation: it is the [[poset]] of subsimplex inclusions whose [[nerve]] is the [[barycentric subdivision]] of $X$, at least if every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, as for a [[simplicial complex]].

## Definition

Let $X \in $ [[sSet]] be a [[simplicial set]].

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

The **category of simplices** of $X$ is equivalently (in increasing order of explicitness)

* the [[category of elements]] of the [[presheaf]] $X_\bullet : \Delta^{op} \to Set$;

* the [[comma category]] $(\Delta\downarrow X)$, where $\Delta$ denotes the [[Yoneda embedding]] $[n] \mapsto \Delta^n$.

* the [[category]] whose [[objects]] are [[homomorphisms]] of simplicial sets $c : \Delta^n \to X$ from a standard simplicial [[simplex]] $\Delta^n$ to $X$, and whose morphisms $c \to c'$ are morphisms $f : \Delta^n \to \Delta^{n'}$ in the [[simplex category]] $\Delta$ such that the [[diagram]]

  $$
    \array{
      \Delta^n
        &&
        \stackrel{f}{\longrightarrow}
        &&
      \Delta^{n'}
      \\
      & {}_{c}\searrow && \swarrow_{c'}
      \\
      && X
    }
  $$

  [[commuting diagram|commutes]].

=--

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

An $n$-[[simplex]] $x\in X_n$ is said to be *non-degenerate* if it is not in the image of any degeneracy map.

Write

$$
  (\Delta\downarrow X)_{nondeg}\hookrightarrow (\Delta\downarrow X)
$$

for the (non-full) subcategory on the non-degenerate simplices.
Notice that a morphism of $\Delta\downarrow X$ with source
a non-degenerate simplex of $X$ is necessary a [[monomorphism]].

This is called the **category of non-degenerate simplices**.

=--

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

If every non-degenerate simplex in $X$ comes from a [[monomorphism]] $\Delta^n \to X$, then the [[nerve]] $N((\Delta \downarrow X)_{nondeg})$ is also called the **[[barycentric subdivision]]** of $X$.

=--

See at _[barycentric subdivision -- Relation to the category of simplices](subdivision#RelationToCategoryOfSimplices)_.

## Properties

### General

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

If $X$ has the property that every face of every non-degenerate simplex is again non-degenerate, then the inclusion of the category of non-degenerate simplices $(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X)$ has a [[left adjoint]] and is hence a [[reflective subcategory]].

=--

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

The category of simplices is a [[Reedy category]].

=--

### Colimits

Write $(\Delta \downarrow X) \to sSet$ for the canonical functor
that sends $(\Delta^n \to X)$ to $\Delta^n$.

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

The [[colimit]] over the [[functor]] $(\Delta \downarrow X) \to sSet$ is $X$ itself

$$
  X \simeq \underset{\to}{\lim}((\Delta \downarrow X) \to sSet)
$$

=--

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

By the [[co-Yoneda lemma]].

=--

In the textbook literature this appears for instance as ([Hovey, lemma 3.1.3](#Hovey)).

+-- {: .num_cor}
###### Corollary

A colimit-preserving [[functor]] $F\colon sSet \to C$ is uniquely determined by its action on the standard simplices:
$$ F(X) \cong colim_{(\Delta\downarrow X)} F(\Delta^\bullet). $$

=--

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

Important colimit-preserving functors out of [[sSet]] include

* [[geometric realization]] of simplicial sets
* [[subdivision]] of simplicial sets
* the [[left adjoint]] to the [[nerve]] functor $N:Cat \to SSet$
* the category of simplices itself, as a functor $SSet \to Cat$; see [category of elements -- colimit preserving](category+of+elements#ColimitPreserving).

=--

### The nerve and subdivision

Let $N\colon$ [[Cat]] $\to$ [[sSet]] denote the simplicial [[nerve]] functor on [[categories]].

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

The functor $sSet \to sSet$ that assigns [[barycentric subdivision]], def. \ref{BarycentricSubdivision},

$$
  X\mapsto N(\Delta\downarrow X)
$$

preserves [[colimits]].

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

An $n$-[[simplex]] of $N(\Delta\downarrow X)$ is determined by a string of $n+1$ composable morphisms
$$ \Delta^{k_n} \to \dots\to \Delta^{k_0} $$
along with a map $\Delta^{k_0} \to X$, i.e. an element of $X_{k_0}$
Thus, each the functor $X\mapsto N(\Delta\downarrow X)_n$ from $SSet \to Set$ is a coproduct of a family of "evaluation" functors.
Since evaluation preserve colimits, coproducts commute with colimits, and colimits in $SSet$ are levelwise, the statement follows.

=--

Therefore, the simplicial set $N(\Delta\downarrow X)$ itself can be computed as a colimit over the category $(\Delta\downarrow X)$ of the simplicial sets $N(\Delta\downarrow \Delta^n)$.

## References

The terminology "category of simplices" is attributed by [Hovey 1999](#Hovey99) to:

* [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], Section 5.9 of: *[[Model Categories and More General Abstract Homotopy Theory]]* (1997) [[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dhk.pdf), [[DwyerHirschhornKan-ModelCategories.pdf:file]]]

Textbook account:

* {#Hovey99} [[Mark Hovey]], Section 3.1 (specifically [p. 75](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf#page=85)) in: *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

[[homotopy final functor|Homotopy finality]] of the non-degenerate simplices:

* {#Lurie} [[Jacob Lurie]], Prop. 4.2.3.14 ([p. 254](https://www.math.ias.edu/~lurie/papers/HTT.pdf#page=272)) of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)]

More on [[barycentric subdivision]]:

* [[J. F. Jardine]], Section 2 of: *Simplicial approximation*, Theory and Applications of Categories, **12** 2 (2004) 34-72 [[tac:12-02](http://www.tac.mta.ca/tac/volumes/12/2/12-02abs.html)]

[[!redirects category of simplices]]
[[!redirects category of simplicies]]
[[!redirects categories of simplices]]

[[!redirects category of nondegenerate simplices]]
[[!redirects categories of nondegenerate simplices]]
[[!redirects category of non-degenerate simplices]]
[[!redirects categories of non-degenerate simplices]]


category of two-sided ideals in a ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The collection of [[two-sided ideals]] in a [[ring]], which happens to have a structure of a [[concrete category]].

## Definition

Given a [[ring]] $R$ in a [[universe]] $\mathcal{U}$, the **category of [[two-sided ideals]]** in $R$ in $\mathcal{U}$ is a [[large category|$\mathcal{U}$-large]] but [[locally small category|locally $\mathcal{U}$-small]] [[bicomplete category|$\mathcal{U}$-small-bicomplete]] [[concrete category|concrete]] [[thin category]] $\mathrm{TwoSidedIdeals}_\mathcal{U}(R)$ whose [[objects]] are the [[two-sided ideals]] in $R$ in $\mathcal{U}$, the [[subbimodule|sub-$R$-$R$-bimodules]] of $R$ in $\mathcal{U}$, and whose [[morphisms]] are the $R$-$R$-[[bimodule monomorphisms]] $f:I \hookrightarrow J$ between two-sided ideals $I$ and $J$ in $R$ in $\mathcal{U}$.

## Properties

The category comes with a [[faithful functor]] $F_{R}:\mathrm{TwoSidedIdeals}_\mathcal{U}(R) \to \mathrm{Bimod}_{R,R}$.

Given a subset $S \subseteq R$ with an [[injection]] $i:S \hookrightarrow R$, [[generalized the|the]] two-sided ideal $I_S$ **generated by** $S$ is defined as the [[initial object|initial]] two-sided ideal in $R$ with a function $j:S \to I_S$. This object always exists because $\mathrm{TwoSidedIdeals}_\mathcal{U}(R)$ is $\mathcal{U}$-small-complete, which means that the [[pullback]] of any $\mathcal{U}$-small [[family]] of $R$-$R$-bimodule monomorphisms with codomain $R$ exists, and thus arbitrary [[intersections]] of two-sided ideals of $R$ exists. $S$ is called a **[[subbase]]**. If $S$ is a [[singleton]], then $I_S$ is called a **[[principal ideal|principal two-sided ideal]]** in $R$.

The two-sided ideal generated by the [[empty set]] $I_\emptyset$ is the **zero two-sided ideal**, and is both the [[initial object|initial]] two-sided ideal in $R$ and the [[zero object|zero]] $R$-$R$-bimodule $0$.

$R$ is [[generalized the|the]] **unit two-sided ideal** or the **improper two-sided ideal** in $R$, the [[terminal object]] in $\mathrm{TwoSidedIdeals}_\mathcal{U}(R)$. A **proper two-sided ideal** is a two-sided ideal $I$ with a [[function]] from the [[set]] $I \cong R$ of $R$-$R$-[[bimodule isomorphisms]] between $I$ and $R$ to the [[empty set]] $f: (I \cong R) \to \emptyset$.

