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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
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#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
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# Categorical semantics of dependent types
* table of contents
{: toc}

## Idea

In the [[relation between type theory and category theory]], [[dependent types]] (in [[dependent type theory]]) have [[categorical semantics]] given by [[morphisms]] which are the [[bundles]] whose [[fibers]] are the [[indexed family|families]] indexed by the base (see also at [dependent type -- Relation to families of elements](dependent+type#RelationToFamiliesOfElements)).

More concretely: if a [[type]] $A$ interprets as an [[object]] in some [[category]] $\mathcal{C}$, then an $A$-[[dependent type]]

$$
  (x \colon A)
  \;\;\;\vdash\;\;\;
  B(x) \;Type
$$

interprets as a [[morphism]] $B \to A$ in $\mathcal{C}$ (hence as an object in the [[slice category]] $\mathcal{C}_{/A}$).  We may think of this morphism as a [[bundle]] or [[fibration]], whose [[fiber]] over $x \colon A$ is the type $B(x)$.  We can then say that [[type former|type forming operations]] such as [[dependent sum type]] and [[dependent product type]] correspond to category-theoretic operations of [[dependent sum]] and [[dependent product]].

However, this correspondence is not quite precise; in the case of dependent types there are extra [[coherence]] issues.  [[substitution|Substitution]] in [[type theory]] should correspond to [[pullback]] in [[category theory]] (but see at _[substitution -- Categorical semantics](substitution#CategoricalSemantics)_ for more); that is, given a term

$$ (y:C) \;\vdash\; (f(y) : A) $$

corresponding to a morphism $f:C\to A$, the substituted dependent type

$$ (y:C) \;\vdash\; (B(f(y)) \;type) $$

should correspond to the pullback of $B\to A$ along $f$.  However, substitution in type theory is strictly [[associativity|associative]].  That is, given also $g:D\to C$, the dependent type

$$ (z:D) \;\vdash\; (B(f(g(z))) \;type)$$

is [[equality|syntactically the same]] regardless of whether we obtain it by substituting $y \coloneqq g(z)$ into $B(f(y))$, or $x \coloneqq f(g(z))$ into $B(x)$.  In category theory, however, pullback is not generally strictly associative, so there is a mismatch.  Similarly, every [[type formation rule|type-forming operation]] must also be strictly preserved by substitition/pullback.

The way this is generally dealt with is to introduce a category-theoretic structure which does have a "substitution" operation which is strictly associative, hence does correspond directly to type theory, and then show that any category can be "strictified" into such a stricter structure.  Unfortunately, there are many different such structures which have been used, differing only slightly.  On this page we define and compare them all.

One of these structures is called "contextual categories" (definition \ref{ContextualCategory} below).

{#StandInForSyntax} This and other kinds of categories-with-extra-structure may hence be thought of as stand-ins for the [[syntax]] of a [[type theory]]:

> Rather than constructing an interpretation of the syntax directly, we may work via the intermediary notion of contextual categories, a class of algebraic objects abstracting the key structure carried by the syntax. The plain definition of a contextual category corresponds to the structural core of the syntax; further syntactic rules (logical constructors, etc.) correspond to extra algebraic structure that contextual categories may carry. Essentially, contextual categories provide a completely equivalent alternative to the syntactic presentation of type theory.

> Why is this duplication of notions desirable? The trouble with the syntax is that it is mathematically tricky to handle. Any full presentation must account for (among other things) variable binding, capture-free substitution, and the possibility of multiple derivations of a judgement; and so a direct interpretation must deal with all of these, at the same time as tackling the details of the particular model in question. By passing to contextual categories, one deals with these subtleties and bureaucracy once and for all, and obtains a clear framework for subsequently constructing models. Conversely, why not work only with contextual categories, dispensing with syntax entirely?

> The trouble on this side is that working with higher-order logical structure in contextual categories quickly becomes unreadable. The reader preferring to take contextual categories as primary may regard the syntax as essentially a notation for working within them: a powerful, flexible, and intuitive notation, but one whose validity requires non-trivial work to establish. (The situation is comparable to that of string diagrams, as used in monoidal and more elaborately structured categories.)

> (from [Kapulkin-Lumsdaine 12](#KapulkinLumsdaine12))

## Definitions

In all the definitions, $C$ will be a [[category]].  Generally, we will regard the objects of $C$ as [[contexts]] in a type theory.

So far, we do not assume anything about $C$ as a category.  Usually, one at least wants $C$ to have a [[terminal object]], representing the empty context, although this is not always included in the definitions.  The additional structures we impose on $C$ below will imply in particular that certain [[pullbacks]] exist in $C$.

Sometimes, we want to consider $C$ as a [[strict category]], that is, we consider its objects up to [[equality]] rather than [[isomorphism]].  However, for most of the definitions below (until we get to contextual categories), it is still sensible to treat $C$ in an ordinary category-theoretic way, with the strictness living in the additional structure.

All of this could be made more precise by assembling the structures considered below into [[categories]], [[2-categories]], and/or [[strict 2-categories]].

### Comprehension categories

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

A **comprehension category** consists of a strictly [[commuting diagram|commutative triangle]] of [[functors]]
$$ \array{ E && \to && C^I\\ & \searrow && \swarrow {\scriptsize cod} \\ && C } $$
where $C^I$ is the [[arrow category]] of $C$ and $cod \colon C^I \to C$ denotes the codomain projection (which is a [[codomain fibration|fibration]] if $C$ has pullbacks), and such that

1. $E\to C$ is a [[Grothendieck fibration]],

1. $E\to C^I$ takes [[cartesian morphisms]] in $E$ to cartesian morphisms in $C^I$ (i.e. to [[pullback]] squares in $C$).

=--

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

In def \ref{ComprehensionCategory} we do not ask that $C^I \to C$ be a [[Grothendieck fibration|fibration]] (that would require $C$ to have all [[pullbacks]]), only that the *particular* morphisms in the image of $E$ are cartesian.

=--

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

A comprehension category (def. \ref{ComprehensionCategory}) is called

* **full** if $E\to C^I$ is a [[fully faithful functor]].

* **split** if $E\to C$ is a [[split fibration]].

In the latter case, we must consider $E$ at least to be a [[strict category]] (that is, we consider its objects up to equality rather than isomorphism) for the notion to make sense.

=--

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

The interpretation of def. \ref{ComprehensionCategory} is as follows:

In a comprehension category, we may regard the objects of $C$ as [[contexts]], and the [[fiber]] $E^\Gamma$ of $E\to C$ over an object $\Gamma$ as the category of [[dependent types]] in context $\Gamma$.  If the comprehension category is split (def. \ref{FullAndSplitComprehensionCategory}), then such dependent types have strictly [[associativity|associative]] [[substitution]].

The functor $E\to C^I$ assigns to each "type $A$ in context $\Gamma$" a new context which we think of as $\Gamma$ *extended* by a fresh [[variable]] of type $A$, and write as $\Gamma.A$.  This new context $\Gamma.A$ comes with a projection $\Gamma.A\to \Gamma$ (which forgets the fresh variable), and all substitutions in $E$ are realized as pullbacks in $C$.

=--

### Display map categories

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

A **[[display map category]]** consists of a [[category]] $C$ together with a [[class]] $D$ of [[morphisms]] in $C$, called _[[display maps]]_, such that all [[pullbacks]] of display maps exist and are themselves display maps.

=--

If $C$ is a display map category, then by defining $E$ to be the [[full subcategory]] of $C^I$ whose objects are display maps, we obtain a full comprehension category (def. \ref{FullAndSplitComprehensionCategory}).  Thus, we have:

+-- {: .num_lemma #DMCasCmpC}
###### Lemma

Display map categories (def. \ref{DisplayMapCategory}) may be identified with those comprehension categories, def. \ref{ComprehensionCategory}, for which the functor $E\to C^I$ is the inclusion of a [[full subcategory]].

=--

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

Working up to [[equivalence of categories]], as is usual in [[category theory]], it is natural to consider display map categories to also be equivalent to *full* comprehension categories, def. \ref{FullAndSplitComprehensionCategory}, (those where $E\to C^I$ is merely fully faithful).

However, this breaks down when we are interested in *split* comprehension categories, def. \ref{FullAndSplitComprehensionCategory}, for modeling substitution with strict associativity, since then $E$ at least must be regarded as a [[strict category]] and considered up to [[isomorphism]] rather than equivalence.  Thus, display map categories may be said to be equivalent to non-split full comprehension categories, but "split display map categories" are not equivalent to split full comprehension categories.  (In fact, split display map categories are not very useful; usually in order to make pullbacks strictly associative we have to introduce extra "names" for the same objects.)

=--

### Categories with attributes

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

A **category with attributes** is a comprehension category, def. \ref{ComprehensionCategory}, for which $E\to C$ is a [[discrete fibration]].

=--

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

That is, in a category with attributes (def. \ref{CategoryWithAttributes}) we demand only that each context comes with a [[set]] of [[dependent types]] in that context, rather than a category of such.  The intent is that the morphisms between two types in context $\Gamma$ should be determined by the morphisms in $C$ between the extended contexts over $\Gamma$.

Another way to convey this same intent with a comprehension category would be to ask that it be *full*, def. \ref{FullAndSplitComprehensionCategory}, i.e. that the functor $E\to C^I$ be fully faithful.

=--

In fact, any category with attributes gives rise to a full comprehension category by factoring the functor $E\to C^I$ into a [[bijective on objects functor]] followed by a [[fully faithful functor]].  In this way, we obtain:

+-- {: .num_lemma #CWAasCmpC}
###### Lemma
The category $\mathbf{CwA}$ of categories with attributes (def. \ref{CategoryWithAttributes}) is equivalent to the category $\mathbf{CompCat}_{\text{full,split}}$ of full split comprehension categories (def. \ref{FullAndSplitComprehensionCategory}).

(These are, however, quite different as subcategories of $\mathbf{CompCat}$.)
=--

### Categories with families

A category with attributes specifies for each "context" only a set of "types" in that context.  A comprehension category, by contrast, specifies a whole *category* of "types" in each context.  If $A,B\in E^\Gamma$, then we may think of a morphism $f:A\to B$ in $E^\Gamma$ as a term

\[ (\vec{x} : \Gamma), (a:A(\vec{x})) \;\vdash\; (f(\vec{x},a) : B(\vec{x})) \label{cmpcterm} \]

in type theory.
A *category with families* specifies instead, for each context and each type in that context, a set of "terms belonging to that type".  These should be thought of as terms

\[ (\vec{x} : \Gamma) \;\vdash\; (f(\vec{x}) : B(\vec{x})) \label{cwfterm} \]

in type theory.

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

A [[term]] of the form (eq:cmpcterm) can equivalently be regarded as a term of the form (eq:cwfterm) by replacing $\Gamma$ by the extended context $\Gamma.A$, and $B$ by its substitution along the projection $\Gamma.A \to \Gamma$.

=--

The additional insight in the following definition is that if we expect all of these terms to be determined by the morphisms in $C$, as in a category with attributes or a full comprehension category, then instead of specifying the functor $E\to C^I$ and then asking either that it be fully faithful or that $E$ be discrete (removing the information about extra morphisms in $E$), if we specify the terms of the form (eq:cwfterm), then the functor $E\to C^I$ is determined by a universal property.

