total category

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#### Yoneda lemma
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## Idea

A **total category** (also called a **totally cocomplete category**, or an **absolutely cocomplete category**) is a [[category]] whose [[Yoneda embedding]] has a [[left adjoint]]. This totality condition implies various strong [[cocomplete category|cocompleteness]] and [[complete category|completeness]] properties that are especially convenient in the context of an [[adjoint functor theorem]].

In the words of [Street](#Street84), "Total categories are precisely the categories, algebraic and topological, at which traditional category theory was aimed". In other words, categories like [[Set]], [[Ab]], [[Grp]], [[Top]], [[TopGrp]]: traditional large categories of "mathematical structures". Roughly speaking, for the various species of structure in the zoology of [[Bourbaki]] (structures of algebraic type, topological type, order type, or some hybrid), if we consider species of a given structure type, and homomorphisms preserving that type of structure, as forming a category, then those sorts of categories have a strong tendency to be total. More precise theorems to this effect are given below.

## Definition

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

A [[locally small category]] $C$ is **total** if its [[Yoneda embedding]] $Y \;\colon \;C\longrightarrow [C^{op},Set]$ has a [[left adjoint]] $L$.

If the [[opposite category]] $C^{op}$ is total, $C$ is called **cototal**.

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

The definition above requires some [[set theory|set-theoretic]] assumption to ensure that the [[functor category]] $[C^{op},Set]$ exists, but it can be rephrased to say that the [[colimit]] of $Id_C:C\to C$ [[weighted limit|weighted]] by $W$ exists, for any $W:C^{op}\to Set$.  (This still involves [[quantification]] over large objects, however, so some foundational care is needed.)  This version has an evident generalization to [[enriched category|enriched]] categories.

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###### Remark

Since the [[Yoneda embedding]] is a [[full and faithful functor]], a total category $C$  induces an [[idempotent monad]] $Y \circ L$ on its [[category of presheaves]], hence a [[modality]].  One says that $C$ is a [[totally distributive category]] if this modality is itself the [[right adjoint]] of an [[adjoint modality]].

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

The $(L \dashv Y)$-[[adjunction]] of a total category is closely related to the
$(\mathcal{O} \dashv Spec)$-[[adjunction]] discussed at _[[Isbell duality]]_ and at _[[function algebras on ∞-stacks]]_. In that context the $L Y$-[[modality]] deserves to be called the _[[affine modality]]_.

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## Properties

* Every total category has all [[small colimits]].

* Total categories satisfy a very satisfactory [[adjoint functor theorem]]: any colimit-preserving functor from a total category to a locally small category has a [[right adjoint]].  (See [Wood 1982](#Wood82), Thm. 1 for a clear statement of this fact and [StreetWalters](#StreetWalters), p. 372 for a proof).

* Although the definition refers explicitly only to colimits, every total category is also [[complete category|complete]], i.e. has all small limits (this is a categorification of the fact that every cocomplete [[partial order]] is also complete).  It also has some large limits.  In fact, it has "all possible" large limits that a locally small category can have: if $F\colon D\to C$ is a functor such that $lim_d Hom_C(X,F d)$ is a small set for all $X\in C$, then $F$ has a limit.

* A total category $\mathcal{C}$ is [[cartesian closed category|cartesian closed]] iff $L$ preserves binary products (cf. [Wood 1982](#Wood82), Thm. 9).

* Recall that every functor $f : A \to X$ factors via the [[comprehensive factorisation]] as a [[final functor]] $e$ follows by a [[discrete fibration]] $m$. A locally small category $X$ is total if and only if for every such functor $f$, such that the discrete fibration $m$ has small fibres, has a colimit. (In fact, this definition makes sense even when $X$ is not locally small.)

## Examples

+-- {: .num_prop #Day}
###### Proposition
Any [[cocomplete category|cocomplete]] and [[epi-cocomplete category|epi-cocomplete]] category with a [[generator]] is total. More generally, any cocomplete and $E$-complete category with an $E$-generator is total, for a suitable class $E$.
=--

See ([Day](#Day)), theorem 1, for a proof.  This includes:

* [[locally presentable category|locally presentable categories]], hence in particular [[Grothendieck toposes]], or the category of [[abelian sheaves]] on a small [[site]].

Also, totality lifts along [[solid functors]]; that is, if the [[codomain]] of a solid functor is total, then so is its domain.  See ([Tholen](#Tholen)) for a proof.  This implies that the following types of categories are total:

* any [[reflective subcategory]] of a total category

For example

* any category which is [[monadic]] over [[Set]] (see also ([Kelly](#Kelly)) Thm. 6.17)

* any category admitting a [[topological functor]] to [[Set]]

* The category [[Grp]] of groups as a category monadic over $Set$ is total, but it is not cototal; see below.

* The category of [[topological groups]] is total as well since it is topological over the total category [[Grp]].

* If $C$ is total and $J$ is small, then $C^J$ is total, morally because it is a reflective subcategory of $Set^{C^{op} \times J}$; see section 6 of [Kelly](#Kelly).

Thus, "most naturally-occurring" cocomplete categories are in fact total.

In practice, i.e., in naturally occurring concrete cases, cototality is more rare. For example, it is frequently *not* the case that categories that are monadic over $Set$ are cototal. This is well-illustrated by the following two examples:

* The category of groups [[Grp]] is not cototal; if it were, then any continuous functor $Grp \to Set$ would be representable. To see this is not the case, it suffices to produce a class of [[simple groups]] $G_\alpha$ of unbounded [[cardinality]] (for example, for any [[infinite set]] $X$, the [[alternating group]] $Alt(X)$, consisting of permutations of finite support that are even, is simple and of cardinality equal to that of $X$). For any group $G$, the hom-set $\hom(G_\alpha, G)$ consists of a single element (the trivial homomorphism) as soon as the cardinality of $G_\alpha$ exceeds that of $G$. Thus the class-indexed product $\prod_\alpha \hom(G_\alpha, G)$ is bounded in size, and defines a continuous functor $F = \prod_\alpha \hom(G_\alpha, -): Grp \to Set$. But it is clear this functor is not representable; e.g., for any group $G$, one can find $G_\alpha$ such that $F(G_\alpha)$ is much larger in size than $\hom(G, G_\alpha)$. This example is given in [Wood 1982](#Wood82).

* By a similar construction, the category of commutative rings is not cototal. For each infinite cardinal $\alpha$, choose a field $F_\alpha$ of size $\alpha$, e.g., an algebraically closed field over $\mathbb{Q}$ of transcendence degree $\alpha$. Put $A_\alpha = \mathbb{Z} \times F_\alpha$. Then, for any commutative ring $R$, there is exactly one homomorphism $A_\alpha \to R$ as soon as $\alpha$ exceeds the cardinality of $R$. Then one argues that $\prod_\alpha \hom(A_\alpha, -): CRing \to Set$ is continuous but not representable.

But cototal categories do occur:

* [[Set]] is cototal (as well as total).

* By dualizing Proposition \ref{Day}, [[Ab]] is cototal (as well as total), because it is complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). Similarly, the category of [[modules]] $R Mod$ is cototal (and total) for any [[ring]] $R$. For that matter, any [[well-powered category|well-powered]] [[Grothendieck category]], such as the category of [[abelian sheaves]] on a small [[site]], is cototal.