## See also

* [[ring]]

* [[bimodule]]

* [[two-sided ideal]]

* [[category of subobjects]]

[[!redirects category of two-sided ideals]]
[[!redirects category of two-sided ideals of a ring]]


category of V-enriched categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categories of categories
+--{: .hide}
[[!include categories of categories - contents]]
=--
#### Enriched category theory
+-- {: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

For $\mathcal{V}$ a [[closed monoidal category|closed]] and [[complete category|complete]] [[cosmos for enrichment]] there is a [[2-category|$2$-category]] $\mathcal{V} Cat$ whose

* [[objects]] are $V$-[[enriched categories]];

* [[morphisms]] are $V$-[[enriched functors]]; and

* [[2-morphism|$2$-morphisms]] are $V$-[[enriched natural transformations]].

Sometimes one also considers $\mathcal{V} Cat$ as a mere [[category]] by dropping the $2$-morphisms (and using enriched [[strict categories]]).

## Possible Contexts

* $\mathcal{V}$ can be a [[monoidal category]] with underlying category $\mathcal{V}_0$

* $\mathcal{V}$ can be a [[closed category]] with underlying category $\mathcal{V}_0$

* $\mathcal{V}$ can be a [[multicategory]] with underlying category $\mathcal{V}_0$

* $\mathcal{V}$ can be a [[cosmos]] with underlying category $\mathcal{V}_0$

## Examples

* For $\mathcal{V} = ($[[Set]]$, \times)$, $\mathcal{V}Cat \simeq$ [[Cat]], the $2$-category of [[locally small categories]].

* For $\mathcal{V} = ($[[Cat]]$, \times)$, $\mathcal{V}Cat \simeq$ [[2Cat|Str2Cat]], the $2$-category of [[strict 2-categories]].

## Extra structure

### Monoidal structure
 {#MonoidalStructure}

If $\mathcal{V}$ is [[symmetric monoidal category|symemtric monoidal]] then $V Cat$ becomes itself a ([[very large category|very large]]) [[monoidal category]] with [[tensor product]] given by forming [[enriched product categories]].

If $\mathcal{C}$ is in addition [[closed monoidal category|closed]] and [[complete category|complete]], then $\mathcal{V}Cat$ becomes itself a [[closed monoidal category]] with [[internal hom]] given by forming [[enriched functor categories]].

[[Kelly (1982), §2.3](#Kelly82)]

### Involutions
 {#Involutions}

* If $\mathcal{V}$ is a category $\mathcal{V}_0$ equipped with a [[monoidal category|monoidal]] structure, then $\mathcal{V}$Cat has a [[unit enriched category|unit object]] $\mathcal{I}$, and a designated lax natural transformation $[\mathcal{I},-]^{op}\stackrel{{}^-_0L}{\Rightarrow}[[\mathcal{I},-],\mathcal{V}_0]\colon\mathcal{V}$Cat$\to$Cat, where the former is a $2$-functor flipping $2$-morphisms, and the latter is a $2$-functor flipping $1$-morphisms (c.f. [[contravariant functor]]). For the sake of simplicity, we note that $[\mathcal{I},-]$ is simply the forgetful $2$-functor from $\mathcal{V}$Cat to Cat, and hence abbreviate it as $(-)_0$. Then the above lax natural transformation is given by the following data:

1. For every object (i.e. $\mathcal{V}$-enriched category) $\mathcal{A}$ of $\mathcal{V}$Cat, we have to give a functor $\mathcal{A}_0^{op}\stackrel{{}^{\mathcal{A}}_0L}{\rightarrow}[\mathcal{A}_0,\mathcal{V}_0]$. By the cartesian closed structure of the $1$-category Cat, we define these to be the hom-functors $\mathcal{A}_0^{op}\times\mathcal{A}\stackrel{\mathcal{A}(-,-)}{\rightarrow}\mathcal{V}_0$ which are defined in terms of the monoidal structure $\mathcal{V}$ and the $\mathcal{V}$-enrichment data of $\mathcal{A}$ by setting $\mathcal{A}(B,C)\stackrel{\mathcal{A}(f,g)}{\rightarrow}\mathcal{A}(A,D)$ to be the composites $\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}{\to}\mathcal{A}(A,D)$ in $\mathcal{V}_0$.

2. For every $1$-morphism (i.e. $\mathcal{V}$-enriched functor}) $\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}$, we have to give a natural transformation ${}^F_0L$:
$$
\array{
\mathcal{A}_0^{op}&\stackrel{F_0^{op}}{\rightarrow}&\mathcal{B}_0^{op}\\
{}_0^{\mathcal{A}}L\downarrow&\stackrel{{}^F_0L}{\Rightarrow}&\downarrow{}^{\mathcal{B}}L\\
[\mathcal{A}_0,\mathcal{V}_0]&\stackrel{[F_0,\mathcal{V}_0]}{\leftarrow}&[\mathcal{B}_0,\mathcal{V}_0]
}$$
Since we have defined ${}_0^{\mathcal{A}}L$ to be the hom-functor $\mathcal{A}(-,-)$, to give a natural transformation ${}^F_0L$ is to give a natural transformation $\mathcal{A}(-,-)\Rightarrow\mathcal{B}(F_0^{op}-,F_0-)\colon\mathcal{A}_0^{op}\times\mathcal{A}\to\mathcal{V}_0$. We thus define the $ob(\mathcal{A}_0)$-indexed family of morphisms $\mathcal{A}(A,B)\stackrel{{}_0^FL_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B)$ in $\mathcal{V}_0$ to be simply the family of morphisms $\mathcal{A}(A,B)\stackrel{F_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B)$ in $\mathcal{V}_0$ defining the $\mathcal{V}$-enriched functor $\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}$.

3. The lax naturality of ${}^-_0L$ says that for every $2$-morphism (i.e. a $\mathcal{V}$-enriched natural transformation) $F\stackrel{\alpha}{\Rightarrow} G\colon\mathcal{A}\to\mathcal{B}$ in $\mathcal{V}$Cat, the natural transformations ${}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ F_0^{op}$ and ${}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ G_0$ have to satisfy a compatibility condition with the natural transformations $G_0^{op}\stackrel{\alpha_0^{op}}{\Rightarrow}F_0^{op}$ and $[F_0,\mathcal{V}_0]\stackrel{[\alpha_0,\mathcal{V}_0]}{\Rightarrow}[G_0,\mathcal{V}_0]$. Explicitly, the condition is that the composite natural transformation ${}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\stackrel{[\alpha_0,\mathcal{V}_0].({}^{\mathcal{B}}_0L\circ F_0)}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]$ is the same as the composite natural transformation ${}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ G_0^{op}\stackrel{([G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L).\alpha_0^{op}}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]$. Unraveling the condition leaves us with the requirement that for every pair of objects $A,B$ of $\mathcal{A}_0$ the following diagram in $\mathcal{V}_0$ must commute:
$$
\array{
\mathcal{A}(A,B)&\stackrel{F_{A,B}}{\rightarrow}&\mathcal{B}(F_0A,F_0B)\\
G_{A,B}\downarrow&&\downarrow\mathcal{B}(F_0A,(\alpha_0)_{B})\\
\mathcal{B}(G_0A,G_0B)&\stackrel{\mathcal{B}((\alpha_0)_A,G_0B)}{\rightarrow}&\mathcal{B}(F_0A,G_0B)
}$$
But a $\mathcal{V}$-enriched natural transformation $\alpha$ is by definition a collection of morphisms $\alpha_0$ in $\mathcal{B}_0$ such that the above diagram commutes.

* Supposing that $\mathcal{V}_0$ was a self-enriched category, i.e. isomorphic to the underlying category $\mathcal{V}^e_0$ of a $\mathcal{V}$-enriched category $\mathcal{V}^e$, then it is natural to require that the above lax natural transformation $[\mathcal{I},-]\stackrel{{}^-_0L}{\Rightarrow}[(-)_0,\mathcal{V}_0]$ is in fact the whiskering of a lax natural transformation $[\mathcal{I},-]\stackrel{{}^-L}{\Rightarrow}[-,\mathcal{V}^e]$ with the forgetful $2$-functor $(-)_0=[\mathcal{I},-]$. Such a lax natural transformation should give us most (if not all) of the closed structure on $\mathcal{V}^e_0\cong\mathcal{V}_0$...

...