Let $Fam$ denote the category of families of sets.  Its objects are set-indexed families $(A_i)_{i\in I}$, and its morphisms $(A_i)_{i\in I} \to (B_j)_{j\in J}$ consist of a function $f\colon I\to J$ and functions $g_i \colon A_i \to B_{f(i)}$.  We can equivalently, of course, regard this as the arrow category $Set^I$ of [[Set]], where $(B_j)_{j\in J}$ corresponds to the arrow $\coprod B \to J$.

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

A **category with families** is a category $C$ together with

* A functor $F:C^{op} \to Fam$, written $F(\Gamma) = (Tm(A))_{A\in Ty(\Gamma)}$.

* For each $\Gamma\in C$ and $A\in Ty(\Gamma)$, a [[representing object]] for the functor
  $$
  \begin{aligned}
    (C/\Gamma)^{op} & \to Set\\
    (\Delta \xrightarrow{f} \Gamma) & \mapsto Tm(f^*(A))
  \end{aligned}
  $$
=--
Intuitively, $F(\Gamma)$ is the set of terms in the context $\Gamma$ indexed by their type, and the representing object for the map $(C/\Gamma)^{op} \to Set$ is the projection from context $\Gamma; A$ to context $\Gamma$.

We can then prove:

+-- {: .num_lemma #CwAareCwF}
###### Lemma

Categories with attributes, def. \ref{CategoryWithAttributes} are equivalent to categories with families, def. \ref{CategoryWithFamilies}.

=--
+-- {: .proof}
###### Proof
Given a category with families, let $E\to C$ be the [[Grothendieck construction]] of the functor $Ty:C^{op}\to Set$, and let $E\to C^I$ take each $A\in Ty(\Gamma)$ to the above representing object.  This is a category with attributes.

Conversely, given a category with attributes, let $Ty:C^{op}\to Set$ be the functor corresponding to the discrete fibration $E\to C$, and for $A\in Ty(\Gamma)$ let $Tm(A)$ be the set of [[sections]] of the morphism in $C$ that is the image of $A$ in $C^I$.  These constructions are inverses up to isomorphism.
=--

### Natural Models
 {#NaturalModels}

[Awodey 2016](#Awodey2016) presented the following "[[natural model of homotopy type theory|natural model]]" of type theory as an alternative to categories with families.

+-- {: .num_theorem}
###### Theorem
If we modify Def. \ref{CategoryWithFamilies} by requiring only that the functors $(\Delta \xrightarrow{f} \Gamma) \mapsto Tm(f^*(A))$ be representable (rather than equipped with representing objects), then it is equivalent to giving

1. a category $C$, together with
1. a morphism $Tm \to Ty$ in the category of [[presheaves]] on $C$ which is a [[representable morphism]].
=--

The category $C$ models the category of contexts and substitutions, and the morphism $Tm \to Ty$ models the bundle of (context-dependent) terms over (context-dependent) types. The representability models the extension of a context with a new typed variable.

The condition that $Tm \to Ty$ is a representable morphism in presheaves can be reformulated in terms of adjoints as follows: $Tm \to Ty$ is representable just if the associated functor ${\int Tm} \to {\int Ty}$ has a right adjoint.  Thus a natural model can be equivalently described as a category $C$ equipped with a natural transformation of presheaves $Tm \to Ty$ for which ${\int Tm} \to {\int Ty}$ has a right adjoint.

This specification can be related to that of a \ref{CategoryWithFamilies}, by reformulating the representability of the morphism $Tm \to Ty$ in terms of representable profunctors as follows.  A natural model is equivalently described as:

1. a category $C$,
2. a presheaf $Ty : \hat C$ of types in context.
3. a presheaf $Tm : \widehat {\int Ty}$ of typed terms in context.
4. a functor $ext : \widehat {\int Ty} \to C$ of context extension with data that represents the profunctor ${\int Ty}(ty-,=) \odot C(-,ctx(ty(=))) : \int Ty ⇸ C$ where $ty : \int Tm \to \int Ty$ and $ctx : \int Ty \to C$ are the projections of the [[Grothendieck construction]].

Writing the profunctor as P, it is equivalent to the following definition:

$$P(\Delta, (\Gamma, A)) = (\gamma : \Delta \to \Gamma) \times (a : Tm(\Delta, A[\gamma]))$$

then the universal property of the context extension is that there is a natural isomorphism $\Delta \to ext(\Gamma, A) \cong P(\Delta,(\Gamma,A))$

### Categories with Representable Maps
 {#CwR}

Taichi Uemura introduced [[categories with representable maps]] as a [[functorial semantics]] approach to type theory generalizing the notion of a Category with Families/Natural Model. A category with representable maps (CwR) is a category with finite limits with a specified class of "representable" arrows that are [[pullback-stable]] and [[exponentiable morphism|exponentiable]]. The motivating examples are presheaf categories where the representable arrows are the [[representable morphisms]] in the usual sense. Then any CwR can be considered to be a type theory: for instance, the [[free]] CwR with a representable map $Tm \to Ty$ has as models precisely the natural models, and so can be thought of as the embodiment of the judgments of Martin-Lof Type Theory (with no connectives).

### Contextual categories, or C-systems
 {#ContextualCategoriesOrCSystems}

Recall that

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

If $C$ is a [[comprehension category]] (def. \ref{ComprehensionCategory}), $\Gamma\in C$ is a "context" and $A\in E^\Gamma$ is a "type" in context $\Gamma$, then we denote by $\Gamma.A$ the "extended context" in $C$ (remark \ref{MeaningOfComprehensionCategory}).

=--

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

A **[[contextual category]]** ([Cartmell 86](#Cartmell86), [Streicher 91](#Streicher91)) or **C-system** ([Voevodsky 15](#Voevodsky15a)) is a category with attributes $C$ (def. \ref{CategoryWithAttributes}) together with a *length function* $\ell : ob(C) \to \mathbb{N}$ such that

1. There is a unique object of length $0$, which is a [[terminal object]].
1. For any $\Gamma\in C$ and $A\in E^\Gamma$, we have $\ell(\Gamma.A) = \ell(\Gamma)+1$.
1. For any $\Delta\in C$ with $\ell(\Delta)\gt 0$, there exists a unique $\Gamma\in C$ and $A\in E^\Gamma$ such that $\Delta = \Gamma.A$ (with notation as in def. \ref{NotationForExtendedContexts}).

=--

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

Since def. \ref{ContextualCategory} refers to [[equality]] of objects, a contextual category $C$ must be a [[strict category]].

=--

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

The idea which distinguishes a contextual category is that "every context must be built out of types" in a unique way.

=--

This makes for the closest match with type theory; in fact we have:

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

The category of contextual categories, def. \ref{ContextualCategory}, and (strictly) structure-preserving functors is [[equivalence of categories|equivalent]] to the [[category of dependent type theories and interpretations]].

=--

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

Since contextual categories are [[strict categories]], the category of such is really a [[1-category]], or perhaps a [[strict 2-category]].

=--

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

Given any category with attributes $C$, def. \ref{CategoryWithAttributes},  possessing a [[terminal object]], there is a canonical way to build a contextual category $cxt(C)$, def. \ref{ContextualCategory}, from it.

1. Choose a [[terminal object]] $1\in C$ (the resulting contextual category does not depend on this choice, up to isomorphism).

1. The [[objects]] of $cxt(C)$ are the finite [[lists]]

   $$(A_0,A_1,\dots,A_n)$$

   such that $A_0 \in E^1$ and $A_{k+1} \in E^{1.A_0.A_1.\cdots .A_k}$ for all $k$.

1. The [[morphisms]] $(A_0,\dots,A_n) \to (B_0,\dots,B_m)$ in $cxt(C)$ are the morphisms $1.A_0.A_1.\cdots.A_n \to 1.B_0.B_1.\cdots.B_m$ in $C$.

All the rest of the structure on $cxt(C)$ is induced in an evident way from $C$.  This construction is in fact right adjoint to the inclusion of contextual categories in categories with attributes; see [Kapulkin-Lumsdaine, Proposition 4.4](#KapulkinLumsdaine16).

=--

## Examples

Comprehension categories and display map categories are easy to come by "in nature", but it is more difficult to find examples of the "split" versions of the above structure (which are what is needed for modeling type theory).  Here we summarize some basic known constructions.

However, first we should mention the examples that come from type theory itself.

### Syntactic categories

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

The [[syntactic category]] of any [[dependent type theory]] has all of the above structures.  Its objects are [[contexts]] in the theory, and the types in context $\Gamma$ form the set or category $E^\Gamma$.  The strict associativity of substitution in type theory makes this fibration automatically split.

=--

### Splitting fibrations

There are standard constructions which can replace any [[Grothendieck fibration]] by an equivalent [[split fibration]].  Therefore,

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

Given any comprehension category, def. \ref{ComprehensionCategory},

1. we may replace it by a split comprehension category, def. \ref{FullAndSplitComprehensionCategory},

1. then consider the underlying category with attributes, def. \ref{CategoryWithAttributes},

1. and finally pass to a contextual category by the construction in example \ref{ContextualFromAttributes}.

=--

Of course, comprehension categories are easy to come by; perhaps they arise most commonly as [[display map categories]].  For instance, if $C$ has all [[pullbacks]], then we can take all maps to be [[display maps]].  If $C$ is a [[category of fibrant objects]], we can take the [[fibrations]] to be the display maps.

So, for the record, we have in particular:

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

For $C$ a [[locally cartesian closed category]] $C$, it becomes a model for [[dependent type theory]]  by regarding its [[codomain fibration]] $C^I \to C$ as a comprehension category, def. \ref{ComprehensionCategory}, and then strictifying that as in example \ref{FromComprehensionToContextual}.

=--

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

It turns out that for modeling additional [[type-forming operations]], however, not all splitting constructions are created equal.

Given $E\to C$, one construction due to [[Benabou]] (called the "right adjoint splitting"), defines $E'\to C$, where the objects of $E'$ over $\Gamma\in C$ are functors $C/\Gamma \to E$ over $C$ which map every morphism of $C/\Gamma$ to a cartesian arrow.  Type-theoretically, we can think of such an object as a type together with *specified* compatible substitutions along any possible morphism.  That type-formers may be extended in this case was proven by [[Martin Hofmann]] for [[dependent sums]] and [[dependent products]] and the [[identity types]] of [[extensional type theory]].  But this is not generally possible for the identity types of [[intensional type theory]] (particularly not their [[term elimination|eliminator]]), which do not have a 1-categorical universal property and hence are not stable under pullback up to isomorphism.

A different construction (due to [[John Power]], called the "left adjoint splitting") defines an object of $E'$ over $\Gamma\in C$ to consist of a morphism $f:\Gamma \to \Delta$ in $C$ along with an object $A$ of $E$ over $\Delta$.  Type-theoretically, we can regard $(f,A)$ as a type $A$ with a "delayed substitution" $f$.  This produces a split fibration (the chosen cartesian arrows are given by composition of morphisms in $C$), and it was proven by [Lumsdaine & Warren (2015)](#LumsdaineWarren13) that essentially all type formers can be extended to it from $E$.

=--

### Universes

Suppose given a particular morphism $p:\widetilde{U} \to U$ in $C$.  We can then define a category with attributes, def. \ref{CategoryWithAttributes}, as follows: the discrete fibration $E\to C$ corresponds to the representable presheaf $C(-,U)$, and the functor $E\to C^I$ is defined by pullback of $p$.  We are thus treating $U$ as a "[[universe]]" of types.  We may then of course pass to a contextual category, via example \ref{FromComprehensionToContextual}.