* Similarly, the category $CH$ of [[compact Hausdorff spaces]] is cototal (as well as total, being monadic over $Set$), because like $Ab$ it is complete, well-powered, and has a cogenerator $I = [0, 1]$ (cf. [[Urysohn's lemma]]).

* If $C$ is cototal and $J$ is small, then $C^J$ is cototal.

* Any [[presheaf category]] of a [[small category]] is cototal (as well as total). Indeed, any [[Grothendieck topos]] is both cototal and total.

* Any category admitting a [[topological functor]] to [[Set]] is cototal (as well as total).

* Any [[totally distributive category]] is cototal (as well as total).

* Any [[coreflective subcategory]] of a cototal category is cototal, e.g., the category of [[compactly generated spaces]] is cototal.

## Related pages

* [[totally distributive category]]

* [[lex total category]]

* [[large cocompleteness]]

## References

* [[Syméon Bozapalides]], *Quelques Remarques sur les Cosmos Elementaires*, Δελτίο της Ελληνικής Μαθηματικής Εταιρίας 15.15 (1974) 144-151 [[pdf](https://hdml.di.ionio.gr/pdfs/journals/1.pdf), [[Bozapalides-CosmosElementaires.pdf:file]]]

* {#StreetWalters}[[Ross Street]], [[Bob Walters]], _Yoneda structures on 2-categories_, Journal of Algebra, 50 no. 2 (1978), p. 350-379, [doi:10.1016/0021-8693(78)90160-6](https://doi.org/10.1016/0021-8693%2878%2990160-6),  (contains the original definition of total categories)

*  {#Kelly} [[Max Kelly]], _A survey of totality for enriched and ordinary categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 2 (1986), p. 109-132, [numdam](http://www.numdam.org/item?id=CTGDC_1986__27_2_109_0)

* [[Walter Tholen]], _Note on total categories_, Bulletin of the Australian Mathematical Society [cambridge journals](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4759056)
 {#Tholen}

* {#Day} [[Brian Day]], _Further criteria for totality_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 28 no. 1 (1987), p. 77-78, [numdam](http://www.numdam.org/item?id=CTGDC_1987__28_1_77_0)

* {#Garner13} [[Richard Garner]], _Topological=Total_  ([arXiv:1310.0903](https://arxiv.org/abs/1310.0903))

* {#Street84} [[Ross Street]], _The family approach to total cocompleteness and toposes_ , Trans. A. M. S. **284** (1984) pp.355-369, (AMS)

* {#Wood82} [[Richard J. Wood]], *Some remarks on total categories*, J. Algebra **75** 2 (1982) 538-545 [doi:10.1016/0021-8693(82)90055-2]

* Michael Albert Wendt. _An introduction to totally cocomplete categories_. Master thesis (1985). ([url](https://dalspace.library.dal.ca/handle/10222/80035))

* [[Ross Street]]. _The comprehensive construction of free colimits_. Sydney Category Seminar Reports (Macquarie University). 1979. ([pdf](http://science.mq.edu.au/~street/CCoFC.pdf))

* [[Francisco Marmolejo]], [[Robert Rosebrugh]], and [[Richard Wood]]. _Completely and totally distributive categories I_. Journal of Pure and Applied Algebra 216.8-9 (2012): 1775-1790.

* [[Ross Street]]. _Elementary cosmoi I_. Category Seminar: Proceedings Sydney Category Theory Seminar 1972/1973. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

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[[!redirects cototal category]]
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total complex

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## Idea

For $C_{\bullet, \bullet}$ a [[double complex]] (in some [[abelian category]] $\mathcal{A}$), its _total complex_ $Tot(C)_\bullet$ is an ordinary complex which in degree $k$ is the [[direct sum]] of all components of total degree $k$.

## Definition

Let $\mathcal{A}$ be an [[abelian category]] with arbitrary [[direct sums]].

Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] in $\mathcal{A}$ and $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ for the category of [[double complexes]]. (Hence we use the convention that in a double complex the vertical and horizontal [[differential]] commute with each other.)

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###### Definition

For $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ a [[double complex]], its associated **total complex** $Tot(C)_\bullet \in Ch_\bullet(\mathcal{A})$ is the [[chain complex]] whose components are the [[direct sums]], [[direct products]] or a mixture of those

The sum total complex
$$
  Tot^{\oplus}(C)_n = \bigoplus_{k+l = n} C_{k,l}
$$
the product total complex
$$
  Tot^{\pi}(C)_n = \prod_{k+l = n} C_{k,l}
$$
the product-sum total complex
$$
  Tot^{\pi\oplus}(C)_n = \prod_{k+l = n, k \lt 0} C_{k,l}\oplus\bigoplus_{k+l = n, k\geq 0} C_{k,l}
$$
and the sum-product total complex
$$
  Tot^{\oplus\pi}(C)_n = \bigoplus_{k+l = n, k \lt 0} C_{k,l}\oplus\prod_{k+l = n, k\geq 0} C_{k,l}
$$

and whose [[differentials]] are given by the [[linear combination]]

$$
  \partial^{Tot}
  \coloneqq
  \partial^C_{vert} + (-1)^{vertical\;degree} \partial^C_{hor}
  \,.
$$
Using the four different types of total complexes gives the flexibility to make more nuanced exactness statements for the total complex under various assumptions on the double complex.

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## Properties
 {#Properties}

### Total homology and spectral sequences
 {#TotalHomology}

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

The [[chain homology]] of the total complex $Tot(C)_\bullet$ is sometimes called the **total homology** of the [[double complex]] $C_{\bullet, \bullet}$.

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

A tool for computing the homology of a total complex, hence for computing the total homology of a double complex, is the _[[spectral sequence of a double complex]]_.
See there for more details.

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### Exactness

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

First let $C$ be a double complex in any abelian category

* If $C_{\bullet,\bullet}$ is bounded and has [[exact sequence|exact]] rows or columns then also $Tot(C)_\bullet$ is exact.

Now let $C$ be a double complex of abelian groups.

* If $C_{\bullet,\bullet}$ has exact rows then the product-sum total complex is exact.
* If $C_{\bullet,\bullet}$ has exact rows and kernels (or equivalently) images between row complexes are exact, then the sum and product total complexes are exact
* If $C_{\bullet,\bullet}$ has exact rows and for each $i$ taking $H_i$ of the columns gives an exact complex, then the sum-product total complex is exact.

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

Use the [[acyclic assembly lemma]].

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### Relation to total simplicial sets and homotopy colimits
 {#RelationToTotalSimplicialSets}

The total chain complex is, under the [[Dold-Kan correspondence]], equivalent to the [[diagonal of a bisimplicial set]] -- see [[Eilenberg-Zilber theorem]]. As discussed at _[[bisimplicial set]]_, this is [[weak homotopy equivalence|weakly homotopy equivalent]] to  the _[[total simplicial set]]_ of a bisimplicial set.

## References

For instance:

* [[Charles Weibel]], section 1.2 of: _[[An Introduction to Homological Algebra]]_

[[!redirects total complexes]]
[[!redirects total chain complex]]
[[!redirects total chain complexes]]

[[!redirects total differential]]
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total derivative

## Idea

Given a [[function]] $f$ of $n$ [[variables]], and given $n$ functions $\{g_i\}$ of one variable, then the _total derivative_ of the composite function $t \mapsto f(g_1(t),\cdots, g_n(t))$ is (if it exists) simply its [[derivative]] with respect to $t$, but understood as a [[linear combination]]of the [[partial derivatives]] of $f$, via the [[chain rule]]:

$$
  \frac{d f}{d t}
    =
  \sum_i
  \frac{\partial f}{\partial g_i}
  \,
  \frac{d g_i}{d t}
  \,.
$$

This has various evident generalizations. One is the [[horizontal derivative]] in [[variational calculus]], see at _[[variational bicomplex]]_.