## Related concepts

[[!include categories of categories - contents]]

## References

* {#Kelly82} [[Max Kelly]], Chapter 1 in: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

category: category

[[!redirects VCat]]
[[!redirects V Cat]]

[[!redirects category of V-enriched categories]]
[[!redirects categories of V-enriched categories]]

[[!redirects category of enriched categories]]
[[!redirects categories of enriched categories]]


category over a category

In SGA I.6 [[Grothendieck]] and [[Pierre Gabriel]] presented the formalism of [[Grothendieck fibration|fibered categories]] and descent; the exposition and the terminology emphasises a version of the [[relative point of view]] for categories. Thus they start with preliminaries over the [[overcategory|slice]] [[2-category]] $Cat/C$ of functors with fixed target category $C$. Objects in $Cat/C$ are thus called **categories over $C$** and the morphisms are called **functors over $C$**. Standard passages between different slice categories are via the [[base change]] and the cobase change 2-functors.

An important sub-2-category of $Cat/C$ is the 2-category of [[fibered categories]] over $C$; this is not a [[full subcategory|full]] sub-2-category: one restricts only to **[[cartesian functor]]s** among fibered categories. The 2-categories $Cat/C$ are fibers in a [[codomain fibration|codomain 2-fibration]] $Cat^2$ over $Cat$.

+--{: .query}
[[Mike Shulman]]: Is that really a 2-fibration?  You can pull back along $Cat/C \to Cat/D$ along a functor $f\colon D\to C$, but can you do anything with a natural transformation $f\to f'$?

Zoran: sorry there is indeed an error there, but I will not yet erase it, as there is (in my memory) a partial repair of the statement for which I need to sit down to formulate precisely.
=--

Its restriction to the sub-2-category $Fib$ of fibered categories is then also a Grothendieck 2-fibration $Fib\to Cat$ which has been studied by [[Claudio Hermida]].

* A. Grothendieck, M. Raynaud et al. _Revêtements étales et groupe fondamental_ ([[SGA1|SGA I]]), Lecture Notes in Mathematics __224__, Springer 1971 (retyped as [math.AG/0206203](http://arxiv.org/abs/math/0206203))

* C. Hermida, _Some properties of $Fib$ as a fibred 2-category_, Journal of Pure and Applied Algebra __134__ (1), 83--109, 1999 (earlier draft version can be found at [jpaa.ps.gz](http://maggie.cs.queensu.ca/chermida/papers/jpaa.ps.gz))

[[!redirects categories over a category]]
[[!redirects categories over categories]]


category over an operad

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

## Idea

A _category over an operad_ is the [[horizontal categorification]]
of an [[algebra over an operad|algebra]] over an [[operad]].  It is like an [[enriched category]]
in which the composition operation is not necessarily binary, but
parameterized by the [[operad]].

## Definition

Given an [[operad]] $O$ in some [[symmetric monoidal category]]
$C$, a **category over the operad** $O$, or **$O$-category** $D$ is

* a set/class/whatever $D_0$, called the set of **objects** of $D$;

* for each pair $x,y \in D_0$ an object $D(x,y) \in C$, called the
object of **morphisms** from $x$ to $y$ in $D$;

* for each [[natural number]] $n$ and each sequence $x_0, x_1, \cdots, x_n$
of objects of $D_0$ a morphism
$$ comp_{(x_0, \cdots, x_n)} : \left(D(x_0,x_1) \otimes D(x_1,x_2)
\otimes \cdots \otimes D(x_{n-1},x_n) \right) \otimes O(n) \to D(x_0, x_n)$$
called the **$n$-ary composition** operation;

* such that the composition operations satisfy the obvious
compatibility conditions with the operad composition operation,
directly analogous to those for $O$-algebras.

## Examples

* Let $Associative$ be the ordinary associative operad in [[Set]].
An $Associative$-category is an ordinary [[Set]]-[[enriched category]]
i.e. a [[locally small category]].

* Let $A$ be the ordinary associative operad in vector spaces. An
$A$-category is a [[Vect]]-[[enriched category]].

* We may regard the operad $A$ as a dg-operad i.e. an operad in
the category of [[cochain complex]]es. As such, $A$ has a
resolution, the [[A-infinity-operad]] operad $A_\infty$. An
$A_\infty$-category is an [[A-infinity-category]] (see there).

* A class of definitions of [[infinity-category|infinity-categories]] is operadic in this sense, or in a generalization thereof. See

  * [[Trimble n-category]]

  * [[Eugenia Cheng]], _Comparing operadic definitions of $n$-category_ ([arXiv](http://arxiv.org/abs/0809.2070))


category theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

\section{Idea}

_Category theory_ is a toolset for describing the general abstract structures in [[mathematics]].

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

As opposed to [[set theory]],  category theory focuses not on elements $x,y, \cdots$ -- called [[object]]s -- but on the **relations** between these objects: the ([[homomorphism|homo]])[[morphism]]s between them

$$
  x \stackrel{f}{\to} y
  \,.
$$

Later this will lead naturally on to an infinite sequence of steps:  first [[2-category theory]] which focuses on relation between relations, morphisms between morphisms: [[2-morphism]]s, then 3-category theory, etc. and to various variants, [[bicategories]], [[Gray categories]] .... Eventually this leads to [[higher category theory]], where one considers $k$-morphisms in all dimensions and to a wealth of interacting intuitions and concepts.

=--

The concept that formalizes this is that of a [[category]]: a collection of arrows/morphism that can be [[composition|composed]] if they are adjacent

$$
  \left(
    \array{
       x
       \\
       \downarrow^{\mathrlap{f}}
       \\
       y
       \\
       \downarrow^{\mathrlap{g}}
       \\
       z
    }
  \;\right)
  \;\;\;\;
  \mapsto
  \;\;\;\;

  \left(
    \array{
       x
       \\
       \downarrow^{\mathrlap{g \circ f}}
       \\
       z
    }
  \;\;\;\right)
  \,.
$$

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

The archetypical example of a category is the category [[Set]] of [[set]]s and [[function]]s between sets.

The classical examples of categories are [[concrete category|concrete categories]] whose [[object]]s are [[stuff, structure, property|sets with extra structure]] and whose [[morphism]]s are structure preserving functions of sets, such as [[Top]], [[Grp]], [[Vect]].  These are the examples from which the term _category_ derives: these categories literally _categorize_ mathematical structures by packing structures of the same _type_ (same category) and structure preserving mappings between them into a single whole structure, a category.

But it is far from the case that all categories are of this type. Categories are much more versatile than these classical examples suggest. After all, a [[category]] is just a [[quiver]] (a [[directed graph]]) with a notion of composition of its edges. As such it generalizes the concepts of [[monoid]] and [[poset]]. If the category is a [[groupoid]], it generalizes the concept of [[group]] (in a sense called [[horizontal categorification]]) and also the concept of [[equivalence relation]].  Thinking of a category as a generalized poset is particularly useful when studying [[limits]] and [[adjoint functors|adjunctions]].

Archetypical examples of non-[[concrete category|concrete]] categories are the [[fundamental groupoid]] of a [[topological space]] and the [[fundamental category]] of a [[directed space]].

=--

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

Categories were named after the examples of [[concrete category|concrete categories]]. As [[Saunders Mac Lane]] writes

>Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: "Category" from Aristotle and Kant, "Functor" from Carnap (Logische Syntax der Sprache), and "natural transformation" from then current informal parlance.

>([[Saunders Mac Lane]], *[[Categories for the Working Mathematician]]*, 29–30).

However, the [[category|categories]] of category theory are way more general than these [[concrete category|concrete categories]], and the way Aristotle and Kant use the [[category (philosophy)|term]] in philosophy is not particularly related to what Eilenberg & Mac Lane did with it.

=--

+-- {: .un_remark}
###### The basic trinity of concepts ####

Category theory reflects on itself. Categories are about collections of morphisms. And there are evident morphisms between categories: [[functor]]s. And there are evident morphisms between functors: [[natural transformation]]s.

This trinity of concepts

1. [[category]]

1. [[functor]]

1. [[natural transformation]]

is what category theory is built on.

In [[higher category theory]] this continues with

* $k$-[[transfor]]s for all $k \in \mathbb{N}$.

=--

+-- {: .un_remark}
###### Conceptual unification ####

A major driving force behind the development of category theory is its ability to abstract and unify concepts. General statements about categories apply to each specific [[concrete category]] of mathematical structures. The general notion of [[universal constructions]] in categories, such as [[representable functors]], [[adjoint functors]]  and [[limits]], turns out to prevail throughout mathematics and manifest itself in myriads of special examples.

=--

+-- {: .un_remark}
###### Abstract nonsense ####

This abstraction power of category theory has led Norman Steenrod to coin the term _abstract nonsense_ or _general abstract nonsense_ for it. It is being used as in "This property is not specific to this context, it already follows from general abstract nonsense". Peter Freyd expressed a similar feeling by his witticism:

> {#WhatIsTriviallyTrivial} "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial."