Type-forming operations may be extended strictly in this case by performing them once in the "universal" case, then acting by composition.  This construction is due to [[Voevodsky]].  It also meshes quite well with type theories that contain internal universes -- a [[type of types]]-- , and in particular for modeling the [[univalence axiom]].

Particular important universes include:

* Any [[Grothendieck universe]] in [[Set]], or more generally a [[universe in a topos]].  The resulting display maps are those with "$U$-small fibers".

* In the category [[Gpd]], the groupoid of small groupoids.  The resulting display maps are the [[split opfibrations]] with small fibers.  Similarly, we can consider the groupoid of small sets, whose display maps are the [[discrete opfibrations]] with small fibers.

* In the category [[sSet]] of [[simplicial sets]], there is a [[universal Kan fibration]] $p:\widetilde{U} \to U$ which classifies all [[Kan fibrations]] with small fibers.

### Simple fibrations

Let $C$ be any category with finite products, and define $E\to C$ to be the [[discrete fibration]] corresponding to the presheaf $C^{op}\to Set$ which is constant at $ob(C)$.  Thus, the objects of $E$ are pairs $(\Gamma,A)$ of objects of $C$, with the only morphisms being of the form $(\Gamma,A) \to (\Delta,A)$ induced by a morphism $\Gamma\to\Delta$ in $C$.

Define the functor $E\to C^I$ to take $(\Gamma,A)$ to the projection $\Gamma\times A \to \Gamma$.  It is straightforward to check that this defines a category with attributes.  The corresponding (split) full comprehension category is called the *simple fibration* of $C$.

The dependent type theory which results from this structure "has no nontrivial dependency".  That is, whenever we have a dependent type $\Gamma \vdash (A \;type)$, it is already the case that $A$ is a type in the empty context (that is, we have $\vdash (A\; type)$), and so it cannot depend nontrivially on $\Gamma$.  In effect, it is not really a dependent type theory, but a *simple* (non-dependent) type theory --- hence the name "simple fibration".

### The big model of locally cartesian closed categories

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

Ignoring coherence issues, the CwF induced by a locally cartesian closed (lcc) category $\mathcal{C}$ (example \ref{TypeTheoryModelFromLocallyCartesianClosedCategory}) is given explicitly by the "Explicit" column of the following table:

|                   | In $\mathcal{C}$: Explicit                                                             | In terms of slices of $\mathcal{C}$                                                                                | Big model                                                                         |
| ----------------- |----------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------|
| Contexts          | Objects $\Gamma \in \mathcal{C}$                                     | Slice categories $\mathcal{C}_{/ \Gamma}$                                                         | Lcc categories $\Gamma$                                                           |
| Context Morphisms | Morphisms $f : \Gamma \rightarrow \Delta$                            | Lcc functors $f^* : \mathcal{C}_{/ \Delta} \rightarrow \mathcal{C}_{/ \Gamma}$ over $\mathcal{C}$ | Lcc functors $f : \Delta \rightarrow \Gamma$                                      |
| Types             | Morphisms $\sigma : \operatorname{dom} \sigma \rightarrow \Gamma$    | Objects $\sigma \in \mathcal{C}_{/ \Gamma}$                                                       | Objects $\sigma \in \Gamma$                                                       |
| Terms             | Sections $s : \Gamma \leftrightarrows \operatorname{dom} \sigma : \sigma$ | Morphisms $s : \mathrm{id}_\Gamma \rightarrow \sigma$ in $\mathcal{C}_{/ \Gamma}$                 | Morphisms $s : 1 \rightarrow \sigma$ in $\Gamma$, where $1$ is a terminal object. |
| Substitution      | Pullback along $f$                                                   | Application of $f^*$                                                                              | Application of $f$

The next column describes the same CwF in the terminology of slice categories:
Every object of $\mathcal{C}$ corresponds to a slice category $\mathcal{C}_{/ \Gamma}$ over $\mathcal{C}$, and $\mathcal{C}_{/ \Gamma}$ is also lcc.
Every morphism $f : \Gamma \rightarrow \Delta$ in $\mathcal{C}$ induces a pullback functor $f^* : \mathcal{C}_{/ \Delta} \rightarrow \mathcal{C}_{/ \Gamma}$.
$f^*$ preserves the finite limits and dependent products of $\mathcal{C}_{/ \Delta}$ (i.e. it is an lcc functor), and the diagram
$$
  \array{
    && \mathcal{C}
    \\
    & {}_{\Delta^*} \swarrow && \searrow_{\Gamma^*}
    \\
    \mathcal{C}_{/ \Delta} && \stackrel{\to}{f^*} && \mathcal{C}_{/ \Gamma}
}
$$
commutes (up to unique isomorphism).
Conversely, every lcc functor $\mathcal{C}_{/ \Delta} \rightarrow \mathcal{C}_{/ \Gamma}$ under $\mathcal{C}$ is uniquely isomorphic to the pullback functor along some morphism $\Gamma \rightarrow \Delta$.

The last column describes the **big model** of type theory in the opposite category of lcc categories and lcc functors.
Since the contexts of this model are itself models of type theory, it can be understood as a "model of models".
The definition of the big model can be arrived at by removing all reference to the fixed lcc category $\mathcal{C}$ in the previous column.
Instead of just the slice categories of $\mathcal{C}$, now all lcc categories are allowed as contexts.
Context morphisms are generalized to general lcc functors instead of just the pullback functors of $\mathcal{C}$.
The identity morphism on an object $\Gamma$ of $\mathcal{C}$ is a terminal object in the slice category $\mathcal{C}_{/ \Gamma}$ and is thus generalized to a terminal object in any lcc category.

The usual model in a fixed lcc category $\mathcal{C}$ can be recovered from the big model by slicing:
The contextual core (example \ref{ContextualFromAttributes}) of the slice of the big model over $\mathcal{C}$ is equivalent to $\mathcal{C}$.

However, the naive definition of the big model above suffers from the same coherence issues as the standard models in individual lcc categories:
Lcc functors preserve lcc structure (i.e. type formers) up to isomorphism, but not necessarily up to equality.
These coherence problems can be resolved by working with a suitable model categorical presentation of the $(2, 1)$-category of lcc categories, lcc functors and natural isomorphisms.
Note that the [[model category]] presenting a higher category is unique up to zig-zag of Quillen equivalences, but that the underlying 1-categories of these model categories can vary non-trivially.
Because coherence issues are ultimately about equations in the underlying 1-category, we can thus hope that some model categories presenting the category of lcc categories will be better behaved than others for our purpose.

One possible way to construct such a well-behaved model category is as follows (see ([Bidlingmaier 2020](#Bidlingmaier20)) for details):

1. First one defines a model category $\mathrm{Lcc}$ of lcc sketches.
   An **lcc sketch** is a category with some diagrams [marked](#Isaev2016) as finite limits cones and evalution maps of dependent products.
   These marked diagrams do not need to satisfy the corresponding universal property, however.
   The model category structure is set up such that every object is cofibrant, and the fibrant objects are precisely the lcc categories with diagrams marked if and only if they satisfy the corresponding universal property.
   Thus the subcategory of fibrant objects of $\mathrm{Lcc}$ corresponds to the usual category of lcc categories.
   Note however, that lcc categories in the sense of fibrant lcc sketches "have" finite limits and dependent products in the sense that these universal objects _merely exist_; there are no distinguished/assigned choices of universal objects.
   Thus preservation of universal objects by functors up to equality (i.e. strictness of substitution) cannot even be stated in $\mathrm{Lcc}$.
2. The model category $\mathrm{sLcc}$ of **strict lcc categories** is defined as the category of [[algebraically fibrant objects]] of $\mathrm{Lcc}$; it is Quillen equivalent to $\mathrm{Lcc}$.
   Assigned lifts against the trivial cofibrations of $\mathrm{Lcc}$ correspond to distinguished choices of universal objects, and these choices are preserved by the morphisms of $\mathrm{sLcc}$, the **strict lcc functors**.
   Thus $\mathrm{sLcc}$ supports substitutions that preserve type formers up to equality.

   However, it does not support some dependent type formers in any obvious way, notably not $\Sigma$ and $\Pi$ types.
   The problem is that the context extension $\Gamma.\sigma$ of some $\Gamma \in \operatorname{Ob} \mathrm{sLcc}$ by a type (i.e. object) $\sigma$ of $\Gamma$ is given by freely (in the 1-categorical sense) adjoining a morphism $1 \rightarrow \sigma$ to $\Gamma$.
   To interpret $\Sigma$ and $\Pi$ types, one has to relate $\Gamma.\sigma$ with the slice category $\Gamma_{/ \sigma}$ in some way, and it appears that this is not generally possible.
   Note that $\Gamma_{/ \sigma}$ has the universal property of a context extension in the higher/bicategorical sense, but that $\Gamma.\sigma$ is defined purely 1-categorically.
3. To reconcile context extensions $\Gamma.\sigma$ with slice categories $\Gamma_{/ \sigma}$, one can work with the [algebraically cofibrant objects](#ChingRiehl2014) of $\mathrm{sLcc}$.
   An algebraically cofibrant object of $\mathrm{sLcc}$ is a coalgebra for a fixed cofibrant replacement comonad.
   The category of such coalgebras has model category structure, and this model category is again Quillen equivalent to $\mathrm{sLcc}$.

   A cofibrant object $\Gamma$ of $\mathrm{sLcc}$ has the property that every non-strict lcc functor (i.e. morphism of underlying lcc sketches) out of $\Gamma$ is isomorphic to a strict lcc functor.
   This property turns out to be sufficient to construct a weak equivalence $\Gamma.\sigma \rightarrow \Gamma_{/ \sigma}$ in $\mathrm{sLcc}$.
   For _algebraically_ cofibrant $\Gamma$, this weak equivalence is structure and hence preserved by coalgebra morphisms up to equality.
   This turns out to be sufficient to endow the category of algebraically cofibrant objects of $\mathrm{sLcc}$ with CwF structure that supports finite limit, $\Sigma$ and $\Pi$ type constructors.