[[!redirects total derivatives]]


total derived functor

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## Idea

In the context of [[derived functors in homological algebra]] one often says "$k$th derived functor" for the $k$th [[chain homology]] of a functor applied to a suitable [[resolution]]. Omitting the passage to homology here is then called forming the _total derived functor_. In more general contexts of [[homotopy theory]], that total derived functor would simply be called the _[[derived functor]]_ itself.

See at _[[derived functor in homological algebra]]_ for a detailed discussion.

## Related concepts

* [[hyper-derived functor]]

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[[!redirects total right derived functor]]
[[!redirects total left derived functor]]


total order

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# Total orders
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{: toc}

## Idea

A *total order* or *linear order* or *chain order* on a set is a way of ordering its elements to say that some elements precede others, with the understanding that any two elements can be compared one way or the other.

## Definitions

### With a relation

Given a [[set]] $S$, a **total order** or **linear order** or **chain order** on $S$ is a (binary) [[relation]] $\leq$ with the following properties:

* [[reflexive relation|reflexivity]]: for any element $x$ of $S$, $x \leq x$;
* [[transitive relation|transitivity]]: whenever $x \leq y \leq z$, then $x \leq z$;
* [[antisymmetric relation|antisymmetry]]: whenever $x \leq y \leq x$, then $x = y$;
* [[total relation|totality]]: for any $x$ and $y$ in $S$, $x \leq y$ or $y \leq x$.

A **totally ordered set** or **toset** or **linearly ordered set** or **loset** is a set equipped with a total order/linear order.

### With a semigroup operation

A **totally ordered set** or **toset** or **linearly ordered set** or **loset** is a set $S$ equipped with a [[commutative semigroup|commutative]] and [[idempotent]] [[semigroup]] $\min:S \times S \to S$ such that for all $a \in S$ and $b \in S$, $\min(a, b) = a$ or $\min(a, b) = b$.

## Relation to simplices

The category of finite nonempty totally ordered sets and order-preserving maps is called $\Delta$, the [[simplex category]].

The category of *all* finite totally ordered sets and order-preserving maps is called $\Delta_a$, the **augmented** simplex category.

## Relation to pseudolattices

Due to the totality of the order relation $\leq$, every pair of elements $a$ and $b$ has a [[join]] and [[meet]], such that $a \wedge b = a$ and $a \vee b = b$, or $a \wedge b = b$ and $a \vee b = a$. This means that meets distribute over joins and joins distribute over meets, and additionally that both operations are [[associative]], [[commutative]], and [[idempotent]], and so every total order is a [[pseudolattice]], and every bounded total order is a [[lattice]].

## Relation to strict total orders

A [[strict total order]] is much like a total order, except that it is based on an irreflexive relation $\lt$.

Using [[excluded middle]], one can move between strict total orders and total orders using [[negation]]; that is, the negation of a total order is a strict total order and vice versa.  Actually one usually swaps the order too, as follows:

* $x \leq y$ iff $y \nless x$;
* $x \lt y$ iff $y \nleq x$.

One often sees $x \lt y$ defined as $x \le y$ but $x \ne y$; this is equivalent, but doesn\'t show the duality explicitly.  Similarly, one often sees $x \leq y$ defined as $x \lt y$ or $x = y$; this is not even equivalent constructively, although it is classically.

In [[classical mathematics]], the distinction between total orders and strict total orders is merely a terminological technicality, which is not always observed; more precisely, there is a [[natural isomorphism|natural bijection]] between the set of total orders on a given set $S$ and the set of strict total orders on $S$, and one distinguishes them by their notation.  In [[constructive mathematics]], however, they are irreducibly different.

For more, including why strict total orders are more often useful in constructive mathematics, see [[strict total order]].

## Classifying topos

There is a [[classifying topos]] for [[inhabited set|inhabited]] linear orders. It is given by the [[category]] of [[cosimplicial sets]], hence by the [[presheaf topos]] over the [[opposite category]] of the [[simplicial category]].

For more see at _[[classifying topos]]_ the section _[For (inhabited) linear orders](http://ncatlab.org/nlab/show/classifying+topos#ForLinearOrders)_.

## Related concepts

* [[order topology]]

* [[total preorder]]

* [[linear order]]

## References

Total orders are defined in the section of chapter 3 titled "Chains":

* [[Eric Schechter]], *[[Handbook of Analysis and its Foundations]]*, Academic Press (1996), (ISBN 0-12-622760-8, [doi:10.1016/B978-0-12-622760-4.X5000-6](https://doi.org/10.1016/B978-0-12-622760-4.X5000-6))

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total preorder

\tableofcontents

## Definition

### Total preorders

A **total preorder** or **linear preorder** or **preference relation** or **(non-strict) weak order** is a [[preorder]] whose [[posetal reflection]] is a [[total order]], or equivalently it is a [[preorder]] which is also a [[total relation]]:

* for all elements $x$ and $y$, $x \leq y$ or $y \leq x$

In [[category theory]], a total preorder is a [[thin category]] $C$ for which given two [[objects]] $x \in C$ and $y \in C$, there exists a [[morphism]] in either $\mathrm{hom}(x, y)$ or $\mathrm{hom}(y, x)$. In fact, every total preorder is an unbounded [[prelattice]], a thin [[locally cartesian category]] whose [[opposite category]] is also locally cartesian.

### Cotransitive preorders

A **cotransitive preorder** on a set $S$ is a preorder $\leq$ which satisfies [[cotransitivity]]/[[weak linearity]]:

* for all $x \in S$, $y \in S$, and $z \in S$, $x \leq z$ implies that $x \leq y$ or $y \leq z$.

### Relation between the two definitions

\begin{theorem}
Cotransitive preorders are total preorders.
\end{theorem}

\begin{proof}
Cotransitivity of $\leq$ says that for all $x \in S$ and $y \in S$, $x \leq x$ implies that $x \leq y$ or $y \leq x$, and reflexivity says that for all $x$, $x \leq x$ is always true. This implies that for all $x$ and $y$, $x \leq y$ or $y \leq x$ is always true, which is precisely the condition of totality. Since cotransitive preorders are preorders, this implies that cotransitive preorders are total preorders.
\end{proof}

\begin{lemma}
Cotransitive [[partial orders]] are [[total orders]].
\end{lemma}

[TBD: figure out if total preorders are cotransitive preorders.]

\begin{theorem}
[[total order|Total orders]] are cotransitive [[partial orders]].
\end{theorem}

## Strict total preorders

Similar to [[total orders]], one could make a distinction between the usual notion of total preorder, and [[strict total preorders]], which are the irreflexive version of total preorders, and are defined as a [[strict preorder]] which is [[weakly linear]] and [[asymmetric]].

Using [[excluded middle]], one can move between total preorders and strict total preorders using negation; that is, the negation of a total preorder is a strict total preorder and vice versa. Actually one usually swaps the order too, as follows:

* $x \leq y$ iff $y \nless x$;
* $x \lt y$ iff $y \nleq x$.