But abstract nonsense still tends to meet with some resistance. In the preface of [Mitchell (1965)](#Mitchell65) it says:

> A number of sophisticated people tend to disparage category theory as consistently as others disparage certain kinds of classical music. When obliged to speak of a category they do so in an apologetic tone, similar to the way some say, "It was a gift – I've never even played it" when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisite that the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind.

=--

+-- {: .un_remark}
###### The nPOV ####

The vast applicability and expressiveness of category theory leads to the observation that most structures in mathematics are best understood from a category theoretic or higher category theoretic viewpoint. This is the [[nPOV]].

=--

\section{The central constructions}

\subsection{Presheaves}

Much of the power of category theory rests in the fact that it reflects on itself. For instance that [[functors]] between two categories form themselves a category: the [[functor category]].

This leads to the notion of [[presheaf categories]] and [[sheaf toposes]]. Much of category theory is [[topos theory]].

Under [[Isbell duality]] this sets the stage for everything in mathematics related to [[space]] and [[algebra]] and their duality.

\subsection{Universal constructions}

Elementary as it is, the definition of a [[category]] supports a powerful set of constructions: [[universal constructions]]. These include

* [[limit]] and [[colimit]];

* [[adjunction]];

* [[Kan extension]]

* [[end]] and [[coend]].

All these are special cases of each other and thus reflect different aspect of one single phenomenon. Applying category theory means applying these constructions in specific situations and using general abstract theorems for deducing statements about concrete contexts.

\subsection{The central theorems}

Category theory has a handful of central lemmas and theorems. Their proof is typically easy, sometimes almost tautological. Their power rests in the fact that they apply over and over again all over mathematics. Many concrete constructions get simplified by observing that they are but special realizations of these general abstract results in category theory. Among these central theorems are

* first and foremost: the [[Yoneda lemma]];

* [[Isbell duality]];

* the [[Grothendieck construction]];

* the [[adjoint functor theorem]];

* the [[monadicity theorem]];

* [[Tannaka duality]].

\section{Applications}

For a detailed list of applications see

* [[applications of (higher) category theory]].

\subsection{In pure mathematics}

Apart from its general role in mathematics, category theory provides the high-level language for

* [[logic]] / [[type theory]]

* [[higher algebra]]

* [[higher geometry]] .

\subsection{Outside of mathematics}

Outside of pure mathematics, category theory finds major applications in

* fundamental [[physics]] -- see [[higher category theory and physics]]

* theoretical [[computer science]]

and is increasingly finding applications in such diverse areas as chemistry, network theory and natural language processing. Further information can be found on the [[applied category theory]] page.

\section{Contrast to theories of other objects}

\subsection{Category theory vs. set theory}

Here set theory is assumed to be a theory of the usual concept of sets, that is *material* [[set theory]].

No one of these is more fundamental than the other as a foundation of mathematics. Category theory is a holistic (structural) approach to mathematics that can (through such methods as Lawvere's [[ETCS]]) provide [[foundations]] of mathematics and (through [[algebraic set theory]]) reproduce all the different axiomatic set theories; elementary category theory does not need the concept of set to be formulated. Set theory is an analytic approach (element-wise) and can reproduce category theory by simply defining all the concepts in the usual way, as long as one include a technique to handle large categories (for instance by using [[class]]es instead of sets, or by including as an axiom that an uncountable [[inaccessible cardinal]] exists or even that [[Grothendieck universe]]s exist).

|Set theory| |Category theory|
|----------|-|---------------|
|membership relation| |-|
|sets| |categories|
|elements| |objects|
|-| |morphisms|
|functions| |functors|
|equations between elements| |isomorphisms between objects|
|equations between sets| |equivalences between categories|
|equations between functions| |natural transformations between functors|

Lawvere pointed out that set theory is axiomatized by a binary membership relation while category theory is axiomatized by a ternary composition relation.

The process of going from sets to categories is a special case of [[categorification]] and the reverse process is a special case of [[decategorification]].

For a philosophical consideration of foundations covering and comparing sets, structuralism (a la Bourbaki?) and categories, see the article

* Sets, categories and structuralism - Costas Drossos

\subsection{Category theory vs. order theory}

A category may be thought of as a [[categorification]] of a [[poset]] rather than of a [[set]]; much (but by no means all) of category theory also appears in [[order theory]].

See [[category theory vs order theory]] for more discussion.

\section{Theorems}
 {#Theorems}

* [[Yoneda lemma]]

* [[Isbell duality]]

* [[Grothendieck construction]]

* [[adjoint functor theorem]]

* [[monadicity theorem]]

* [[adjoint lifting theorem]]

* [[Tannaka duality]]

* [[Gabriel-Ulmer duality]]

* [[small object argument]]

* [[Freyd-Mitchell embedding theorem]]

\section{Related concepts}

* [[formal category theory]]

* [[enriched category theory]]

* [[internal category]] [[internalization|theory]]

[[!include table of category theories]]

* [[type theory]]

  * [[relation between category theory and type theory]]

* [[computer science]]

  * [[computational trinitarianism]]

\section{References}
 {#References}

For more see the references at *[[category]]*.

\subsection{History}
 {#History}

The concepts of _[[category]]_, _[[functor]]_ and _[[natural transformation]]_ were introduced in

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], *[[General Theory of Natural Equivalences]]*,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

apparently (see [there](Hans+Freudenthal#AsAForerunnerOfCategoryTheory)) taking inspiration from:

* {#Freudenthal37} [[Hans Freudenthal]], *Entwicklungen von Räumen und ihren Gruppen*, Comp. Math. **4** (1937) 145-234. [[numdam](http://www.numdam.org/item/CM_1937__4__145_0/)]

The rational for introducing the concept of [[categories]] was to introduce the concept of [[functors]], and the reason for introducing functors was to introduce the concept of [[natural transformations]] (more specifically [[natural equivalences]]) in order to make precise the meaning of "natural" means in mathematics:

> If [[topology]] were publicly defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology. The mathematical correctness of such a definition reveals nothing about topology except that its basic axioms can be made quite simple. And with category theory we are confronted with the same pedagogical problem. The basic axioms, which we will shortly be forced to give, are much too simple.

> A better (albeit not perfect) description of topology is that it is the study of continuous maps; and category theory is likewise better described as the theory of functors. Both descriptions are logically inadmissible as initial definitions, but they more accurately reflect both the present and the historical motivations of the subjects.

> {#CategoriesForNaturalTransformations} It is not too misleading, at least historically, to say that categories are what one must define in order to define natural transformations.

> (from [Freyd 64, page 1](#Freyd64))

and

> {#CategoriesForNaturalTransformationsII} Category theory is an embodiment of Klein's dictum that it is the maps that count in mathematics. If the dictum is true, then it is the functors between categories that are important, not the categories. And such is the case. Indeed, the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations. And it was in order to define these that Eilenberg and MacLane first defined functors.

> (from [Freyd 65, beginning of Part Two](#Freyd65))

The paper [Eilenberg-Maclane 45](#EilenbergMacLane45) was a clash of ideas from abstract [[algebra]] (Mac Lane) and [[topology]]/[[homotopy theory]] (Eilenberg). It was first rejected on the ground that it had no content but was later published. Since then category theory has flourished into almost all areas of mathematics, has found many applications outside mathematics and even attempts to build a [[foundations]] of mathematics.