=--

## Related pages

* [[fibration category]]

  * [[clan]], [[tribe]]

  * [[type-theoretic model category]]

* [[categorical semantics]]

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

* [[initiality conjecture]]

* [[Awodey's conjecture]]

* [[model of type theory in an (infinity,1)-topos]]

## References
 {#References}

The general idea of categorical semantics of dependent types goes back at least to [Seely 1984](#Seely84), a first formalization via *categories with [[display maps]]* ("[[clans]]") is due to:

* {#Taylor87} [[Paul Taylor]], §4.3.2 in: *Recursive Domains, Indexed Category Theory and Polymorphism*, Cambridge (1983-7) [[[Taylor-IndexedCategoryTheory.pdf:file]]]

* {#Taylor99} [[Paul Taylor]], Section 8.3 in: _[[Practical Foundations of Mathematics]]_, Cambridge University Press (1999) [[webpage](http://www.paultaylor.eu/~pt/prafm/index.html)]

General overview:

* {#Hofmann97} [[Martin Hofmann]], *Syntax and semantics of dependent types*, in *Semantics and logics of computation*, Publ. Newton Inst. **14**, Cambridge Univ. Press (1997) 79-130 [[doi:10.1017/CBO9780511526619.004](https://doi.org/10.1017/CBO9780511526619.004)]

which originates as

* {#Hofmann95} [[Martin Hofmann]], chapter 2 of: *Extensional concepts in intensional type theory*, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

Texbook accounts:

* [[Bart Jacobs]], *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

Comprehension categories are defined in

* [[Bart Jacobs]], *Comprehension categories and the semantics of type dependency*, Theoret. Comput. Sci. **107** 2 (1993) 169-207 [[MR1201808](http://www.ams.org/mathscinet-getitem?mr=1201808), <a href="https://doi.org/10.1016/0304-3975(93)90169-T">doi:10.1016/0304-3975(93)90169-T</a>]

A 2-categorical treatment of variant kinds of comprehension category is given in

* [[Paul-André Melliès]], Nicolas Rolland, _Comprehension and quotient structures in the language of 2-categories_, ([arXiv:2005.10015](https://arxiv.org/abs/2005.10015))

A correspondence with orthogonal factorization systems is discussed in

* Clemens Berger, Ralph M. Kaufmann, *Comprehensive factorisation systems* ([pdf](https://arxiv.org/abs/1710.09438))

Categories with attributes are discussed in

* [[John Cartmell]], *Generalised algebraic theories and contextual categories*, Ph.D. thesis, Oxford (1978), Ann. Pure Appl.  Logic **32** 3 (1986) 209–243 [<a href="https://doi.org/10.1016/0168-0072(86)90053-9">doi:10.1016/0168-0072(86)90053-9</a>[MR865990](http://www.ams.org/mathscinet-getitem?mr=865990)]

* [[Eugenio Moggi]], *A category-theoretic account of program modules*, Math. Structures Comput. Sci. **1** 1 (1991) 103-139 [[MR1108806 ](http://www.ams.org/mathscinet-getitem?mr=1108806), <a href="https://doi.org/10.1017/S0960129500000074">doi:10.1017/S0960129500000074</a>]

* [[Andrew M. Pitts]], *Categorical logic*, in *[[Handbook of Logic in Computer Science]]* **5** Oxford Univ. Press (2000) 39-128 [[doi:10.1093/oso/9780198537816.003.0002](https://doi.org/10.1093/oso/9780198537816.003.0002), [pdf](https://www.cl.cam.ac.uk/~amp12/papers/catl/catl.pdf), [MR1859470](http://www.ams.org/mathscinet-getitem?mr=1859470)]

Categories with families are defined in

* [[Peter Dybjer]], *Internal type theory*, Types for proofs and programs (Torino, 1995), Lecture Notes in Comput. Sci. **1158** Springer (1996) 120-134 [[doi:10.1007/3-540-61780-9_66](https://doi.org/10.1007/3-540-61780-9_66), [pdf](http://www.cse.chalmers.se/~peterd/papers/InternalTT.pdf), [MR1474535](http://www.ams.org/mathscinet-getitem?mr=1474535)]

and shown to be equivalent to categories with attributes in [Hofmann (1997)](#Hofmann97)

The formulation in terms of representable natural transformations ([[natural models of homotopy type theory]]) is due to:

* {#Awodey2016} [[Steve Awodey]], *Natural models of homotopy type theory*, Mathematical Structures in Computer Science, **28** 2 (2016) 241-286 [[arXiv:1406.3219](https://arxiv.org/abs/1406.3219), [doi:10.1017/S0960129516000268](https://doi.org/10.1017/S0960129516000268)]

See also

* Clive Newstead, *Algebraic models of dependent type theory* [[arXiv:2103.06155](https://arxiv.org/abs/2103.06155)]

A proof of initiality for dependent type theory is claimed in

* Simon Castellan, _Dependent type theory as the initial category with families_, 2014 ([pdf](http://iso.mor.phis.me/archives/2011-2012/stage-2012-goteburg/report.pdf))

This was formalized inside type theory with set quotients of [[higher inductive types]] in:

* {#AltKap2015} [[Thorsten Altenkirch]], Ambrus Kaposi, _Type Theory in Type Theory using Quotient Inductive Types_, (2015) ([pdf](http://www.cs.nott.ac.uk/~txa/publ/tt-in-tt.pdf)), ([formalisation in Agda](https://bitbucket.org/akaposi/tt-in-tt)).

Contextual categories were defined in

* {#Cartmell86} [[John Cartmell]], _Generalised algebraic theories and contextual categories_, Annals of Pure and Applied Logic Volume 32, 1986, Pages 209-243 ([doi:10.1016/0168-0072(86)90053-9](http://dx.doi.org/10.1016/0168-0072%2886%2990053-9))

* {#Streicher91} [[Thomas Streicher]], *Semantics of type theory -- Correctness, completeness and independence results*, Progress in Theoretical Computer Science, Birkhäuser (1991) [ISBN:0-8176-3594-7, [doi:10.1007/978-1-4612-0433-6](https://doi.org/10.1007/978-1-4612-0433-6), [MR1134134 ](http://www.ams.org/mathscinet-getitem?mr=1134134)]

Review:

* {#KapulkinLumsdaine12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], section 1.2 and appendix B of _The Simplicial Model of Univalent Foundations (after Voevodsky)_ ([arXiv:1211.2851](https://arxiv.org/abs/1211.2851))

Contextual categories as models for [[homotopy type theory]] are discussed in

* {#KapulkinLumsdaine16} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The homotopy theory of type theories*, Advances in Mathematics **337** (2018) 1-38 [[arXiv:1610.00037](https://arxiv.org/abs/1610.00037), [doi:10.1016/j.aim.2018.08.003](https://doi.org/10.1016/j.aim.2018.08.003)]

* {#Joyal17} [[André Joyal]], _Notes on Clans and Tribes_ [[arXiv:1710.10238](https://arxiv.org/abs/1710.10238)]

Further discussion of contextual categories is in

* {#Voevodsky15a} [[Vladimir Voevodsky]], _A C-system defined by a universe category_, Theory Appl. Categ. 30 (2015), No. 37, 1181–1215, arXiv:1409.7925 ([arXiv:1409.7925](https://arxiv.org/abs/1409.7925))

* {#Voevodsky15b} [[Vladimir Voevodsky]], _Martin-Löf identity types in the C-systems defined by a universe category_ ([arXiv:1505.06446](https://arxiv.org/abs/1505.06446))

* {#Voevodsky15c} [[Vladimir Voevodsky]], _Products of families of types in the C-systems defined by a universe category_ ([arXiv:1503.07072](https://arxiv.org/abs/1503.07072))

* {#Voevodsky15d} [[Vladimir Voevodsky]], _Subsystems and regular quotients of C-systems_ ([arXiv:1406.7413](https://arxiv.org/abs/1406.7413))

Discussion of B-systems, another essentially algebraic model, is in

* {#Voevodsky14} [[Vladimir Voevodsky]], _B-systems_ ([arXiv:1410.5389](https://arxiv.org/abs/1410.5389))

The model of [[dependent type theory]] with [[dependent product types]]/[[dependent function types]] in [[locally cartesian closed categories]] was first made explicit in

* {#Seely84} [[R. A. G. Seely]], _Locally cartesian closed categories and type theory_, Math. Proc. Camb. Phil. Soc. **95** (1984) 33-48  [[doi:10.1017/S0305004100061284](https://doi.org/10.1017/S0305004100061284), [pdf](http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf)]

The problem with strict substitution compared to weak pullback in this argument was discussed and fixed in

* {#Curien} [[Pierre-Louis Curien]], *Substitution up to isomorphism*, Fundamenta Informaticae, **19** 1-2 (1993) 51-86 [[doi:10.5555/175469.175471](https://dl.acm.org/doi/10.5555/175469.175471)] and: Diagrammes **23** (1990) 43-66 [[numdam:DIA_1990__23__43_0](http://www.numdam.org/item/DIA_1990__23__43_0)]

* {#Hofmann} [[Martin Hofmann]], *On the interpretation of type theory in locally cartesian closed categories*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** (1994) 427–441 [[doi:10.1007/BFb0022273](https://doi.org/10.1007/BFb0022273)]

but in the process the equivalence of categories was lost. This was finally all rectified in

* {#ClairambaultDybjer} [[Pierre Clairambault]], [[Peter Dybjer]], *The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories*, in *Typed lambda calculi and applications*, Lecture Notes in Comput. Sci. **6690** Springer (2011) [[arXiv:1112.3456](http://arxiv.org/abs/1112.3456), [doi:10.1007/978-3-642-21691-6_10](https://doi.org/10.1007/978-3-642-21691-6_10)]

and

* {#CurienGarnerHofmann} [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], *Revisiting the categorical interpretation of dependent type theory*, Theoretical Computer Science **546** 21 (2014) 99-119 [[doi:10.1016/j.tcs.2014.03.003](https://doi.org/10.1016/j.tcs.2014.03.003), [[CurienGarnerHofmann.pdf:file]]]

See also:

* {#LumsdaineWarren13} [[Peter LeFanu Lumsdaine]], [[Michael Warren]], *The local universes model: An overlooked coherence construction for dependent type theory*, ACM Trans. Comput. Log., **16** 3 no 23 (2015) 1-31 [[arXiv:1411.1736](https://arxiv.org/abs/1411.1736), [doi:10.1145/2754931](https://doi.org/10.1145/2754931)]

* [[Vladimir Voevodsky]], [Notes on type systems](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/expressions_current.pdf)

Comparisons of various models can be found in

* [[Benedikt Ahrens]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], *Categorical structures for type theory in univalent foundations* [[arxiv:1705.04310](https://arxiv.org/abs/1705.04310)]

* {#KapulkinLumsdaine16} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The homotopy theory of type theories*[[arXiv:1610.00037](https://arxiv.org/abs/1610.00037)]

* [[Daniel Gratzer]], [[Jonathan Sterling]], *Syntactic categories for dependent type theory: sketching and adequacy* [[arXiv:2012.10783](https://arxiv.org/abs/2012.10783)]

Recent work on abstract definitions of (models of) type theory include:

* [[Valery Isaev]], _Algebraic Presentations of Dependent Type Theories_ ([arXiv:1602.08504](https://arxiv.org/abs/1602.08504))

* [[Taichi Uemura]], _A General Framework for the Semantics of Type Theory_ ([arXiv:1904.04097](https://arxiv.org/abs/1904.04097))

A category with families structure is constructed on the $(2,1)$-category of all locally cartesian closed categories, which since locally presentable may be treated via model categories, in:

* {#Bidlingmaier20} Martin Bidlingmaier, _An interpretation of dependent type theory in a model category of locally cartesian closed categories_, ([arXiv:2007.02900](https://arxiv.org/abs/2007.02900))

* {#ChingRiehl2014} [[Michael Ching]], [[Emily Riehl]], _Coalgebraic models for combinatorial model categories_ [arXiv:1403.5303](https://arxiv.org/abs/1403.5303)

* {#Isaev2016} Valery Isaev, _Model category of marked objects_ [arXiv:1610.08459](https://arxiv.org/abs/1610.08459)

An equivalence between contextual categories and [B-systems](#Voevodsky14) is in:

* [[Benedikt Ahrens]], Jacopo Emmenegger, [[Paige Randall North]], [[Egbert Rijke]], _B-systems and C-systems are equivalent_ ([arXiv:2111.09948](https://arxiv.org/abs/2111.09948))

[[!redirects category-theoretic semantics of dependent types]]

[[!redirects categorical model of dependent types]]

[[!redirects categorical model of dependent type theory]]
[[!redirects categorical models of dependent type theory]]

[[!redirects categorical semantics of dependent types]]
[[!redirects categorical semantics of dependent type theory]]

[[!redirects categorical semantics for dependent types]]
[[!redirects categorical semantics for dependent type theory]]

[[!redirects categorical model of dependent types]]
[[!redirects categorical models of dependent types]]
[[!redirects category-theoretic model of dependent types]]
[[!redirects category-theoretic models of dependent types]]
[[!redirects split model of type theory]]
[[!redirects split models of type theory]]

[[!redirects comprehension category]]
[[!redirects comprehension categories]]
[[!redirects display map category]]
[[!redirects display map categories]]
[[!redirects category with attributes]]
[[!redirects categories with attributes]]
[[!redirects category with families]]
[[!redirects categories with families]]
[[!redirects contextual category]]
[[!redirects contextual categories]]
[[!redirects C-system]]
[[!redirects C-systems]]


categorical set theory

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### Context
#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
{: toc}

## Idea

There are two general approaches to [[structural set theory]]; those that attempt to axiomatise the [[category]] [[Set]] of [[sets]] and [[functions]], and those that attempt to axiomatise the [[allegory]] [[Rel]] of [[sets]] and [[relations]]. The former is called **categorical set theory**.