To prove this, it\'s enough to see that the properties of a strict total preorder are [[de Morgan duality|dual]] to the properties of a total preorder, as follows:

| strict total preorder |   | total preorder |
| --------------------- | - | -------------- |
| irreflexivity         |   | reflexivity    |
| asymmetry             |   | totality       |
| transitivity          |   | weak linearity |
| weak linearity        |   | transitivity   |

In [[classical mathematics]], the distinction between total preorders and strict total preorders is merely a terminological technicality, which is not always observed; more precisely, there is a [[natural isomorphism|natural bijection]] between the set of total preorders on a given set $S$ and the set of strict total preorders on $S$, and one distinguishes them by their notation.

In [[constructive mathematics]], however, they are irreducibly different. To be specific, if one starts with a total preorder $\leq$ and defines $\lt$ as above, then weak linearity does not follow; and if one starts with a strict total preorder $\lt$ and defines $\leq$ as above, then totality does not follow. Nevertheless, at least $\leq$ will be a [[preorder]], and least $\lt$ will be a [[strict preorder]].

## Examples

An example of a set with a total preorder are the [[dual number|dual]] [[rational numbers]] $\mathbb{Q}[\epsilon]/\epsilon^2$. The dual rational numbers have a strict weak order $\lt$ given by
$$(a + b \epsilon \lt c + d \epsilon) \iff (a \lt c)$$
for rational numbers $a$, $b$, $c$, and $d$. This strict weak order is not [[connected relation|connected]] because $0 \neq \epsilon$, and thus the negation of the strict weak order, $a \leq b \coloneqq \neg(b \lt a)$, is not [[antisymmetric]]. However, $\leq$ is a total preorder, because the strict weak order $\lt$ is irreflexive, weakly linear, transitive, and asymmetric, which implies that $\leq$ is reflexive, transitive, and total.

More generally, any [[ordered local ring]] with strict weak order $\lt$ has a total preorder $\leq$ defined by negation of $\lt$. The quotient [[ordered field]] by the [[ideal]] of non-invertible elements results in a total preorder $\leq$ which is also a [[total order]]. In [[constructive mathematics]] one has to make sure that $\lt$ is also decidable.

## Related concepts

* [[preorder]]

* [[prelattice]]

* [[total relation]]

* [[total order]]

## References

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

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

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[[!redirects linear preorder]]
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[[!redirects linear preordering]]
[[!redirects linear preorderings]]

[[!redirects linearly preordered]]
[[!redirects linearly preordered set]]
[[!redirects linearly preordered sets]]


total relation

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# Total relations
* table of contents
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## Idea

A _total relation_ or _linear relation_ is a [[binary relation]] on a set with the property that distinguishes the [[total orders]]/[[linear orders]] among the [[partial orders]].

## Terminology

An [[entire relation]] is sometimes called 'total', but these are unrelated concepts.  The 'total' there is in the sense of a total (as opposed to [[partial function|partial]]) [[function]], while the 'total' here is in the sense of a total (as opposed to partial) [[order]].

A total relation is necessarily [[reflexive relation|reflexive]].  For an [[irreflexive relation|irreflexive]] version, see _[[connected relation]]_.  In [[classical mathematics]], total relations and connected relations are equivalent concepts, since either may be constructed from the other by changing every truth value of the form $x \sim x$, and so the distinction is not always maintained.

## Definitions

A [[binary relation]] $R$ on a set $A$ is **total** if any two elements are related in one order or the other:
$$ \forall (x, y: A),\; x \sim_R y \;\vee\; y \sim_R x .$$
In other words, $R$ is **total** if its [[union]] with its [[reverse relation|reverse]] is the [[universal relation]]:
$$ A \times A \subseteq R \cup R^{op} .$$
(Of course, this containment is in fact an equality.)  In this way, the concept can be generalized from [[Rel]] to any $2$-[[2-poset|poset]]-[[2-poset with duals|with-duals]]; part of the definition then becomes the requirement that the union $R \cup R^{op}$ exists.

In [[constructive mathematics]], we can distinguish between a **strongly total** relation, defined as above, and a **weakly total** or **linear** relation:
$$ \forall (x, y: A),\; \neg(x \sim_R y) \;\Rightarrow\; y \sim_R x .$$
(In theory, this should be conjoined with $\neg(y \sim_R x) \Rightarrow x \sim_R y$, but this follows because the original definition is symmetric in $x$ and $y$.)
This can also be expressed in $Rel$ using the weak union $\parr$ instead of $\cup$ and generalized to other $2$-posets with the structure to support this operation.
We can define an even weaker notion:
$$ \forall (x, y: A),\; \neg(x \sim_R y) \;\Rightarrow\; \neg\neg(y \sim_R x) ,$$
which is based on using the [[double negation]] of union in $Rel$, but I\'m not sure that anyone uses this in practice.

We can also let $R$ be a [[disjoint pair]] of relations, denoted $R^+$ and $R^-$ or (more suggestively) $\sim_R$ and $\nsim_R$, related according to the [[antithesis interpretation]] of [[linear logic]] within constructive mathematics, which requires that
$$ \forall (x, y: A),\; x \nsim_R y \;\Rightarrow\; \neg(x \sim_R y) $$
(but not the converse).  Then $R$ is **weakly total** or **linear** if additionally
$$ \forall (x, y: A),\; (x \nsim_R y \;\Rightarrow\; y \sim_R x) .$$
It\'s **strongly total** if, in addition to all of the above,
$$ \forall (x, y: A),\; x \sim_R y \;\vee\; y \sim_R x .$$
This can all be expressed in a $2$-poset of disjoint pairs of relations equipped with strong ($\uplus$) and weak ($\parr$) notions of union.

## Properties

A [[total order]] is precisely a [[partial order]] that is total in the sense of this article.

In classical logic, $R$ is total if and only if $R = \dot{R} \cup \Delta_A$ for some [[connected relation]] $\dot{R}$, and then $\dot{R}$ must be $R \setminus \Delta_A$.  (Here, $\Delta_A$ is the [[equality relation]] on the set $A$.)

Totality is antithetical to [[asymmetric relation|asymmetry]].  In [[classical logic]], this means that $R$ is total if and only if its [[negation]] is asymmetric.  More generally, in terms of the [[antithesis interpretation]], the disjoint pair $R = (R^+,R^-)$ is strongly/weakly total if and only if its formal negation $R^\bot = (R^-,R^+)$ is strongly/weakly asymmetric.

In particular, $R^-$ is asymmetric in its own right whenever a disjoint pair $(R^+,R^-)$ is weakly total.  Conversely, a disjoint pair $(R^+,R^-)$ is strongly total if (and only if) $R^+$ is strongly total in its own right and $R^-$ is asymmetric in its own right.

## Related concepts

* [[cotransitive relation]]/[[weakly linear relation]]

* [[total order]]

## References

For linearity of relations as weak totality, see section 11.2.1 of:

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

[[!redirects total relation]]
[[!redirects total relations]]
[[!redirects totality]]

[[!redirects weakly total]]
[[!redirects weakly total relation]]
[[!redirects weakly total relations]]
[[!redirects weak totality]]

[[!redirects linear relation]]
[[!redirects linear relations]]
[[!redirects linearity]]


totality space

[[!redirects totality spaces]]

# Totality spaces

## Definition

A **totality space** is an [[arity space]] for $\kappa=\{1\}$.  At least some references add the additional condition that $X = \bigcup \mathcal{U} = \bigcup \mathcal{U}^\perp$.