This and much more history is recalled in

* {#MacLaneHistory} [[Saunders MacLane]], _Concepts and Categories in Perspective_, AMS ([pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf))

* [[Ralf Krömer]], *Tool and object: A history and philosophy of category theory*, Science Networks. Historical Studies **32** (2007) [[doi:10.1007/978-3-7643-7524-9](https://doi.org/10.1007/978-3-7643-7524-9)]

See also:

* [[Jean-Pierre Marquis]], *What is Category Theory?*, in G. Sica (ed.) *What is Category Theory?*, Polimetrica (2006) 221-256 [[[Marquis-CategoryTheory.pdf:file]], [semanticscholar](https://www.semanticscholar.org/paper/What-is-category-theory-Sica/7737401202b220f89ae9c11cf9af65995a4dcf50)]

* [[Jean-Pierre Marquis]], _From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory_, Springer (2009) [[doi:10.1007/978-1-4020-9384-5](https://doi.org/10.1007/978-1-4020-9384-5)]

\subsection{Textbooks}
 {#TextBooks}

\subsubsection{Basic category theory}
 {#BasicTextBooks}

* {#Freyd64} [[Peter Freyd]], _Abelian Categories -- An Introduction to the theory of functors_, Harper and Row (1964), Reprints in Theory and Applications of Categories, **3** (2003)  [[tac:tr3](http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf)]

* {#Freyd65} [[Peter Freyd]], _The theories of functors and models_, in: Proceedings of _Symposium on the Theory of Models_, North Holland (1965) [[doi:10.1016/C2013-0-11897-1](https://doi.org/10.1016/C2013-0-11897-1)]

* {#Mitchell65} [[Barry Mitchell]], *Theory of categories*, Pure and Applied Mathematics **17**, Academic Press (1965) [[ISBN:978-0-12-499250-4](https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4)]

* [[Saunders MacLane]] (notes by [[Ellis Cooper]]), *[[Lectures on category theory]]*, Bowdoin Summer School (1969)

* {#Hilton69} [[Peter Hilton]] (ed.) _Category Theory, Homology Theory and Their Applications_,

  vol 1: Lecture Notes in Mathematics **86**, Springer (1969) [[doi:10.1007/BFb0079380](https://doi.org/10.1007/BFb0079380)]

  vol 2: Lecture Notes in Mathematics **92**, Springer (1969) [[doi:10.1007/BFb0080761](https://doi.org/10.1007/BFb0080761)]

  vol 3: Lecture Notes in Mathematics **99**, Springer (1969) [[doi:10.1007/BFb0081959](https://doi.org/10.1007/BFb0081959)]

* {#Pareigis70} [[Bodo Pareigis]], *Categories and Functors*, Pure and Applied Mathematics **39**, Academic Press (1970) [[doi:10.5282/ubm/epub.7244](https://doi.org/10.5282/ubm/epub.7244), [pdf](https://epub.ub.uni-muenchen.de/7244/1/7244.pdf)]

* {#MacLane97} [[Saunders MacLane]], _[[Categories for the Working Mathematician]]_, Graduate texts in mathematics **5**, Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Gray74} [[John Gray]], *[[Adjointness for 2-Categories|Formal category theory: adjointness for $2$-categories]]*, Lecture Notes in Mathematics, **391**, Springer 1974 ([doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280))

   ([[formal category theory]] in the [[2-category]] [[Cat]])

* [[Horst Schubert]], *Categories*, Springer (1972) [[doi:10.1007/978-3-642-65364-3](https://doi.org/10.1007/978-3-642-65364-3)]

* {#Geroch85} [[Robert Geroch]], *Mathematical Physics*, University of Chicago Press (1985) [[ISBN:9780226223063](https://press.uchicago.edu/ucp/books/book/chicago/M/bo4158035.html)]

  > (introduces categories by examples arising in [[mathematical physics]])

* [[Jiri Adamek]], [[Horst Herrlich]], [[George Strecker]], *[[Abstract and Concrete Categories]]*, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 [[tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/)]

* [[Peter Freyd]], [[Andre Scedrov]], *[[Categories, Allegories]]*, Mathematical Library Vol 39, North-Holland (1990) ([ISBN 978-0-444-70368-2](https://www.elsevier.com/books/categories-allegories/freyd/978-0-444-70368-2))

* [[Benjamin Pierce]], _Basic category theory for computer scientists_, 1991 20 ([publisher](https://mitpress.mit.edu/books/basic-category-theory-computer-scientists))

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)

* [[Roy L. Crole]], *Categories for types*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139172707](https://doi.org/10.1017/CBO9781139172707)]

  > ([[categorical semantics]] of [[type theory]])

* {#BarrWells95} [[Michael Barr]], [[Charles Wells]], *Category theory for computing science*, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories **22** (2012) 1-538 [[pdf](http://www.math.mcgill.ca/barr/papers/ctcs.pdf), [tac:tr22](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html)]

  > (aimed at [[computer science]], see *[[computational trilogy]]*)

* {#Awodey06} [[Steve Awodey]], *Category theory*, Oxford University Press (2006, 2010) [[doi:10.1093/acprof:oso/9780198568612.001.0001](https://doi.org/10.1093/acprof:oso/9780198568612.001.0001), [ISBN:9780199237180](https://global.oup.com/ukhe/product/category-theory-9780199237180), [pdf](http://englishonlineclub.com/pdf/Category%20Teory%20%5BEnglishOnlineClub.com%5D.pdf)]

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, 2006

* {#LawvereSchanuel09} [[F. William Lawvere]] and [[Stephen Schanuel]], _Conceptual Mathematics: A first introduction to categories_, $2^{nd}$ Edition, Cambridge University Press 2009 ([pdf](https://img.4plebs.org/boards/tg/image/1460/05/1460059215690.pdf))

* [[David Spivak]], _Category theory for scientists_ ([arXiv:1302.6946](http://arxiv.org/abs/1302.6946))

* [[Tom Leinster]], _[Basic Category Theory](http://www.maths.ed.ac.uk/~tl/bct/)_, 2014

* {#RiehlCTInContext} [[Emily Riehl]], _[[Category Theory in Context]]_, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf)]

* [[Martin Brandenburg]], *Einführung in die Kategorientheorie*, Springer 2017

  ([doi:10.1007/978-3-662-53521-9](https://link.springer.com/book/10.1007/978-3-662-53521-9))

* {#FongSpivak18} [[Brendan Fong]], [[David Spivak]], _An invitation to applied category theory_, 2018 ([web](http://math.mit.edu/~dspivak/teaching/sp18/), [pdf](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf))

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], *Categories for Quantum Theory*, Oxford University Press 2019 ([ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616))

  > (emphasis on [[monoidal category]]-theory with an eye towards [[finite quantum mechanics in terms of dagger-compact categories]] and [[quantum computation]])

* [[Marco Grandis]], _Category Theory and Applications: A Textbook for Beginners_, World Scientific, 2021 ([doi:10.1142/12253](https://doi.org/10.1142/12253))

On category theory in [[computer science]]/[[programming languages]] (such as for [[monads in computer science]]):

* [[Bartosz Milewski]] (compiled by Igal Tabachnik), *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

\subsubsection{Topos theory}
 {#ReferencesToposTheory}

Monographs with focus on [[topos theory]]:

* [[Peter Johnstone]], _Topos theory_, 1977.

* [[Michael Barr]], [[Charles Wells]], _[[Toposes, Triples, and Theories]]_, Grundlehren der math. Wissenschaften 278 Springer 1983 ([TAC reprints](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html))
  > (Here "triple" means" [[monad]]").

* [[Robert Goldblatt]], _Topoi, the categorial analysis of logic_, 1984 ([gbnooks](https://books.google.us/books/about/Topoi.html?id=AwLc-12-7LMC))

* [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

* [[Ieke Moerdijk]], [[Saunders Mac Lane]], _[[Sheaves in Geometry and Logic]]_ Springer 1992

* [[Peter Johnstone]], _[[Elephant|Sketches of an Elephant]]_, vol 1-2, Oxford University Press 2002

\subsubsection{Higher category theory}

* [[Carlos Simpson]], _[[Homotopy Theory of Higher Categories]]_ ([pdf](http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf))

* Project description: higher categorical structures and their applications ([pdf] (http://www.math.uchicago.edu/~may/NCATS/ForWeb.pdf))

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf))

* [[Tom Leinster]], _Higher operads, higher categories_, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)

* [[Eugenia Cheng]], [[Aaron Lauda]], _Higher-dimensional categories: an illustrated guide book_ [free online] (http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf)

Towards [[homotopy theory]]:

* [[Birgit Richter]], _From categories to homotopy theory_, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 ([doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf))

\subsection{Foundations}

The [[foundation]] of category theory in [[homotopy type theory]] (see at _[[internal category in homotopy type theory]]_) is discussed in

* [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], _Univalent categories and the Rezk completion_ ([arXiv:1303.0584](http://arxiv.org/abs/1303.0584))

\subsection{Course notes}
 {#CourseNotes}

* [[Jaap van Oosten]], _Basic category theory_ (2002) [[pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf)]

* [[Thomas Streicher]], *Introduction to Category Theory and Categorical Logic* (2003) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf), [[Streicher-CategoryTheory.pdf:file]]]

* [[Bodo Pareigis]], _Category theory_ (2004) [[dvi](http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/04SS/Cats1.dvi), [ps](http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/04SS/Cats1.ps), [pdf](http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/04SS/Cats1.pdf)]

* [[Tom Leinster]], _Notes on basic category theory_ (2007) [[web](http://www.maths.ed.ac.uk/~tl/msci)]

* [[David Spivak]], _Category theory for scientists_ [[arXiv:1302.6946](http://arxiv.org/abs/1302.6946)]

* [[Peter Johnstone]], _Category Theory_ , Lecture notes taken by David Mehrle, University of Cambridge (2015) [[pdf](http://pi.math.cornell.edu/~dmehrle/notes/partiii/cattheory_partiii_notes.pdf)]

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_ (2019) [[arXiv:1912.10642](http://arxiv.org/abs/1912.10642)&rbrack

* [[André Joyal]], *[[joyalscatlab:Categories]]*

* [[Benedikt Ahrens]] and Kobe Wullaert, _Category Theory for Programming_ (2022)  [[arXiv:2209.01259](https://arxiv.org/abs/2209.01259)]

\subsection{Videos}

* The Catsters, Videos on various topics in category theory. ([YouTube link](https://www.youtube.com/user/TheCatsters))

Videos at an introductory level that cover basic concepts and constructions of category theory. The Catsters are [[Eugenia Cheng]] and [[Simon Willerton]] (anyone else?).