## Types of categorical set theories

One distinguishes between simply sorted categorical set theories, where membership is a [[relation]], and dependently sorted categorical set theories, where membership is a typing [[judgment]].

### Simply sorted categorical set theories

Simply sorted categorical set theories have a sort of possible sets and a sort of possible elements, and a membership relation between the sort of possible sets and the sort of possible elements. By possible set we mean that some of the terms in the sort might be a set; and by possible element we mean that some of the terms in the sort might be an element.

* In unsorted categorical set theory, there is only a sort or judgment of [[functions]], and the sort/judgment of functions plays the role of both the sort of possible sets and the sort of possible elements.
* In two-sorted categorical set theory, there are sorts for [[functions]] and [[sets]], the sort of sets plays the role of the sort of possible sets, and the sort of functions plays the role of the sort of possible elements.
* In three-sorted categorical set theory, there are sorts for [[functions]], [[sets]], and [[elements]], and the sort of sets plays the role of the sort of possible sets, and the sort of elements plays the role of the sort of possible sets.

#### Unsorted categorical set theory

Unsorted categorical set theory only consists of [[functions]] as primitive [[judgments]] or [[sorts]]. The elements are defined as functions with singleton domain ($\mathrm{element}(a) \coloneqq (\mathrm{dom}(a) = 1)$) However, there are multiple ways of defining sets in unsorted categorical set theory, each which lead to a different categorical set theory; examples include but are not limited to

* [[identity functions]] ($\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = a$ or $\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = a$)
* [[functions]] with [[singleton]] [[codomain]] ($\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = 1$)
* [[functions]] with [[empty set]] [[domain]] ($\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = 0$)

In any unsorted first-order categorical set theory, the [[membership relation]] is a defined global relation on the functions, rather than a primitive global relation as in [[material set theory]]. For each definition of set, there is a definition of membership relation:

* for sets as identity functions, given functions $a$ and $b$, the membership relation is defined as $a$ being an element, $b$ being a set, and the codomain of $a$ being equal to $b$:
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = b$$

* for sets as functions into the singleton, given functions $a$ and $b$, the membership relation is defined as $a$ being an element, $b$ being a set, and the codomain of $a$ being equal to the domain of $b$:
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = \mathrm{dom}(b)$$

* for sets as functions out of the empty set, given functions $a$ and $b$, the membership relation is defined as $a$ being an element, $b$ being a set, and the codomain of $a$ being equal to the codomain of $b$:
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = \mathrm{codom}(b)$$

While that membership relation can be proven to satisfy the [[axiom of weak extensionality]], it cannot in general be proven to satisfy the [[axiom of strong extensionality]]. In addition, depending on the definition of the set, the membership relation has different properties:

* for sets as identity functions, every set is a [[Quine atom]] with respect to $\in$
* for sets as functions with singleton codomain, only sets in bijection with the singleton is a [[Quine atom]] with respect to $\in$
* for sets as functions with empty set domain, no set is a [[Quine atom]] with respect to $\in$

The canonical example of an unsorted first-order categorical set theory is [[fully formal ETCS]]. An [[axiom schema of collection]] or [[axiom schema of replacement]] could be added to fully formal ETCS to get an unsorted categorical set theory which is equivalent in strength to [[ZFC]].

#### Two-sorted categorical set theory

Two-sorted categorical set theory consists of [[sets]] and [[functions]] as primitive [[judgments]] or [[sorts]]. The elements are defined as functions with singleton domain.

The canonical example of this is the two-sorted first-order theory [[ETCS]] of sets, functions, identity functions, and composition of functions. An [[axiom schema of collection]] or [[axiom schema of replacement]] could be added to ETCS to get an two-sorted categorical set theory which is equivalent in strength to [[ZFC]].

#### Three-sorted categorical set theory

Three-sorted categorical set theory consists of [[sets]], [[functions]], and [[elements]] as primitive [[judgments]] or [[sorts]]. An example of a three-sorted categorical set theory is the simply-sorted presentation of [[structural ZFC]].

### Dependently sorted categorical set theory

Two-sorted categorical set theory can also be expressed as a dependently sorted categorical set theory, where there are [[sets]], and there are [[functions]] which depend on a pair of sets.

An example of this is the dependently sorted version of [[ETCS]].

Three-sorted categorical set theory can also be expressed as a dependently sorted categorical set theory, where there are [[sets]], there are [[elements]] which depend on a set, and there are [[functions]] which depend on a pair of sets.

An example of this is [[ETCS with elements]], which defines the theory of a [[concrete category|concrete]] [[well-pointed topos]] with [[natural numbers object]] and the [[axiom of choice]].

An [[axiom schema of collection]] or [[axiom schema of replacement]] could be added to ETCS with elements to get an two-sorted categorical set theory which is equivalent in strength to [[ZFC]].

## See also

* [[structural set theory]]

* [[categorical set theory]], [[allegorical set theory]]

[[!redirects categorical set theory]]
[[!redirects categorical set theories]]


categorical shape theory

In the original form of [[shape theory|abstract shape theory]], the restrictions that (i) it should refer to a _subcategory_ of 'nice' or 'good' objects, and (ii) that subcategory should be [[pro-reflective subcategory|pro-reflective]] ([[dense subcategory|shape theorist's 'dense']]) were unnecessarily restrictive, eliminating interesting situations where similar constructions could be useful, and tended to obscure the nature of certain of the results, which were more much general than was initially apparent.

This was explored by Deleanu and Hilton in the early 1970s.

## Definition##
Let $K : D\to C$ be a [[functor]]. The **shape category** of $K$ is the category with objects the same objects as those of $C$ but with morphisms from $c$ to $c^\prime$ being the functors between the corresponding [[comma categories]] $(c^\prime)/K)$ and $(c/K)$ (note reversal of order) that are compatible with the codomain functors to $D$.

(More to come later!)


categorical trace

The term _categorical trace_ usually refers to the concept introduced in

* Nora Ganter and Mikhail Kapranov, _Representation and character theory in 2-categories_ ([arXiv](http://arxiv.org/abs/math.KT/0602510), [blog](http://golem.ph.utexas.edu/string/archives/000757.html))

#Definition#

Let $C$ be a [[2-category]] and $F : B \to B$ a 1-[[endomorphism]] on one of its objects. Then the **categorical trace** $Tr(F)$ of $F$ is the collection of 2-morphisms from the identity $Id_B$ on $B$ into $B$
$$
  Tr(F) = Hom_C(Id_B, F)
  \,.
$$

#Examples#

* For $C = KV2Vect$, the 2-category of Kapranov-Voevodsky [[2-vector space]]s, a 1-endomorphism $Vect^n \stackrel{F}{\to} Vect^n$ is an $n \times n$-matrix $(V_{i j})_{i,j}$ of vector spaces $V_{i j} \in Vect$. The categorical trace on that is the direct sum of the diagonal entries, $Tr(F) = \oplu_{i} V_{i i}$. Hence under the decategorification functor from $KV2Vect$ to vectors and matrices with entries in $\mathbb{N}$, this becomes the ordinary trace of matrices.

#Remarks#

* The categorical trace is closely related to the [[span trace]].


categorical wreath product

> for disambiguation see _[[wreath product]]_

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

## Definition

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

Let $A$ be a [[small category]]. Its _categorical wreath product_  with the [[simplex category]] is the category $\Delta \wr A$ whose

* objects are $k$-tuples $([k], (a_1, \cdots, a_k))$ of objects of $A$, for any $k \in \mathbb{N}$;

* morphisms are tuples

  $$
    (\phi, \phi_{i j}) : ([k],(a_1, \cdots, a_k)) \to ([l],(b_1, \cdots, b_l))
  $$

  consisting of

  * a morphism $\phi: [k] \to [l]$ in $\Delta$;

  * morphisms $\phi_{i j} : a_i \to b_j$ for $0 \lt i \leq k$ and $\phi(i-1) \lt j \leq \phi(i)$.

=--

([Berger, def. 3.1](#Berger)).

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

An object of $\Delta \wr A$ is to be thought of as a sequence of morphisms labeled by objects of $A$

$$
  \array{
    0
    \\
    \downarrow \mathrlap{a_1}
    \\
    1
    \\
    \downarrow \mathrlap{a_2}
    \\
    \downarrow
    \\
    \vdots
    \\
    \downarrow \mathrlap{a_n}
    \\
    n
  }
$$

and morphisms are given by maps between these linear orders equipped with morphisms from the $k$th object in the source to all the objects in the target that sit in between the image of the $k$th step.

=--

## Examples

* The $k$-fold wreath product of the [[simplex category]] with itself is the
  $n$th [[Theta-category]] $\Theta_n = \Delta^{\wr n}$.

* Other applications are discussed at _[[club]]_ and at _[[terminal coalgebra of an endofunctor]]_.

## References

Section 3 of

* [[Clemens Berger]], _Iterated wreath product of the simplex category and iterated loop spaces_ ([arXiv:math/0512575](http://arxiv.org/abs/math/0512575)),
  {#Berger}

[[!redirects categorical wreath products]]


categoricity

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

#Contepts#
* table of contents
{:toc}

## Idea

In [[model theory]], given a [[cardinal]] $\kappa$, a [[theory]] is __$\kappa$-categorical__ (or _categorical in cardinality $\kappa$_), if it has precisely one [[isomorphism class]] of [[models]] of [[cardinality]] $\kappa$.

## Properties

The __Morley categoricity theorem__ says that a [[first-order theory]] $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.

The main interest is beyond first order.
There is much work there, for non-elementary classes, centered around (various versions of) Shelah's Categoricity Conjecture especially in the setting of [[abstract elementary class]]es.
Also, very interesting program is initiated by Zilber in early 2000s.

## Related concepts

Study of categoricity lead historically to the development of the [[stability theory]] in model theory, see also [[geometric stability theory]].

## References

* John Baldwin, _Categoricity_, Amer. Math. Soc. 2011, [pdf](http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf)

* John T. Baldwin, _What is a complete theory_, talk, [pdf](http://homepages.math.uic.edu/~jbaldwin/nd08.pdf)

Morley's categoricity theorem is from

* M. Morley, _Categoricity in power_, Trans. Amer. Math. Soc.
__114__:514–538, 1965.