## References

* Loader, Ralph. *Linear logic, totality and full completeness.* Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994): 292-298.


totalization

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## Idea

The _totalization_ of a [[cosimplicial object]] is the [[duality|dual]] concept to the [[geometric realization]] of a [[simplicial object]].

## Definition

### As an end

For $A : \Delta \to C$ a [[cosimplicial object]] in a [[category]] $C$ which is [[powering|powered]] over [[simplicial sets]] and for

$$
  \Delta : [n] \mapsto \Delta[n]
$$

the canonical cosimplicial simplicial set of [[simplices]], the **totalization** of $A$ is the [[end]]

\[
  \label{AsAnEnd}
  \int_{[k]\in \Delta}
  (A_k)^{\Delta[k]}
  \,\,\,
  \in
  C
  \,.
\]

### As the homotopy limit
 {#AsTheHomotopyLimit}

For a [[cosimplicial object]] $A \colon \Delta \to \mathcal{C}$ in a suitable [[model category]] such that $A$ is a [[fibrant object]] with respect to the [[Reedy model structure]] on $Func(\Delta, \mathcal{C})$, totalization in terms of the [[end]]-construction above in (eq:AsAnEnd) is a model for the [[homotopy limit]] over $A$.

([Hirschhorn 15, theorem 9.2](#Hirschhorn15))

## Properties

### Homotopy and homology

The [[homotopy groups]] of the totalization of a [[cosimplicial space]] are computed by a [[Bousfield-Kan spectral sequence]].

The [[homology groups]] by an [[Eilenberg-Moore spectral sequence]].

## Related concepts

Formally the dual to totalization is [[geometric realization]]: where totalization is the [[end]] over a [[power]]ing with $\Delta$, realization is the [[coend]] over the [[tensoring]].

But various other operations carry names similar to "totalization". For instance a [[total chain complex]] is related under [[Dold-Kan correspondence]] to the [[diagonal of a bisimplicial set]] -- see at *[[Eilenberg-Zilber theorem]]*. As discussed at _[[bisimplicial set]]_, this is [[weak homotopy equivalence|weakly homotopy equivalent]] to the operation that is often called $Tot$ and called the _total simplicial set_ of a bisimplicial set.

To a cosimplicial chain complex we can assign a double complex by taking the alternating sum of the coface maps. Then the totalization of this cosimplicial object and the totalization of the double complex as defined in homological algebra coincide. Moreover, the associated [[Bousfield-Kan spectral sequence]] and [[spectral sequence of a double complex]] coincide.

## References

The concept for [[cosimplicial spaces]] originates with:

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], chapter X.3 of _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, **304** Springer (1972)

Introductory notes:

* {#Eldred08} [[Rosona Eldred]], *Tot primer* (2008) [[[Eldred-Totalization.pdf:file]]]

Quick review:

* [[Daniel Dugger]], section 5.3 of _A primer on homotopy colimits_ ([pdf](https://pages.uoregon.edu/ddugger/hocolim.pdf))

The generalization to [[cosimplicial objects]] in more general [[model categories]] is discussed in

* {#Bousfield03} [[Aldridge Bousfield]], _Cosimplicial resolutions and homotopy spectral sequences in model categories_, Geom. Topol. 7 (2003) 1001-1053 ([arXiv:math/0312531](http://arxiv.org/abs/math/0312531))

Review of this includes

* {#Levine13} [[Marc Levine]], _The Adams-Novikov spectral sequence and Voevodsky's slice tower_, Geom. Topol. 19 (2015) 2691-2740 ([arXiv:1311.4179](http://arxiv.org/abs/1311.4179))

Discussion of totalizations as [[homotopy limits]] includes

* {#Hirschhorn15} [[Philip Hirschhorn]], Section 9 of: _The diagonal of a multicosimplicial object_ ([arXiv:1506.06837](https://arxiv.org/abs/1506.06837))

* [[Akhil Mathew]], [[Vesna Stojanoska]], _Fibers of partial totalizations of a pointed cosimplicial space_, Proc. Amer. Math. Soc. 144 (2016), no. 1, 445--458 ([arXiv:1408.1665](https://arxiv.org/abs/1408.1665))

[[!redirects totalizations]]


totally bounded space

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# Totally bounded spaces
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## Idea

A [[topological space]] is _totally bounded_ if it may be [[covering|covered]] by finitely many sets of arbitrarily small size.

The [[Heine-Borel theorem]], which states that a [[closed subset|closed]] and [[bounded subset|bounded]] [[subset]] of the [[real line]] is [[compact space|compact]] (in the finite open subcover sense), applies to all [[Euclidean spaces]] but not to general [[metric spaces]].  However, if we use two facts about the real line (which hold for all cartesian spaces) ---that a subset is [[closed subspace|closed]] if and only if it is [[complete space|complete]] and that a subset is [[bounded space|bounded]] if and only if it is _totally bounded_---, then we get a theorem that *does* apply to all metric spaces (at least assuming the [[axiom of choice]]): that a complete and totally bounded space is compact.

The concept (and the Heine--Borel theorem, in this sense) apply not only to metric spaces but to [[uniform spaces]]; like completeness, total boundedness is a uniform property.

## Definition

### Basic definitions

In the following definitions, '[[finite set|finite]]' means Kuratowski-finite, or finitely indexed, for the purposes of [[constructive mathematics]].  All of these definitions are constructively equivalent.

The slickest definition for [[uniform spaces]] is probably this one:

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

A uniform space $X$ is __totally bounded__ if every [[uniform cover]] of $X$ has a finite subcover.
=--

Since uniform covers are not a common approach to uniform spaces, we unwrap the definition of uniform covers in terms of entouranges to get this definition:

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

A uniform space $X$ is __totally bounded__ if, for every [[entourage]] $U$ of $X$, there is a finite [[open cover]] $\mathcal{C}$ of $X$ such that every set $G$ in $\mathcal{C}$ satisfies $G \times G \subseteq U$.
=--

In fact, it is enough to consider only basic entourages for some [[base]] of the uniformity.  Thus, we may specialise to [[gauge spaces]]:

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

A gauge space $X$ is __totally bounded__ if, for every gauging distance $d$ of $X$, there is a finite open cover $\mathcal{C}$ of $X$ such that every set in $\mathcal{C}$ has $d$-diameter less than $1$.
=--

In fact, it is enough to consider only basic gauging distances for some base of the gauge, or even for some [[subbase]] of the gauge if we make the requirement for arbitrarily small diameters (rather than the fixed diameter $1$ as above).  Thus, we may specialise to [[metric spaces]]:

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

A metric space $X$ is __totally bounded__ if, for every positive number $\epsilon$, there is a finite open cover $\mathcal{C}$ of $X$ such that every set in $\mathcal{C}$ has diameter less than $\epsilon$.
=--

### Proximity spaces

The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of [[proximity spaces]] and proximally continuous functions.  Thus, proximity spaces can be considered yet another axiomatization of "totally bounded space" that doesn't rely on a pre-existing kind of "space".

## Properties

All of these results hold constructively unless otherwise noted.