* {#LaGatta} [[Tom LaGatta]], _Category theory_ ([video](http://www.youtube.com/watch?v=o6L6XeNdd_k))

Enthusiastic, mostly nontechnical talk given by a probability theorist, made for an audience innocent of any exposure to category theory.

\subsection{Relation to philosophy}
 {#ReferencesRelationToPhilosophy}

Discussion of the relation to and motivation from the [[philosophy of mathematics]] includes

*  [[Colin McLarty]], _The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics_,Brit. J. Phil. Sci. 2007 ([pdf](http://www.cwru.edu/artsci/phil/BJPSMacLane.pdf))

\subsection{Further resources and links}

There are several networks of category theorists organised, initially, at a national level and aiming to join forces to organise conferences, online seminars, etc.:

* _[[networks of category theorists]]_.

[[!redirects abstract nonsense]]
[[!redirects general abstract nonsense]]


category theory - contents

**[[category theory]]**

## Concepts {#sidebar_concepts}

* [[category]]

* [[functor]]

* [[natural transformation]]

* [[Cat]]

## Universal constructions {#sidebar_universal_constructions}

* [[universal construction]]

  * [[representable functor]]

  * [[adjoint functor]]

  * [[limit]]/[[colimit]]

  * [[weighted limit]]

  * [[end]]/[[coend]]

  * [[Kan extension]]

## Theorems {#sidebar_theorems}

* [[Yoneda lemma]]

* [[Isbell duality]]

* [[Grothendieck construction]]

* [[adjoint functor theorem]]

* [[monadicity theorem]]

* [[adjoint lifting theorem]]

* [[Tannaka duality]]

* [[Gabriel-Ulmer duality]]

* [[small object argument]]

* [[Freyd-Mitchell embedding theorem]]

* [[relation between type theory and category theory]]

## Extensions {#sidebar_extensions}

* [[sheaf and topos theory]]

* [[enriched category theory]]

* [[higher category theory]]

## Applications {#sidebar_applications}

* [[applications of (higher) category theory]]

[Edit this sidebar](/nlab/edit/category+theory+-+contents)


category theory and foundations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

\tableofcontents

## Overview

There are two big questions about [[category theory]] and the logical [[foundations of mathematics]]:

 1. What foundations are adequate to formulate category theory?
 2. How can category theory provide a foundation for mathematics?

These questions also apply to [[higher category theory]], which also involves the relation between them. (2-categories as a foundation for categories, for example.)

## Mathematical foundations of category theory
{#MFCT}

The problem with mathematical foundations of category theory is that in category theory we frequently speak of [[large category|large categories]], which it is tricky to deal with rigorously in the usual sort of [[set theory|set theories]].

One common view seems to be to found category theory on a theory of sets and classes; see [the English Wikipedia\'s definition](https://en.wikipedia.org/wiki/Category_%28mathematics%29#Definition), for example. But the standard reference, Saunders Mac Lane\'s _Categories for the Working Mathematician_, assumes the existence of a universe (an inaccessible cardinal) instead. Both of these approaches rely on a distinction between *small* and *large* categories. There is a category of all [[small category|small categories]], but this category is not itself small; there is no category of all categories.

[[Alexander Grothendieck]] used more (although [apparently](http://permalink.gmane.org/gmane.science.mathematics.categories/6847) he did not need to); he used what we now call [[Grothendieck universes]]\. He assumed that every set is contained within a universe; that is, for every cardinal number $\kappa$, there is a cardinal inaccessible from $\kappa$. (This is still a rather moderate axiom, compared to some of the large-cardinal axioms studied by set theorists.) Now one has a relative notion of small and large; the category of all $U$-small categories (where $U$ is some universe) is $U$-large but must be $U'$-small for some other universe $U'$, and there exists a category (which is both $U$-large and $U'$-large) of all $U'$-small categories.

If one does not accept the [[axiom of choice]], then there are additional complications in general category theory. In particular, one must distinguish between a universal *property* (for example, having all [[product]]s) and having a universal *structure* realising that property (in the example, a functor taking each pair $(x,y)$ of objects to a specific product cone $x \leftarrow x \times y \rightarrow y$). This difficulty was overcome by Michael Makkai using [[anafunctor]]s, but these have not been widely adopted, even by constructivists.

For a discussion of set theoretic foundations of category theory see the references [below](#ReferencesSetTheoreticFoundationsOfCategoryTheory).

### The concept of _identity_
{#Identity}

One way to think of category theory is as a framework in which the idea is formalized that every kind of [[equality]] is really secretly a choice of [[isomorphism]] or [[equivalence]]. In some sense the notion of identity potentially breaks the [[principle of equivalence]].

[[Michael Makkai]] works on a language, [[FOLDS]] ('first-order logic with dependent sorts'), which is designed to make it impossible to formulate any statements that break the [[principle of equivalence]].

## Categorial foundations of mathematics
{#CFM}

[[Bill Lawvere]] proposed to found mathematics on [[ETCC]] (for ' Elementary Theory of the Category of Categories'), a first-order axiomatisation of [[Cat|the category of categories]]. This has not been very successful,
but his other proposal, a first-order axiomatisation of [[Set|the category of sets]], works well. These and related approaches to foundations may be called _structural_ or _categorial_ (or _categorical_, which is more common but clashes with another sense of 'categorical' in logic).

Lawvere\'s system [[ETCS]] (for 'the Elementary Theory of the Category of Sets') essentially states that the category of sets is a [[topos]] with certain properties, in particular a [[well-pointed topos]]. This can be stated in elementary (first-order) terms; indeed, the system of axioms for ETCS in 1965 was retrospectively an important step in Lawvere's quest to give a first-order axiomatization for the topos concept that was originally formulated in higher-order terms by the Grothendieck school, resulting in 1970 in the by the now-default notion of *elementary* topos that subsumes the original notion, now called [[Grothendieck topos]], as an important special case.

It is also possible to found mathematics on the [[internal logic|internal language]] of a topos. In this case, the topos need *not* be well-pointed (and indeed, the condition that a topos be well-pointed cannot be stated in its own internal language; or if you prefer, *every* topos is well-pointed *internally*). This is equivalent to a certain formulation of [[type theory]], so it is (in a sense) nothing new, although it leads to new perspectives, as in the next paragraph.

Categories (not just toposes) can serve as models of [[type theory|type theories]], each type theory corresponding to a certain class of categories. Toposes correspond directly to a constructive but impredicative type theory; to make the theory predicative (in the constructivists\' sense) you generalise to a [[pretopos]] (maybe [[locally cartesian closed category|locally cartesian closed]]), to make the theory nonconstructive you specialise to a [[Boolean topos]], and so on. More specifically, every category\'s internal language is a type theory (with many odd constants), and every type theory (of appropriate form) defines a category (its free model); this is an [[adjunction]] between categories and type theories. [[Paul Taylor]]\'s book _[[Practical Foundations|Practical Foundations of Mathematics]]_ is essentially all about this subject, as is (at a more advanced level) most of the career of [[Michael Makkai]].

[[Jim Lambek]] proposed to use the [[free topos]] as ambient world to do mathematics in. Being syntactically constructed, but universally determined, with higher-order [[intuitionistic type theory]] as [[internal language]] he saw it as a reconciliation of the three classical schools of [[philosophy of mathematics]], namely [[formalism]], [[platonism]], and [[intuitionism]]. His latest views on this variant of categorical foundations can be found in ([Lambek-Scott 2011](#LS11)).

Certain 'strong' axioms of [[set theory]] (those involving [[quantification]] over all sets) are difficult to state in category-theoretic (or type-theoretic) terms, but this can be overcome in a theory like ETCS; talk to Mike Shulman. (Ironically, this makes it harder to do foundations with categorial foundations!)