* wikipedia: [Morley categoricity theorem](http://en.wikipedia.org/wiki/Morley%27s_categoricity_theorem)

* [[Saharon Shelah]], _Classification theory and the number of non-isomorphic models_,  2nd ed. North Holland Amsterdam, 1990.
* Saharon Shelah, _Categoricity of abstract classes with amalgamation_, Annals of Pure and Applied Logic 98(1-3) 241–294, 1999.
* [[Boris Zilber]], _Uncountably categorical theories_, Transl. Math. Monog., Amer. Math. Soc. 1993
* Rami Grossberg, Monica Vandieren, _Shelah's categoricity conjecture from a successor for tame AECs_, [pdf](https://www.math.cmu.edu/~rami/vp,.pdf)
* [[Michael Makkai]], Saharon Shelah, _Categoricity of theories $L_{\kappa\omega}$ with $\kappa$ a compact cardinal_, Annals of Pure and Applied Logic 47, 41-97, 1990
* Christian Espíndola, _An extension of Shelah’s eventual categoricity conjecture to accessible categories with directed colimits_, [arxiv/1906.09169](https://arxiv.org/abs/1906.09169); _A short proof of Shelah’s eventual categoricity conjecture for AEC’s with interpolation, under GCH_, [arxiv/1909.13713](https://arxiv.org/abs/1909.13713); _A topos-theoretic proof of Shelah’s eventual categoricity conjecture_, Brno, April 30, 2020, video [yt](https://www.youtube.com/watch?v=P8mmP40kSgM), slides [pdf](http://www.math.muni.cz/~bourkej/1ESP/1ESPslides.pdf)
* Adam Harris, _Categoricity and covering spaces_, PhD thesis, Merton college [arXiv:1412.3484](https://arxiv.org/pdf/1412.3484)
* C. Daw, A. Harris, _Categoricity of modular and Shimura curves_, J. Inst. Math. Jussieu, 16(5) (2017) 1075--1101 [doi](https://doi.org/10.1017/S1474748015000365)
* [[Boris Zilber]], Chris Daw, _Modular curves and their pseudo-analytic cover_, [arXiv:2107.11110](https://arxiv.org/abs/2107.11110); _Canonical models of modular curves and the Galois action on CM-points_, [arXiv:2106.06387](https://arxiv.org/abs/2106.06387)

category: model theory


categories and logic - table

[[!redirects logic and categories - list]]
| [[category]] | [[functor]] | [[internal logic]] | [[theory]] | [[hyperdoctrine]] | [[subobject poset]] | [[coverage]] | [[classifying topos]]
|-----------|---------------------|-----------------|-----------------|------------------|---------------|---------------|---------------|
| [[finitely complete category]] | [[cartesian functor]] | [[cartesian logic]] | [[essentially algebraic theory]] | | | | |
| [[lextensive category]] |  | [[disjunctive logic]] |  | | | | |
| [[regular category]] | [[regular functor]] | [[regular logic]] | [[regular theory]] | [[regular hyperdoctrine]] | [[infimum]]-[[semilattice]] | [[regular coverage]] | [[regular topos]] |
| [[coherent category]] | [[coherent functor]] | [[coherent logic]] | [[coherent theory]] | [[coherent hyperdoctrine]] | [[distributive lattice]] | [[coherent coverage]] | [[coherent topos]] |
| [[geometric category]] | [[geometric functor]] | [[geometric logic]] | [[geometric theory]] | [[geometric hyperdoctrine]] | [[frame]] | [[geometric coverage]] | [[Grothendieck topos]] |
| [[Heyting category]] | [[Heyting functor]] | [[intuitionistic logic|intuitionistic]] [[first-order logic]] | intuitionistic [[first-order theory]] | [[first-order hyperdoctrine]] | [[Heyting algebra]] | | |
| [[De Morgan Heyting category]] |  | [[intuitionistic logic|intuitionistic]] [[first-order logic]] with [[weak excluded middle]] |  |  | [[De Morgan Heyting algebra]] | | |
| [[Boolean category]] |  | [[classical logic|classical]] [[first-order logic]] | classical [[first-order theory]] | [[Boolean hyperdoctrine]] | [[Boolean algebra]] | | |
|[[star-autonomous category]] | | [[multiplicative classical linear logic]] | | | | | |
|[[symmetric monoidal closed category]] | | [[multiplicative intuitionistic linear logic]] | | | | | |
|[[cartesian monoidal category]] | | [[fragment]] $\{\&, \top\}$ of [[linear logic]] | | | | | |
|[[cocartesian monoidal category]] | | [[fragment]] $\{\oplus, 0\}$ of [[linear logic]] | | | | | |
|[[cartesian closed category]] | | [[simply typed lambda calculus]] | | | | | |


Categories and Sheaves

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

This entry provides links related to the book

* [[Masaki Kashiwara]], [[Pierre Schapira]],

  **Categories and Sheaves**

  Grundlehren der Mathematischen Wissenschaften __332__

  Springer (2006)

  [doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4)

  [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)

on basics of [[category theory]] and the foundations of [[homological algebra]] and [[abelian sheaf cohomology]].

See also the related lecture notes:

* [[Pierre Schapira]], _Categories and homological algebra_ (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))

* [[Pierre Schapira]], *An Introduction to Categories and Sheaves*, lecture notes (2023) [[pdf](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf)]

**Summary**

The book discusses the theory of [[presheaf|presheaves]] and [[sheaf|sheaves]] with an eye towards their application in [[homological algebra]] and with an outlook on [[stack]]s.

A self-contained introduction of the basics of [[presheaf]]-categories with detailed discussion of [[representable functor]]s and the corresponding notions of [[limit]]s, [[colimit]]s, [[adjoint functor]]s and [[ind-object]]s forms the first third of the book.

The second part describes central concepts and tools of modern category-theoretic [[homological algebra]] in terms of [[derived category|derived]] [[triangulated category|triangulated categories]].

The last part merges these two threads in a discussion of [[sheaf|sheaves]] in general and [[abelian sheaf|abelian sheaves]] in particular. This provides the machinery for the consideration of [[abelian sheaf cohomology]] conceptually embedded into the general notion of [[cohomology]] and [[(infinity,1)-category of (infinity,1)-sheaves|higher stacks]], on which the last section provides an outlook.

The organization and emphasis of the book (for instance of the [[category of sheaves]] as a [[localization]] of the category of [[presheaf|presheaves]]) makes it a suitable 1-categorical preparation for the [[infinity-category|infinity-categorical]] discussion of sheaves in

* J. Lurie, _[[Higher Topos Theory]]_

and of triangulated categories, i.e. [[stable (infinity,1)-category|stable infinity-categories]], in

* J. Lurie, _[[Stable Infinity-Categories]]_

On the other hand, [[topos]]-theoretic aspects of the [[category of sheaves]] are not emphasized, here

* Moerdijk-MacLane, _[[Sheaves in Geometry and Logic]]_

is the natural complementary reading. In particular sections V and VII there are directly useful for supplementing the concept of [[geometric morphism]] and its relation to [[localization]].

#Contents#
* table of contents
{:toc}

The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

For a pedagogical motivation of the general topic under consideration here see

* [[motivation for sheaves, cohomology and higher stacks]]

## 1 The language of categories

* [[foundations]]

  * [[ETCS]]

  * [[Grothendieck universe]]

* [[category]]

  * [[morphism]]

    * [[hom-set]]

    * [[identity morphism]]

    * [[epimorphism]]

    * [[monomorphism]]

    * [[inverse]]

    * [[isomorphism]]

    * [[endomorphism]]

    * [[automorphism]]

  * [[object]]

    * [[initial object]]

    * [[terminal object]]

    * [[zero object]]

  * [[small category]]

  * [[locally small category]]

  * [[finite category]]

  * [[opposite category]]

  * [[subcategory]]

  * general examples

    * [[discrete category]]

    * [[partial order|poset]]

    * [[groupoid]]

    * [[monoid]]

    * [[group]]

    * [[arrow category]]

      * [[comma category]]

        * [[category of elements]]

        * [[generalized universal bundle]]

      * [[subobject]]

  * [[partially ordered category]]

  * concrete examples

    * classical examples: sets with [[stuff, structure, property|extra structure]]

      * [[Set]]

      * [[Top]]

      * [[Vect]]

      * [[simplex category]]

    * [[point]]

    * [[globe]]

    * [[fundamental groupoid]]

* [[functor]]

  * [[stuff, structure, property]]

    * [[forgetful functor]]

    * [[essentially surjective functor]]

    * [[full functor]]

    * [[faithful functor]]

    * [[subcategory]]

  * [[natural transformation]]

    * [[functor category]]

    * [[skeleton]]

    * [[equivalence|equivalence of categories]]

  * [[contravariant functor]]

* [[presheaf]]

  * [[representable functor]]

  * [[Yoneda lemma]]

    * [[Yoneda embedding]]

    * [[Yoneda extension]]

    * [[free cocompletion]]

    * [[co-Yoneda lemma]]

  * [[adjoint functor]]

    * [[adjunction]]

    * [[examples of adjoint functors]]

* functor and presheaf categories

  * [[simplicial set]]

  * [[presheaf on open subsets]]

## 2 Limits

* [[limit]]

  * [[colimit]]

  * examples

    * [[limits and colimits by example]]

    * [[product]]

    * [[coproduct]]

    * [[pullback]]

    * [[pushout]]

    * [[fiber product]]

    * [[equalizer]]

    * [[coequalizer]]

    * [[kernel]]

    * [[cokernel]]

  * [[weighted limit]]

  * [[continuous functor]]

* [[Kan extension]]

  * [[Yoneda extension]]

* [[cofinal functor]]

* [[cofinally small category]]

## 3 Filtrant Limits

* [[filtered category]]

* [[filtered limit]]

* [[exact functor]]

  * [[flat functor]]

## 4 Tensor Categories

* [[monoidal category]]

  * [[braided monoidal category]]

  * [[closed monoidal category]]

* [[enriched category theory]]

  * [[enriched category]]

## 5 Generators and Representability

* [[image]]

  * [[coimage]]

  * [[strict morphism]]

    * [[strict epimorphism]]

    * [[regular epimorphism]]

* [[subobject]]

* [[generator]]

## 6 Indization of Categories

* [[ind-object]]

  * [[pro-object]]

  * [[finitely presentable object]]

  * [[compact object]]

## 7 Localization

* [[homotopical category]]

  * [[category with weak equivalences]]

  * [[category of fibrant objects]]

  * [[Waldhausen category]]

  * [[model category]]

* [[localization]]

  * [[calculus of fractions]]

    * [[multiplicative system]]

    * [[category of fractions]]

  * [[geometric embedding]]

  * [[homotopy category]]

  * [[derived functor]]

## 8 Additive and Abelian Categories

* [[group object]]

* [[Ab]]

* [[Ab-enriched category]]

* [[additive and abelian categories]]

  * [[pre-additive category]]

  * [[additive category]]

  * [[pre-abelian category]]

  * [[abelian category]]

  * [[semi-abelian category]]

  * [[Grothendieck category]]

* [[biproduct]]

  * [[direct sum]]

  * [[matrix calculus]]

* [[chain complex]]

  * [[kernel]]

  * [[cokernel]]

  * [[homology]]

## 9 $\pi$-accesible Objects and $F$-injective Objects

## 10 Triangulated Categories

* [[stable (infinity,1)-category]]

* [[triangulated category]]

* [[cohomological functor]]

* [[null system]]

## 11 Complexes in Additive Categories

* [[homological algebra]]