Every [[compact space]] is totally bounded; this is immediate from Definition \ref{slick}, since every uniform cover is an open cover.  Conversely, if one assumes the [[ultrafilter principle]], then every [[complete space|complete]] and totally bounded space is compact.  In [[constructive mathematics]], "complete and totally bounded" is sometimes taken as a definition of "compact" -- see [[Bishop-compact space]].

Any [[product]] of totally bounded spaces is totally bounded.  The totally bounded subspaces of a given space $X$ form an [[ideal]] in the [[power set]] of $X$.

A [[subspace]] of a [[Cartesian space]] is totally bounded if and only if it is [[bounded space|bounded]].  Every totally bounded metric space is [[separable space|separable]].

### Relation to precompact spaces

Every [[precompact space|precompact]] [[uniform space]] is totally bounded; using Definition \ref{slick}, this may be proved by checking that any uniform cover of $X$ generates a uniform cover of $\overline{X}$. The converse, that every totally bounded space is precompact, is equivalent to the [[ultrafilter principle]]. Of course, many totally bounded spaces may be proved precompact on weaker assumptions; in particular, that a bounded subset of a cartesian space is precompact is equivalent to the [[fan theorem]] (and so also follows from the principle of [[excluded middle]]), a fact related to the [[Heine–Borel theorem]].

## Related concepts

* [[bounded subset]]
* [[precompact space]]

## References

*  _[[HAF]]_, Sections 19.14--20

[[!redirects totally bounded space]]
[[!redirects totally bounded spaces]]
[[!redirects totally bounded metric space]]
[[!redirects totally bounded metric spaces]]
[[!redirects totally bounded gauge space]]
[[!redirects totally bounded gauge spaces]]
[[!redirects totally bounded uniform space]]
[[!redirects totally bounded uniform spaces]]
[[!redirects totally bounded uniformity]]
[[!redirects totally bounded uniformities]]


totally connected geometric morphism

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# Totally connected geometric morphism
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## Idea

**Totally connectedness** is a stronger form of _local connectedness_ that arose in the work of [[Marta Bunge|M. Bunge]] and [[Jonathon Funk|J. Funk]] on _topos distributions_.

The properties of totally connected geometric morphisms are largely dual to those of [[local geometric morphisms]].

## Definition

A [[geometric morphism]] $f\colon E\to S$ is **totally connected** if

1. It is [[locally connected geometric morphism|locally connected]], i.e. its [[inverse image functor]] $f^*$ has a left adjoint $f_!$ which is $S$-[[indexed functor|indexed]], and

1. The functor $f_!$ is [[left exact functor|left exact]], i.e. preserves [[finite limits]].

When thinking of $E$ as a topos over $S$ via $f$, we say that it is a **totally connected $S$-topos**.  In particular, when $S=Set$ and $f = (L Const, \Gamma)$ is the unique [[global sections]] geometric morphism, we call $E$ a **totally connected topos**.

## Properties

* Of course, any totally connected geometric morphism is [[connected geometric morphism|connected]], since the [[terminal object]] is a particular finite limit.  It is also [[strongly connected geometric morphism|strongly connected]], since finite products are also finite limits.

* Totally connected geometric morphisms are closed under composition and stable under pullback.

* $f$ is totally connected iff $f$ is connected and has a right adjoint in $\mathfrak{Top}$, the [[topos|2-category of toposes]] and geometric morphisms.

* Totally connected geometric morphisms are orthogonal to [[grouplike geometric morphism|grouplike geometric morphisms]] (cf. Johnstone (2002)).

* A connected, locally connected $p:\mathcal{E}\to\mathcal{S}$ that satisfies the [[Nullstellensatz|Lawvere Nullstellensatz]] (i.e. the canonical $\theta:p_*\to p_!$ is epic) is totally connected precisely iff it is a [[quality type]] over $\mathcal{S}$ (i.e. $\theta$ is iso; cf. Johnstone (2011)).

## Examples

*  A topos $Sh(X)$ of sheaves on a [[topological space]] is totally connected iff $X$ has a [[dense subset|dense]] point (a single point whose closure is all of $X$).

*  A presheaf topos $Psh(C)$ is totally connected iff $C$ is [[cofiltered category|cofiltered]].

## Totally connected sites

A small [[site]] $C$ is called **totally connected** if

1. $C$ is [[cofiltered category|cofiltered]], and

1. Every [[cover|covering]] [[sieve]] in $C$ is [[connected category|connected]], when regarded as a subcategory of a [[slice category]].

The second condition implies that all constant presheaves are sheaves, and hence that the left adjoint $Colim\colon Psh(C) \to Set$ of $Const\colon Set\to Psh(C)$ restricts to $Sh(C)$ to give a left adjoint of $L Const$.  Cofilteredness of $C$ is exactly what is needed for left exactness of $Colim\colon Psh(C) \to Set$, essentially by definition.  Hence the topos of sheaves on any totally connected site is totally connected.

Conversely, one can show that any totally connected topos can be (but need not be) presented by some totally connected site.

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * **totally connected site** / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]] /  [[∞-cohesive site]]

## References

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos_ , JPAA **113** (1996) pp.1-38.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos II_ , JPAA **130** (1998) pp.49-84.

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. (vol. 2, Section C.3.6, pp.707-710)

* {#JS11} [[Peter Johnstone|P. Johnstone]], _Remarks on Punctual Local Connectedness_ , TAC **25** no.3 (2011) pp.51-63. ([pdf](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

[[!redirects totally connected]]
[[!redirects totally connected geometric morphisms]]
[[!redirects totally connected topos]]
[[!redirects totally connected toposes]]
[[!redirects totally connected topoi]]
[[!redirects totally connected site]]
[[!redirects totally connected sites]]


totally convex space

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# Totally convex spaces
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## Idea

The notion of totally convex spaces is essentially what one obtains when formulating an [[algebraic theory]] of [[Banach spaces]].

## Definitions

+-- {: .num_defn #slick}
###### Definition
(abstractly)

A __totally convex space__ is a [[module for a monad|module]] for the [[monad]] on [[Set]] which sends a set $X$ to the [[closed ball|closed unit ball]] in the [[Banach space]] $\ell^1(X)$.
=--

This is an example of an [[algebraic theory]] (in the strong sense that it is [[monadic category|bounded monadic]]) but not finitary, hence not a [[Lawvere theory]].  The [[free object|free]] totally convex space on $X$ is the unit ball of the Banach space $\ell^1(X)$, and thus an operation in this theory is a formal sum $\sum_{x \in X} a_x x$ for $a_x \in \mathbb{R}$ with the property that $\sum_{x \in X} {|a_x|} \le 1$.  The finiteness of this sum forces it to have only countably many non-zero terms, and thus it factors through an operation on $\mathbb{N}$.  Hence there is a presentation of this theory with operations in $\ell^1(\mathbb{N})$ (which is why the theory is *bounded* monadic).  The corresponding identities are simply the substitution rules -- namely that substituting a sum into another works as expected -- and the reordering rules.