In contrast, many of the optional or controversial axioms of set theory (such as the [[axiom of choice]]) can be stated quite directly in ETCS. These can be examined quite well in a naïve set-theoretic language that never need be precise about whether one\'s foundations are traditional (membership-based), categorial, or whatever.

## Categorial foundations of category theory
{#CFCT}

Relevant topics: [[ETCC]], [[FOLDS]], [[homotopy type theory]]

## Related topics

* [[foundations of mathematics]]

* [[category theory]]

* [[univalent foundations of mathematics]]

## References

### General

* [[William Lawvere]], _Foundations and applications: axiomatization and education_, Bulletin of Symbolic Logic 9 (2003), 213-224 ([ps](http://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps), [Euclid](http://projecteuclid.org/euclid.bsl/1052669290))

The following contains a careful discussion of [[Gödel's incompleteness theorem]] in the context of categorical foundations using the free topos:

* [[Jim Lambek]], [[Phil Scott]], _Reflections on Categorical Foundations of Mathematics_ , pp.171-185 in  Sommaruga (ed.), _Foundational Theories of Classical and Constructive Mathematics_,  Springer New York 2011. ([draft](https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf)) {#LS11}

Some old discussions about category theory and foundations are archived [here](http://nforum.ncatlab.org/discussion/4172/foundation-of-mathematics/?Focus=33992#Comment_33992)

### Set theoretic foundations of category theory
 {#ReferencesSetTheoreticFoundationsOfCategoryTheory}

Without assuming  [[Grothendieck universes]]:

* {#Rao06} [[Vidhyanath K. Rao]], _On Doing Category Theory within Set Theoretic Foundations_ in: Giandomenico Sico (ed.), _What is category theory?_, Polimetrica (2006) [[[Rao-CategoryTheory.pdf:file]], [semanticscholar](https://www.semanticscholar.org/paper/What-is-category-theory-Sica/7737401202b220f89ae9c11cf9af65995a4dcf50)],

Via [[Grothendieck universes]]:

* [[Horst Schubert]], §3 in: *Categories*, Springer (1972) [[doi:10.1007/978-3-642-65364-3](https://doi.org/10.1007/978-3-642-65364-3)]

* [[Mike Shulman]], _Set theory for category theory_ [[arXiv:0810.1279](http://arxiv.org/abs/0810.1279)]

* {#Murfet06} [[Daniel Murfet]], _Foundations for category theory_ (2006) [[pdf](http://therisingsea.org/notes/FoundationsForCategoryTheory.pdf), [[Murfet-GrothendieckUniverses.pdf:file]]]

* [[Zhen Lin Low]], _Universes for category theory_ [[arxiv/1304.5227](http://arxiv.org/abs/1304.5227)]

* [[Paul Blain Levy]], _Formulating Categorical Concepts using Classes_ [[arXiv:1801.08528](https://arxiv.org/abs/1801.08528)]


Category Theory conference

The **Category Theory** (or simply **CT**) conference is an international annual conference on the subject of [[category theory]].

*The list below is incomplete.*

|Year|Location|Website|Abstracts|
|----|--------|-------|---------|
|2023|Louvain-la-Neuve|[website](https://sites.uclouvain.be/ct2023/), [recordings](https://www.youtube.com/playlist?list=PLu4STGsfbix9l6rPxGsjG6Gl822k1hj2X)| [pdf](https://sites.uclouvain.be/ct2023/img/abstracts.pdf)|
|2022|*no event due to lack of organisers*| | |
|2021|Genoa|[website](https://sites.google.com/view/ct2021/)| |
|2020|*no event due to COVID-19 pandemic*| | |
|2019|Edinburgh|[website](https://conferences.inf.ed.ac.uk/ct2019/)| |
|2018|Ponta Delgada|[website](http://www.mat.uc.pt/~ct2018/)| |
|2017|Vancouver|[website](http://www.mat.uc.pt/~ct2017/)| |
|2016|Halifax|[archive](https://web.archive.org/web/20160602211513/http://mysite.science.uottawa.ca/phofstra/CT2016/CT2016.htm)| |
|2015|Aveiro|[website](https://sweet.ua.pt/dirk/ct2015/)| |
|2014|Cambridge|[website](http://julia-goedecke.de/CT2014/)| |
|2013|Sydney|[website](http://web.science.mq.edu.au/groups/coact/seminar/ct2013/)| |
|2012|*Data missing*|*Data missing*| |
|2011|Vancouver|[website](http://www.mat.uc.pt/~ct2011/)| |
|2010|Genoa|[archive](https://web.archive.org/web/20120115044855/http://ct2010.disi.unige.it/)| |
|2009|Cape Town|[website](http://137.158.44.70/~janelidze/ct2009/Index.html)| |
|2008|Calais|[website](http://www-lmpa.univ-littoral.fr/CT08/)| |
|2007|Carvoeiro|[website](https://www.mat.uc.pt/~categ/ct2007/)| |
|2006|Nova Scotia|[website](https://www.mscs.dal.ca/~selinger/ct2006/)| |
|2005|*Data missing*|*Data missing*| |
|2004|Vancouver|[archive](https://web.archive.org/web/20050209223404/http://www.pims.math.ca/science/2004/CT04/)| [pdf](https://web.archive.org/web/20070710174825/http://www.pims.math.ca/science/2004/CT04/ct04_abstracts.pdf)|
|2003|*Data missing*|*Data missing*| |
|2002|*Data missing*|*Data missing*| |
|2001|*Data missing*|*Data missing*| |
|2000|Como|[archive](https://web.archive.org/web/20060114193307/https://www.disi.unige.it/conferences/ct2000/)| |
|1999|Coimbra|[website](http://www.mat.uc.pt/~ct99/)|[invited abstracts](https://www.mat.uc.pt/~ct99/invited.html), [contributed abstracts](https://www.mat.uc.pt/~ct99/talks.html)|
|1998|*no event*|*Data missing*| |
|1997|Vancouver|*Data missing*| |
|1996|Isle of Thorns|*Data missing*| |
|1995|Halifax|[website](https://www.mta.ca/~cat-dist/ct95.html)|[pdf](https://www.mta.ca/~cat-dist/ct95abs.pdf)|
|1994|Tours|*Data missing*| |
|1993|*no event*|*Data missing*| |
|1992|Isle of Thorns|*Data missing*| |
|1991|Montreal|*Data missing*| |
|1990|Como|*Data missing*| |
|1989|Bangor|*Data missing*| |
|1988|Isle of Thorns|*Data missing*| |
|1987|Louvain-la-Neuve|*Data missing*| |
|1986|Cambridge|*Data missing*| |
|1985|Isle of Thorns (University of Sussex)|*Data missing*| |

The following is a quote from [Rota 1993](#Rota) on the proceedings of the Category Theory conference in held 1991:
> It is good to know that category theory is alive and well after all these years. We were turned off to category theory by the excesses of the sixties, when a small but loud crowd pretended to reduce all of mathematics to the language of categories. Now at least they have toned down their claims, and category theory has taken its modest place in the mathematical spectrum side by side with lattice theory, more pretentious perhaps than the latter, but with a good pedigree. We wish there were two versions of each of the papers presented at this conference: one for the specialist, as one finds it in this book, and one for the would-be specialist, who needs some direction and encouragement before he or she swallows this morass of definitions.

## Related pages

* [[Peripatetic Seminar on Sheaves and Logic]]

* [[Category Theory Octoberfest]]

* [Séminaire Itinérant des Catégories](http://www-lmpa.univ-littoral.fr/SIC/)

* [[events of interest for the working category theorist]]

* [[Oberwolfach]]

## References

* {#CT81} [[Klaus Heiner Kamps]], [[Dieter Pumplün]], [[Walter Tholen]] (Eds), _Category Theory: Applications to Algebra, Logic and Topology. Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981_, Lecture Notes in Mathematics **962** (1982) Springer–Verlag, doi:[10.1007/BFb0066878](https://doi.org/10.1007/BFb0066878)

* {#CT89} [[R. Brown]], _International category theory meeting. Bangor, Wales, July 2-7 1989. Cambridge, England, March 23-25, 1990_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 32, No. 1, 1991, page 3. [[numdam](http://www.numdam.org/item/CTGDC_1991__32_1_3_0/)]

* {#CT90} [[Aurelio Carboni]], [[Maria Cristina Pedicchio]], [[Guiseppe Rosolini]] (Eds), _Category Theory: Proceedings of the International Conference held in Como, Italy, July 22-28, 1990_, Lecture Notes in Mathematics **1488** (1991) Springer–Verlag, doi:[10.1007/BFb0084207](https://doi.org/10.1007/BFb0084207)