  * [[category with translation]]

  * [[differential object]]

    * [[complex]]

    * [[double complex]]

  * [[mapping cone]]

  * [[category of chain complexes]]

## 12 Complexes in Abelian Categories

* [[snake lemma]]

* [[quasi-isomorphism]]

## 13 Derived Categories

* [[derived category]]

* [[derived functor on a derived category]]

## 14 Unbounded Derived Categories

* [[injective object]]

* [[projective object]]

## 15 Indization and Derivation of Abelian Categories

## 16 Grothendieck Topologies

* [[coverage]]

  * [[Grothendieck topology]]

    * [[Lawvere-Tierney topology]]

  * [[sieve]]

  * [[local epimorphism]]

  * [[local isomorphism]]

  * [[dense monomorphism]]

* [[site]]

## 17 Sheaves on Grothendieck Topologies

* [[category of sheaves]]

  * [[sheaf]]

  * [[sheafification]]

    * [[geometric morphism]]

    * sheafification with respect to a  [[Lawvere-Tierney topology]]

    * [[IPC-property]]

  * operations

    * [[direct image]]

    * [[inverse image]]

    * [[restriction and extension of sheaves]]

## 18 Abelian Sheaves

* hindsight motivation

  * [[simplicial presheaf|simplicial sheaf]]

  * [[Dold-Kan correspondence]]

  * [[abelian sheaf]]

* [[closed monoidal structure on sheaves]]

* [[ringed site]]

* [[abelian sheaf cohomology]]

  * [[Deligne cohomology]]

## 19 Stacks and Twisted Sheaves

* [[higher category theory]]

  * [[2-category]]

  * [[2-groupoid]]

* [[stack]]

* outloook and further reading

  * [[infinity-stack homotopically]]

    * [[simplicial presheaf]]

    * [[model structure on simplicial presheaves]]

    * [[descent for simplicial presheaves]]

  * [[Higher Topos Theory]]

    * [[(infinity,1)-category of (infinity,1)-sheaves]]

    * [[descent]]

category: reference


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

This page compiles pointers related to:

* [[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)

which is a classical textbook on [[category theory]].

The iconic title probably refers to the declared ambition of demonstrating the prevalence of examples of [[adjoint functors]] occurring throughout [[mathematics]]:

> [p. 2:] "[[adjoint functor|Adjoints]], as we shall see, occur throughout [[mathematics]]."

> [p. 103:] "The multiple examples, here and elsewhere, of [[adjoint functors]] tend to show that adjoints occur almost everywhere in many branches of [[mathematics|Mathematics]]. It is the thesis of this book that a systematic use of all these [[adjunctions]] illuminates and clarifies these subjects."

See also:

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

* Wikipedia, *[Categories for the Working Mathematician](http://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician)*

and the list of category theory textbooks [here](category+theory#section_Textbooks).

{#Contents} **Contents**

The original edition has the following chapters:

*  **I. Categories, functors, and natural transformations**

   on the basic notions of [[categories]], [[functors]] and [[natural transformations]]

*  **II. Constructions on categories**

   on forming [[opposite categories]], [[product categories]], [[functor categories]], [[comma categories]], [[free categories]] and the ([[very large category|very large]]) [[category of categories]].

*  **III. Universals and limits**

   on [[universal constructions]] and [[limits]]

*  **IV. Adjoints**

   on [[adjoint functors]]

*  **V. Limits**

   on [[limits]] and [[colimits]]

*  **VI. Monads and algebras**

   on [[monads]] and [[algebras over a monad]]

*  **VII. Monoids**

   on [[monoid objects]]

*  **VIII. Abelian categories**

   on [[abelian categories]]

*  **IX. Special limits**

   on [[filtered limits]], [[final functors]], [[ends]], [[coends]]

*  **X. Kan extensions**

   on [[Kan extensions]]

The 2nd edition was published in 1997 with two additional chapters:

* **XI. Symmetry and braidings in monoidal categories**

  on [[braided monoidal categories|braided]] and [[symmetric monoidal categories]]

  also [[monoidal functors]], [[strict monoidal categories]] and the [[braid group]]

* **XII. Structures in categories**

  on [[internal categories]], [[simplicial nerves]], [[2-categories]]/[[bicategories]], [[crossed modules]] and [[2-groups]]

category: reference

[[!redirects Categories for Working Mathematician]]
[[!redirects CWM]]
[[!redirects Categories Work]]


Categories in Continuum Physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

This page collects some links related to the book

* [[William Lawvere]], [[Stephen Schanuel]] (eds.),

  _Categories in Continuum Physics_

  Lectures given at a Workshop held at SUNY, Buffalo 1982

  Lecture Notes in Mathematics 1174

  1986

  [doi:10.1007/BFb0076928](https://link.springer.com/book/10.1007/BFb0076928)

on the [[foundations]] of [[continuum mechanics]] (also [[rational thermodynamics]]), [[variational calculus]] and [[equations of motion]] in [[synthetic differential geometry]], [[diffeological spaces]], [[Frölicher spaces]] and of the notion [[intensive and extensive]] properties by [[mapping spaces]] in [[ring objects]],  and related constructions in [[topos theory]].

Related texts by Lawvere include

* [[William Lawvere]], _[[Toposes of laws of motion]]_

For related discussion see also at _[[geometry of physics]]_ and _[[higher category theory and physics]]_.

category: reference


categories in SEAR

# Categories in SEAR
* table of contents
{: toc}

## Outline

In response to a [request](http://golem.ph.utexas.edu/category/2009/09/towards_a_computeraided_system.html#c027419), this page explains how to define a [[small category]] using [[SEAR]] as [[foundations]].  Note that, while SEAR respects the [[principle of equivalence]] in that one cannot compare arbitrary sets for equality, it has no provision for small types with no notion of equality.  Thus we will be formalising small [[strict category|strict categories]], and it will be possible to make statemens about them that do not respect the [[principle of equivalence]].

There are actually two ways to interpret this programme.  One is to accept every set in SEAR as 'small' and consider categories that are small in that sense.  So the category of *all* SEAR sets cannot be formalised this way, but any category whose set of objects and hom-sets are sets in SEAR can be formalised in this way.  Another way is to declare a [[family of sets]] within SEAR to be the family of 'small' sets; then only categories with a set of morphisms that belongs to this family can be formalised in this way.  (This way is even more [[principle of equivalence]]-breaking in that one can even consider equality of categories, not only equality of objects within a given category.)  There is small category (in the first sense) of all small categories (in the second sense).

We call these two interpretations the 'large universe' and 'small universe' interpretations.

## Large universe version

A __category__ is a syntactic quintuple that consists of

*  a set $O$ (called the set of _objects_),
*  a set $M$ (called the set of _morphisms_),
*  a function $s: M \to O$ (the _source_ map),
*  a function $t: M \to O$ (the _target_ map),
*  and a partial function $c: M \times M \dashrightarrow M$ (the _composition map_)

satsifying these conditions:

*  For $f, g$ elements of $M$, $c(f,g)$ is defined if and ony if $s(f) = t(g)$.
*  In that case, $s(c(f,g)) = s(g)$ and $t(c(f,g)) = t(f)$.
*  For every element $x$ of $O$, there exists an element $i$ of $M$ such that
   *  $s(i) = x$ and $t(i) = x$,
   *  $c(i,f) = f$ for every element $f$ of $M$ such that $x = t(f)$, and
   *  $c(f,i) = f$ for every element $f$ of $M$ such that $s(f) = x$.
*  For $f, g, h$ elements of $M$ such that $s(f) = t(g)$ and $s(g) = t(h)$, we have $c(c(f,g),h) = c(f,c(g,h))$.

Note that a partial function from $p: X \dashrightarrow Y$ is formalised as a [[functional relation]], that is a relation $\phi: X \to Y$ such that $b = c$ whenever $\phi(a,b)$ and $\phi(a,c)$ both hold (for elements $a$ of $X$ and $b,c$ of $Y$).  We say that $p(x)$ is _defined_ if there is $y$ such that $\phi(x,y)$ holds; in that case, such $y$ is unique and we denote it $p(x)$.  This notation can be eliminated from the language using [[existential quantification]], the same as when $p$ is a function.

## Small universe version

Take a relation $m : U \looparrowright E$ as the _family of small sets_.  Then a __small category__ (relatively to this family) is an element of ... such that ....  Along with the set of small categories, we consider the relations ..., which are respectively the families of _sets of objects_, _sets of morphisms_, _source maps_, _target maps_, and _composition maps_.

+-- {: .standout}
Hopefully it\'s obvious, but this not completed.  It would be nice to carry the development to the point that, given any family of sets, the relatively small categories in this sense form a category in the above sense.
=--

[[!redirects category in SEAR]]
[[!redirects categories in SEAR]]
[[!redirects SEAR category]]
[[!redirects SEAR categories]]


categories mailing list

The *Categories mailing list* is a [mailing list](https://en.wikipedia.org/wiki/Mailing_list) on the topic of *[[category theory]]*.

The list commenced in 1990 and is still (somewhat) active. From 1990 to 2023, the mailing list was hosted by Mount Allison University (categories@mta.ca) and moderated by [[Bob Rosebrugh]]. In 2023, the mailing list was transitioned to Macquarie University (categories@mq.edu.au) and is currently moderated by [[Jean-Simon Lemay]].

The journal *[[Theory and Applications of Categories]]* was [launched on the mailing list in January 1995](https://www.mta.ca/~cat-dist/catlist/1999/elec-journal).

## Archives

The list's messages are archived in a number of different ways, depending on date:

* For June 1994 to November 1999, there is a curated website, organised alphabetically by topic/title: <https://www.mta.ca/~cat-dist/catlist/>

* For 11 March 1990 (the start) to July 2009 (when a change in Outlook led to a halt in archiving), there are compiled monthly digests, organised in yearly/monthly folders here: <https://www.mta.ca/~cat-dist/archive/>

* An dump of the [Gmane](https://en.wikipedia.org/wiki/Gmane) archive (otherwise only accessible via the protocol [nntp](https://en.wikipedia.org/wiki/Network_News_Transfer_Protocol)) can be found on [GitHub](https://github.com/punkdit/categories/), which is searchable, but not sorted or archived in any particularly human-readable way.

## Links

The website for the old categories mailing list (categories@mta.ca):

* [www.mta.ca/~cat-dist](https://www.mta.ca/~cat-dist)

## See also

* [[Theory and Applications of Categories]]
* [[Category Theory Zulip]]


categories of categories - contents

**$(n+1,r+1)$-categories of [[(n,r)-categories]]**

* [[Pos]]

* [[Set]]

  * [[Rel]]

* [[Grpd]], [[∞Grpd]]

* [[Cat]]

  * [[Ho(Cat)]]

  * [[AccCat]]

  * [[PrCat]]

  * [[LexCat]]

  * [[MonCat]]

  * [[VCat]]

  * [[CatAdj]]

  * [[Prof]]

  * [[Operad]]

* [[2Cat]]

* [[ModCat]], [[CombModCat]]

* [[(∞,1)Cat]]

  * [[Pr(∞,1)Cat]]

  * [[(∞,1)Operad]]

* [[(∞,n)Cat]]


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

#Contents#
* table of contents
{:toc}

## Idea

A basic example of [[limits commuting with colimits]] in [[category theory]] is that [[colimits]] over [[opposite categories]] of categories with [[finite products]] preserves [[finite products]]. One says equivalently that categories with [[finite products]] are _[[cosifted categories]]_.