+-- {: .num_defn #concrete}
###### Definition
(concretely)

A __totally convex space__ is a non-empty set $X$ equipped with, for each [[infinite sequence]] $(a_1, a_2, \ldots)$ of [[real numbers]] such that $\sum_i {|a_i|} \leq 1$, an operation from $X^{\mathbb{N}}$ to $X$, written $(x_1,x_2,\ldots) \mapsto \sum_i a_i x_i$, such that:

1.  Reordering:  If $\sigma$ is any [[permutation]] of $\mathbb{N}$, then $\sum_i a_i x_i = \sum_i a_{\sigma(i)} x_{\sigma(i)}$;
2.  Nullary substitution:  If $\delta_i$ is the [[Kronecker delta]] at $0$ (so $\delta_i = 0$ normally, but $\delta_0 = 1$), then $\sum_i \delta_i x_i = x_0$;
3.  Binary substitution:  If the functions $\pi,\rho\colon \mathbb{N} \to \mathbb{N}$ express $\mathbb{N}$ as its own [[product]] with itself (it is enough to pick one pair of functions and state this axiom only for it), then $\sum_i a_i (\sum_j b_{i,j} x_{i,j}) = \sum_k a_{\pi(k)} b_{\pi(k),\rho(k)} x_{\pi(k),\rho(k)}$.
=--

Of course, one would normally write the right-hand side of the last equation as $\sum_{i,j} a_i b_{i,j} x_{i,j}$, but that is not technically an operation in the theory, except as mediated by $\pi$ and $\rho$.  A common choice for $(\pi,\rho)$ is the [[inverse]] of $(i,j) \mapsto \big({i + j + 1 \atop 2}\big) + j$, where for this expression to work we take $0$ to be a natural number.

A totally convex space $X$ has the same axioms as a [[super convex space]] except that a totally convex space $X$ must be a nonempty set, and the condition for countable affine sums $\sum_{i \in \mathbb{N}} a_i x_i$, given by $\sum_{i \in \mathbb{N}} |a_i| = 1$ for super convex spaces, is replaced by the inequality $\sum_{i \in \mathbb{N}} |a_i|\le 1$.

+-- {: .query}
I haven\'t actually checked that this list is complete; but it\'s what I get if I take Andrew at his word that we need only substitution and reordering rules.  I wouldn\'t be terribly surprised if nullary substitution is redundant, but right now I don\'t see how.  ---Toby
=--

## Properties

1. A totally convex space is a [[pointed object|pointed]] [[convex space]].  The operations of a convex space are encoded in the operations $(r, 1 - r, 0, 0, 0, \ldots)$ and the "point" comes from the operation $(0, 0, \ldots)$.  This functor preserves underlying sets and so has a [[left adjoint]]; thus any convex space can be "completed" to a totally convex space.

1. The operations for this theory are [[commutative algebraic theory|commutative]], hence the [[category]] of totally convex spaces is a [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] category.

## Examples

1. Clearly, the closed unit ball $B X$ of any Banach space $X$ is a totally convex space.

1. Bizarrely, the **open** unit ball of a Banach space is a totally convex space.  This is because if a sum, $\sum_{x \in B X} a_x x$ for $\sum_x {|a_x|} \leq 1$, lies on the boundary of $B X$ then every $x$ for which $a_x \ne 0$ must have norm $1$.  Thus if the series only contains terms from the interior of $B X$, the sum remains in the interior.  Hence the open unit ball is a [[subalgebra]] of the closed unit ball.

1. Continuing, the quotient of the closed unit ball by the open unit ball is a totally convex space.

1. In particular, the three-point space $\{-1,0,1\}$ is (assuming [[excluded middle]]) a totally convex space.  The operations on this space are as follows:

   $$
   \sum a_j \epsilon_j = \begin{cases}
   1 & \epsilon_j = \operatorname{sign}(a_j), \sum {|a_j|} = 1 \\
   -1 & \epsilon_j = -\operatorname{sign}(a_j), \sum {|a_j|} = 1 \\
   0 & \;\text{otherwise}
   \end{cases}
   $$

1. Going back one step, $(-1,1)$ is a totally convex space.   It is illuminating to describe this as a [[coequaliser]] of free totally convex spaces.  Consider a functional $f \colon \ell^1(\mathbb{N}) \to \mathbb{R}$ which is bounded of norm $1$ but does not achieve its norm; for example, let $f$ be represented by the sequence $(\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots)$.  Then for any $x \in B\ell^1(\mathbb{N})$, $f(x) \in (-1,1)$.  Thus $(-1,1)$ is the coequaliser of the inclusion $\ker f \to \ell^1(\mathbb{N})$ and the [[zero map]] $\ker f \to \ell^1(\mathbb{N})$.

## Related concepts
* [[super convex space]]
* [[convex space]]
* positively convex space

## References

* [[Dieter Pumplün]], [[Helmut Röhrl]]: _Banach Spaces and totally convex spaces I_, Comm. Alg. **12** 8 (1984) 953-1019 [[doi:10.1080/00927878408823035](https://doi.org/10.1080/00927878408823035)]

* [[Dieter Pumplün]], [[Helmut Röhrl]], _Banach Spaces and totally convex spaces II_, Comm. Alg. **13** 5 (1985) 1047-1113 [[doi:10.1080/00927878508823205](https://doi.org/10.1080/00927878508823205)]

* [[Dieter Pumplün]], [[Helmut Röhrl]], *Totally convex algebras*, Comment. Math. Univ. Carolin. **33** 2 (1992) 205–235
[[pdf](http://emis.maths.tcd.ie/journals/CMUC/pdf/cmuc9202/pump.pdf), [dml:10338.dmlcz/118490](https://dml.cz/handle/10338.dmlcz/118490)]

[[!redirects totally convex spaces]]

[[!redirects totally convex algebra]]
[[!redirects totally convex algebras]]


totally disconnected space

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## Definition

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

A ([[topological space|topological]]) [[space]] whose only [[connected topological space|connected]] [[subspaces]] are [[singletons]] is called *totally disconnected*.
=--

## Examples

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

[[discrete space|Discrete spaces]] are totally disconnected.

=--

+-- {: .num_example #RationalNumbersAreTotallyDisconnected}
###### Example
**(the rational numbers are totally disconnected)**

The [[rational numbers]] $\mathbb{Q} \subset \mathbb{R}$ equipped with their [[subspace topology]] inherited from the [[Euclidean space|Euclidean]] [[metric topology]] on the [[real numbers]], form a totally disconnected space.

=--

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

By construction, a [[base for a topology|base for the topology]] is given by the open subsets which are restrictions of [[open intervals]] of real numbers to the rational numbers

$$
  (a,b)_{\mathbb{Q}} \coloneqq (a,b) \cap \mathbb{Q}
$$

for $a \lt b \in \mathbb{R}$.

Now for any such $a \lt b$ there exists an [[irrational number]] $r \in \mathbb{R}\backslash \mathbb{Q}$ with $a \lt r \lt b$. This being irrational implies that $(a,r)_{\mathbb{Q}} \subset \mathbb{Q}$ and $(r,b)_{\mathbb{Q}} \subset \mathbb{Q}$ are disjoint subsets. Therefore every basic open subset is the disjoint union of (at least) two open subsets:

$$
  (a,b)_{\mathbb{Q}} = (a,r)_{\mathbb{Q}} \cup (r,b)_{\mathbb{Q}}
  \,.
$$

Hence no [[inhabited set|inhabited]] open subspace of the rational numbers is connected.