* {#CT91} [[R.A.G. Seely]] (Ed), _Category Theory 1991: Proceedings of the 1991 Summer Category Theory Meeting, Montreal, Canada_,  Conference Proceedings, Canadian Mathematical Society **13** (1992) American Mathematical Society/Canadian Mathematical Society [AMS bookstore](https://bookstore.ams.org/cmsams-13). (pdf)

* {#Rota} [[Gian-Carlo Rota]], _Book review of "Category Theory 1991"_, Advances in Mathematics, Vol. 102, No. 1, 1993, page 126. [[doi:10.1006/aima.1993.1061](https://doi.org/10.1006/aima.1993.1061)]

* {#CT99} J. Adamek, P.T. Johnstone and M. Sobral (Eds), _Category Theory 1999: selected papers, conference held in Coimbra in honour of the 90th birthday of Saunders Mac Lane_, Journal of Pure and Applied Algebra **168** Issues 2–3 (2002) [journal](https://www.sciencedirect.com/journal/journal-of-pure-and-applied-algebra/vol/168/issue/2)

* {#CT00} [[Aurelio Carboni]], [[Giuseppe Rosolini]] and [[Robert Walters]] (guest editors), _CT2000_, Theory and Applications of Categories, Vol. 9, 2001. [[TAC](http://www.tac.mta.ca/tac/index.html#vol9)]

* {#CT04} [[George Janelidze]], [[John MacDonald]], [[Ross Street]] and [[Walter Tholen]] (guest editors), _CT2004_, Theory and Applications of Categories, Vol. 15, 2005-2006. [[TAC](http://www.tac.mta.ca/tac/index.html#vol15)]

* {#CT06} _CT2006_, Theory and Applications of Categories, Vol. 19, 2007. [[TAC](http://www.tac.mta.ca/tac/index.html#vol19)]

* {#CT11} [[Marino Gran]], [[George Janelidze]], [[Stephen Lack]], [[John MacDonald]] and [[Walter Tholen]] (guest editors), _CT2011_, Theory and Applications of Categories, Vol. 27, 2012. [[TAC](http://www.tac.mta.ca/tac/index.html#vol27)]

A list of category theory conferences appears in the unpublished note:

* [[Peter Johnstone]], *Category Theory Summer Conferences* (2023) [[pdf](https://ncatlab.org/nlab/files/CThistory.pdf)]

category: reference

[[!redirects category theory conference]]
[[!redirects category theory conferences]]
[[!redirects Category Theory conferences]]


category theory for dynamical systems

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On [[category theory]] (and [[higher category theory]]) for the conceptualization and description of [[dynamical systems]].

## Related concepts

* [[lens (in computer science)]]

* [[para construction]]

* [[double category]]

* [[doubly indexed category]]

* [[polynomial functors]]

## References

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], *Dynamical Systems and Sheaves*, Appl Categor Struct **28** (2020) 1–57 [[arXiv:1609.08086](https://arxiv.org/abs/1609.08086), [doi:10.1007/s10485-019-09565-x](https://doi.org/10.1007/s10485-019-09565-x)]

* [[David Jaz Myers]], *Double Categories of Open Dynamical Systems*, EPTCS **333** (2021) 154-167 [[arXiv:2005.05956](https://arxiv.org/abs/2005.05956), [doi:10.4204/EPTCS.333.11](https://doi.org/10.4204/EPTCS.333.11)]

* {#CGHR22} [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], [[Eigil F. Rischel]], *Towards Foundations of Categorical Cybernetics*, EPTCS **372** (2022) 235-248 [[arXiv:2105.06332](https://arxiv.org/abs/2105.06332), [doi:10.4204/EPTCS.372.17](https://doi.org/10.4204/EPTCS.372.17)]

Monographs:

* {#SpivakNiu} [[David Spivak]], [[Nelson Niu]], _Polynomial Functors: A General Theory of Interaction_ [[webpage](https://topos.site/poly-course/), [pdf](https://topos.site/poly-book.pdf)]

  > (via [[polynomial functors]])

* {#MyersBook} [[David Jaz Myers]], *Categorical systems theory*, book project [[github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book), [pdf](http://davidjaz.com/Papers/DynamicalBook.pdf)]

  > (via [[virtual double categories]])

Exposition:

* [[David Jaz Myers]], *[Categorical systems theory](https://topos.site/blog/2021/11/categorical-systems-theory/)*, [Topos Institute Blog](https://topos.site/blog/) (Nov 2021)

* [[David Jaz Myers]], *Double Categories of Dynamical Systems* (2020) [[pdf](http://davidjaz.com/Talks/DJM_Dyn2020.pdf)]

On a [[triple category]]-generalization of the [[para construction]] for systems theory:

* {#Myers22} [[David Jaz Myers]], _The Para construction as a distributive law_, talk at the Virtual Double Categories Workshop (2022) [[slides](https://bryceclarke.github.io/virtual-double-categories-workshop/slides/david-jaz-myers.pdf), [video](https://www.youtube.com/watch?v=zB_ifewP8Yk)]

* {#Capucci23} [[Matteo Capucci]], _Constructing triple categories of cybernetic processes_ [[slides](https://zenodo.org/record/8221550/files/main.pdf), [video](https://youtu.be/jYxnUy1vH7Q)]

* [[Matteo Capucci]], *Para Construction as a Wreath Product*, [talk at](CQTS#CapucciJan2024) *[[CQTS]]* (Jan 2024) [slides: [[Capucci-ParaAsWreathProduct.pdf:file]]]

[[!redirects categorical system theory]]
[[!redirects categorical systems theory]]

[[!redirects cybernetics]]


Category Theory in Context

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

This page collects pointers related to the textbook

* [[Emily Riehl]],

  _Category Theory in Context_

  Dover Publications (2017)

  [pdf](https://math.jhu.edu/~eriehl/context.pdf)

  [webpage](https://math.jhu.edu/~eriehl/context/)

on [[category theory]].

> Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. While the reader will be rewarded for familiarity with these background mathematical contexts, essential prerequisites are limited to basic set theory and logic.

category: reference


Category Theory Octoberfest

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

The *Category Theory Octoberfest* (usually simply called *Octoberfest*) is an annual conference on [[category theory]], held in October.

## Related pages

* [[Category Theory conference]]

* [[Peripatetic Seminar on Sheaves and Logic]]

* [[events of interest for the working category theorist]]

category: reference

[[!redirects Octoberfest]]


category theory vs order theory

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

Consider the following table.  The basic thesis of this page is that, if you are having trouble understanding an item in the left column, then first you should understand the item in the right column.  (Each item on the right is a [[decategorification]], in one way or another, of the item on the left.)

| [[category theory|Category theory]] | [[order theory|Order theory]] |
| - | - |
| [[category]] | [[proset]] |
| [[skeletal category]] | [[poset]] |
| [[groupoid]] | [[set]] |
| [[functor]] | [[monotone function]] |
| [[limit]] | [[meet]] |
| [[colimit]] | [[join]] |
| ([[dual adjunction|dual]]) [[adjunction]]| [[Galois connection]] |
| [[monad]] | [[Moore closure]] |
| [[cartesian closed category|cartesian closed]] [[pretopos]] | [[Heyting algebra]] |
| [[Grothendieck topos]] | [[locale]] |
| [[site]] | [[posite]] |
| ... | ... |

In the first several rows, we trust that people know about the order-theoretic concept before we teach them about category theory, so we can and do use these rows pedagogically.  But after that, we cannot trust that people know about the order-theoretic concepts, so we teach the category-theoretic concepts without them ---but *the order-theoretic concepts are still simpler*, so maybe we should teach (or learn) about them first. This is the approach taken, for example, in [[Paul Taylor]]\'s book _[Practical Foundations](#Taylor)_.

This is an example of [[negative thinking]], but perhaps not more negative than normal.  It is common to go down as far as $(0,0)$-categories, but not so common to move to $(0,1)$-categories before pushing on to $(1,1)$-categories.

See also at *[[relation between preorders and (0,1)-categories]]*.

## References

* [[Paul Taylor]], _[[Practical Foundations of Mathematics]]_, Cambridge University Press ([web](http://www.paultaylor.eu/~pt/prafm/index.html))
{#Taylor}


Category Theory Zulip

The *Category Theory Zulip server*

* [categorytheory.zulipchat.com](https://categorytheory.zulipchat.com/)

is a [Zulip server](https://en.wikipedia.org/wiki/Zulip) on the topic of *[[category theory]]*.

The server was set up in 2020 and is currently active. Messages can be read without an account, but an account is necessary to post. There is an archive of messages available at:

* [Category Theory Zulip archive](https://mattecapu.github.io/ct-zulip-archive/)

## See also

* [[categories mailing list]]

[[!redirects Category Theory Zulip server]]
[[!redirects Zulip]]