## Statement

+-- {: .num_example #CategoriesWithFiniteProductsAreCosifted}
###### Example
**([[categories with finite products are cosifted]])**

Let $\mathcal{C}$ be a [[small category]] which has [[finite products]]. Then $\mathcal{C}$ is a _[[cosifted category]]_, equivalently its [[opposite category]] $\mathcal{C}^{op}$ is a _[[sifted category]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] are _[[sifted colimits]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] _[[limits commuting with colimits|commute]] with [[finite products]]_, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to [[functors]] on the [[opposite category]] of $\mathcal{C}$ (hence two [[presheaves]] on $\mathcal{C}$) we have a [[natural isomorphism]]

$$
  \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
  \left(
    \mathbf{X} \times \mathbf{Y}
  \right)
  \;\simeq\;
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{X}
  \right)
  \times
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{Y}
  \right)
  \,.
$$

=--

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

First observe that for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ two [[presheaves]], their [[Cartesian product]] is a [[colimit]] over [[representable presheaf|presheaves represented]] by Cartesian products in $\mathcal{C}$. Explicity, using [[coend]]-notation, we have:

\[
  \label{OnSiteWithProductsExpandProductOfPresheaves}
  \mathbf{X} \times \mathbf{Y}
  \;\simeq\;
  \int^{c_1,c_2 \in \mathcal{C}}
  y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
  \,,
\]

where $y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set]$ denotes the [[Yoneda embedding]].

This is due to the following sequence of [[natural isomorphisms]]:

$$
  \begin{aligned}
    (\mathbf{X} \times \mathbf{Y})(c)
    & \simeq
    \left(
      \int^{c_1 \in \mathcal{C}}
      \mathcal{C}(c,c_1) \times \mathbf{X}(c_1)
    \right)
    \times
    \left(
      \int^{c_2 \in \mathcal{C}}
      \mathcal{C}(c,c_2) \times \mathbf{Y}(c_2)
    \right)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}}
    \int^{c_2 \in \mathcal{C}}
    \underset{
      \simeq \mathcal{c}(c, c_1 \times c_2)
    }{
    \underbrace{
      \mathcal{C}(c,c_1) \times \mathcal{C}(c,c_2)
    }}
    \times
    \left(
      \mathbf{X}(c_1) \times \mathbf{X}(c_2)
    \right)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}}
    \int^{c_2 \in \mathcal{C}}
    \mathcal{C}(c,c_1 \times c_2)
    \times
       \mathbf{X}(c_1) \times \mathbf{X}(c_2)
    \,,
  \end{aligned}
$$

where the first step expands out both presheaves as colimits of representables separately, via the [[co-Yoneda lemma]], the second step uses that the [[Cartesian product]] of presheaves is a two-variable [[left adjoint]] (by the [[symmetric monoidal category|symmetric]] [[closed monoidal structure on presheaves]]) and [[adjoints preserve (co-)limits|as such preserves colimits]] (in particular [[coends]]) in each [[variable]] separately, and under the brace we use the defining [[universal property]] of the [[Cartesian products]], assumed to exist in $\mathcal{C}$.

Now observe that the [[colimit]] of a [[representable presheaf]] is the [[singleton]].

\[
  \label{ColimitOfRepresentableIsSingleton}
  \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c)
  \;\simeq\;
  \ast
  \,.
\]

One way to see this is to regard the colimit as the [[left Kan extension]] along the unique functor $\mathcal{C}^{op} \overset{p}{\to} \ast$ to the [[terminal category]]. By the formula [there](Kan+extension#PointwiseByCoEnds), this yields

$$
  \begin{aligned}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    y(c)
    & \simeq
    \int^{c_1 \in \mathcal{C}}
      \underset{const_\ast(c_1)}{\underbrace{\ast(-,p(c_1))}} \times y(c)(c_1)
    \\
    & \simeq
    \int^{c_1 \in \mathcal{C}} const_\ast(c_1) \times \mathcal{C}(c_1,c)
    \\
    & \simeq const_\ast(c)
    \\
    & \simeq \ast
  \end{aligned}
$$

where we made explicit the [[constant functor]] $const_\ast \;\colon\; \mathcal{C} \to Set$, constant on the [[singleton]] set $\ast$, and then applied the [[co-Yoneda lemma]].

Using this, we compute for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ the following sequence of [[natural isomorphisms]]:

$$
  \begin{aligned}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \left(
      \mathbf{X} \times \mathbf{Y}
    \right)
    &
    \simeq
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \int^{c_1,c_2 \in \mathcal{C}}
    y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
    \left(
      y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
    \right)
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \left(
      \underset{
        \simeq \ast
      }{
      \underbrace{
        \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim}
        y(c_1 \times c_2)
      }}
    \right)
    \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
    \\
    & \simeq
    \int^{c_1,c_2 \in \mathcal{C}}
    \left(
      \mathbf{X}(c_1) \times \mathbf{Y}(c_2)
    \right)
    \\
    & \simeq
    \left(
      \int^{c_1\in \mathcal{C}}
      \mathbf{X}(c_1)
    \right)
    \times
    \left(
      \int^{c_2\in \mathcal{C}}
      \mathbf{Y}(c_2)
    \right)
    \\
    & \simeq
    \left(
      \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
      \mathbf{X}
    \right)
    \times
    \left(
      \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
      \mathbf{Y}
    \right)
  \end{aligned}
$$

Here the first step is (eq:OnSiteWithProductsExpandProductOfPresheaves), the second uses that [[colimits commute with colimits]], the third uses again that the [[Cartesian product]] respects colimits in each variable separately, the fourth is (eq:ColimitOfRepresentableIsSingleton), the last step is again the respect for colimits of the Cartesian product in each variable separately.

=--

## Applications

* In the definiton of _[[cohesive site]]_ and _[[infinity-cohesive site]]_, the assumption that the site has finite products (more generally: is [[cosifted category|cosifted]]) is what guarantees that the induced [[shape modality]] preserves finite products, as (usually) required for a [[cohesive topos]] or [[cohesive (infinity,1)-topos]]

## Related concepts

* [[commutativity of limits and colimits]]


Categories, Allegories

This page is about the textbook

* [[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)

on [[category theory]] and [[categorical algebra]].

On the [[category|Categories]] side, the book centers on that part of categorical algebra that studies [[exactness property|exactness properties]], or other properties enjoyed by nice or convenient categories such as [[topos|toposes]], and their relationship to [[logic]] (for example, [[geometric logic]]). A major theme throughout is the possibility of representation theorems (aka completeness theorems or embedding theorems) for various categorical structures, spanning back now about five decades (as of this writing) to the original embedding theorems for abelian categories, such as the [[Freyd-Mitchell embedding theorem]].

On the [[allegory|Allegories]] side: it may be said they were first widely publicized in this book. They comprise many aspects of [[relation|relational]] algebra corresponding to the categorical algebra studied in the first part of the book.

The book, while it covers an extraordinary amount of ground in less than 300 pages, is fairly idiosyncratic, especially in the choice of terminology (such as _terminator_ instead of _terminal object_) and in the overall arrangement (designed to be self-contained for the diligent reader). There is no list of references given.

category: reference


categorification - contents

**categorification**

## Background

* [[category theory]]

* [[higher category theory]]

## Contents

* [[vertical categorification]]

* [[horizontal categorification]]

* [[(∞,1)-categorification]]

* [[decategorification]], [[Grothendieck group]]

## Examples

* [[categorification in representation theory]]

* [[Khovanov homology]]
* [[Kazhdan-Lusztig theory]]
* [[categorification via groupoid schemes]]
* The [[geometric Langlands]] program
* [[geometric Satake]]

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


categorification in representation theory

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

#Contents#
* table of contents
{:toc}

## Idea

Many important cases of [[categorification]] (in fact most of those so far studied in $n$Lab) belong to the categorification of basic and general structures in category theory, algebra and geometry like fibered categories, monads, operads, sheaves etc. To find the "correct" categorification one usually needs just clear understanding of foundations and clear categorical strategy.

On the other hand, a number of categorifications of rather special structures in [[representation theory]] on the interface of [[Lie theory]] and low dimensional topology, is emerging from study of rather special and deep phenomena. In those examples special and often advanced structures in quantum group theory, knot theory etc. are starting revealing to be a shadow of more fundamental structures on the categorified level.

## Related concepts

* [[Khovanov homology]]

* [[Langlands program]]

## References
 {#References}

* J. Chuang, [[Raphaël Rouquier|R. Rouquier]], _Derived equivalences for symmetric groups and $sl_2$-categorification_, Ann. Math. __167__ (2008), 245–298.
* S.-J. Kang, [[M. Kashiwara]], _Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras_, [arXiv:1102.4677](http://arxiv.org/abs/1102.4677)
* S.-J. Kang, [[M. Kashiwara]], E. Park, _Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data_, [arxiv/1202.1622](http://arxiv.org/abs/1202.1622)
* [[M. Kashiwara]], _Biadjointness in cyclic Khovanov-Lauda-Rouquier algebras_, [arxiv/1111.5898](http://arxiv.org/abs/1111.5898)
* [[A. Lauda]], M. Vazirani, _Crystals from categorified quantum groups_, Adv. Math., 228, no. 2, (2011), 803–861, [arXiv:0909.1810](http://arxiv.org/abs/0909.1810)

* [[Mikhail Khovanov]], _Nilcoxeter algebras categorify the Weyl algebra_,  Comm. Algebra __29__, No. 11 (2001) 5033–5052, [math.RT/9906166](http://arxiv.org/abs/math/9906166), [MR2002h:16041](http://www.ams.org/mathscinet-getitem?mr=1856929), [doi](http://dx.doi.org/10.1081/AGB-100106800)

* [[Mikhail Khovanov]], V. Mazorchuk, [[Catharina Stroppel]], _A brief review of abelian categorifications_, [math.RT/0702746](http://arxiv.org/abs/math/0702746)

* [[J. Bernstein]], [[I. Frenkel]], M. Khovanov, _A categorification of the Temperley-Lieb algebra and Schur quotients of $U(sl_2)$ via projective and Zuckerman functors_, Selecta Math. (N.S.) __5__ (1999), 199-241, [MR2000i:17009](http://www.ams.org/mathscinet-getitem?mr=1714141), [doi](http://dx.doi.org/10.1007/s000290050047)

* [[Mikhail Khovanov]], _A categorification of the Jones polynomial_, Duke Mathematical Journal __101__ (3): 359–426, 2000, [doi](http://dx.doi.org/10.1215/S0012-7094-00-10131-7), [MR1740682](http://www.ams.org/mathscinet-getitem?mr=1740682)

* [[Igor Frenkel]], [[Catharina Stroppel]], Joshua Sussan, _Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols_, [arXiv:1007.4680](http://arxiv.org/abs/1007.4680)

* [[Catharina Stroppel]], Joshua Sussan, _Categorified Jones-Wenzl Projectors: a comparison_, [arXiv:1105.3038](http://arxiv.org/abs/1105.3038)

* [[Catharina Stroppel]], [[Ben Webster]], _Quiver Schur algebras and q-Fock space_, [arXiv:1110.1115](http://arxiv.org/abs/1110.1115)

* [[Ivan Losev]], Ben Webster, _On uniqueness of tensor products of irreducible categorifications_, [arXiv/1303.1336](https://arxiv.org/abs/1303.1336)

category: algebra

[[!redirects categorification in Lie theory]]