=--

+-- {: .num_example #limits}
###### Example
A [[product]] in $Top$ of totally disconnected spaces is totally disconnected. A subspace of a totally disconnected space is totally disconnected. Hence [[limits]] in $Top$ of diagrams of totally disconnected spaces are totally disconnected.
=--

For example, the [[Baire space]] of [[irrational numbers]] is [[homeomorphism|homeomorphic]] to a [[countable set|countable]] [[product space]] $\mathbb{N}^\mathbb{N}$ (via [[continued fractions]]), so it is totally disconnected. Similarly, [[Cantor space]] $2^\mathbb{N}$ is totally disconnected. Another notable special case of the preceding class of examples is the following.

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

Every [[profinite group]] is totally disconnected and in particular the set of [[p-adic number|p-adic numbers]] is totally disconnected.

=--

See also _[[Stone space]]_.

* [[extremally disconnected topological space|extremally disconnected]] Hausdorff space

## Properties

The general class of examples in Example \ref{limits} may be seen in the following light.

+-- {: .num_theorem}
###### Theorem
The [[category]] of totally disconnected spaces and continuous maps is a [[reflective subcategory]] of [[Top]].
=--

+-- {: .proof}
###### Proof
The reflector takes a space $X$ to the space of connected components, i.e., the [[quotient space]] $X/\sim$ of $X$ where $\sim$-equivalence classes are precisely the connected components of $X$. We check that connected components $C$ of $X/\sim$ are singletons. Let $q: X \to X/\sim$ be the quotient map; it suffices to check that $q^{-1}(C) \subseteq X$ is connected (because then $q^{-1}(C)$ is contained in a single $\sim$-equivalence class, making $C = q q^{-1}(C)$ a single point). So, suppose the (closed) set $q^{-1}(C)$ is a disjoint union $K_1 + K_2$ of closed sets $K_1, K_2$; we are required to show one or the other is empty. For each $c \in C$, the inverse image $q^{-1}(\{c\})$ is connected, hence we must have $q^{-1}(\{c\}) \subseteq K_1$ or $q^{-1}(\{c\}) \subseteq K_2$. Thus we can partition $C$ into sets

$$C_1 \coloneqq \{c \in C: q^{-1}(\{c\}) \subseteq K_1\}, \qquad C_2 \coloneqq \{c \in C: q^{-1}(\{c\}) \subseteq K_2\}$$

and we observe $q^{-1}(C_1) = K_1$ and $q^{-1}(C_2) = K_2$. By definition of quotient topology, since $K_1, K_2$ are closed we infer $C_1, C_2$ are closed. They are also disjoint and $C = C_1 + C_2$, so by connectedness of $C$ either $C_1 = \emptyset$ or $C_2 = \emptyset$, and therefore $K_1 = q^{-1}(C_1)$ or $K_2 = q^{-1}(C_2)$ is empty, as required.

Finally, given a continuous map $f: X \to Y$ with $Y$ totally disconnected, each connected component $C$ of $X$ is mapped to a connected set $f(C)$ of $Y$ which is a singleton by total disconnectedness of $Y$, and so we get a (unique) factoring through a map $X/\sim \to Y$, continuous of course by virtue of the quotient topology. This completes the proof.
=--

## Related concepts

* [[locally connected topological space]]

* [[extremally disconnected topological space]]

* [[connected space]]

[[!redirects totally disconnected spaces]]

[[!redirects totally disconnected topological space]]
[[!redirects totally disconnected topological spaces]]
[[!redirects totally disconnected]]


totally distributive category

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

# Totally distributive categories

* table of contents
{: toc}

## Definition

Recall that a [[locally small category]] $\mathcal{K}$ is [[total category|total]] if its [[Yoneda embedding]] $Y\colon\mathcal{K}\longrightarrow [\mathcal{K}^{op},Set]$ has a [[left adjoint]] $X$.

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

A total category $\mathcal{K}$ is **totally distributive** if $X \colon [\mathcal{K}^{op},Set] \longrightarrow \mathcal{K}$ has a further [[left adjoint]] $W$.

$$
  (W \dashv X \dashv Y)
  \;\colon\;
  \mathcal{K}
   \stackrel{\overset{W}{\hookrightarrow}}{\stackrel{\overset{X}{\leftarrow}}{\underset{Y}{\hookrightarrow}}}
  PSh(\mathcal{K})
  \,.
$$

=--

## Properties

If $\mathcal{K}$ is totally distributive, then since $Y$ is [[full and faithful functor|fully faithful]], then, by the properties of [[adjoint triples]], so is $W$.  Thus, $\mathcal{K}$ is a [[coreflective subcategory]] of $[\mathcal{K}^{op},Set]$, which is cototal (or more precisely, [[pro-cototal category|pro-cototal]], since it is not locally small) --- hence $\mathcal{K}$ is also cototal.

Moreover this means that the induced [[adjoint pair]] of ([[comonad|co]]-)[[monads]]

$$
  W X \dashv  Y X \;\colon\; PSh(\mathcal{K}) \leftrightarrow PSh(\mathcal{K})
$$

is an [[adjoint modality]].

## Examples

* If $C$ is a [[small category]], then its [[presheaf category]] $[C^{op},Set]$ is totally distributive.

* If $\mathcal{K}$ is totally distributive and $\mathcal{L}\subseteq \mathcal{K}$ is a full [[subcategory]] that is both [[reflective subcategory|reflective]] and [[coreflective subcategory|coreflective]], then $\mathcal{L}$ is totally distributive.

## Waves

By the [[adjoint functor theorem]] for total categories, a total category is totally distributive if and only if the functor $X$ preserves all limits.  Moreover, for any total category, it is possible to define the functor $W:\mathcal{K} \to [\mathcal{K}^{op},Set]$ which "wants to be" the left adjoint of $X$ (and is if $\mathcal{K}$ is total).  The elements of $W(A)(K)$ are called **waves** from $K$ to $A$ (just because they are usually denoted by wavy arrows), and as in the case of [[continuous categories]] they form an [[idempotent comonad]] on $\mathcal{K}$ in the [[bicategory]] of [[profunctors]].

## Related pages

* [[distributive category]]

* [[completely distributive category]]

* Totally distributive categories are "almost" an example of [[continuous algebras]] for a [[lax-idempotent 2-monad]].

## References

* [[Robert Rosebrugh]], [[Richard J. Wood]], *An adjoint characterization of the category of sets*. PAMS **122** 2 (1994) 409-413 [[jstor:2161031](https://www.jstor.org/stable/2161031)]

* Rory Lucyshyn-Wright, *Totally distributive toposes*.  [spnet](https://selectedpapers.net/arxiv/1108.4032).

* [[Francisco Marmolejo]], [[Bob Rosebrugh]], and [[Richard Wood]], *Completely and totally distributive categories I*, JPAA 216 no. 8-9 (2012).  [PDF](http://www.mta.ca/~rrosebru/articles/ctd.pdf)

[[!redirects totally distributive categories]]


totally geodesic submanifold

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

#Contents#
* table of contents
{:toc}

## Definition

A [[submanifold]] $\Sigma \subset X$ of a [[Riemannian manifold]] $X$ is called _totally geodesic_ if every [[geodesic]] in $\Sigma$ is also a geodesic when regarded in $X$.

## Examples

+-- {: .num_prop #FixedLociOfSmoothProperActionsAreSubmanifolds}
###### Proposition
**([[fixed loci]] of [[smooth function|smooth]] [[proper actions]] are [[submanifolds]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then the $G$-[[fixed locus]] $X^G \hookrightarrow X$ is a [[smooth manifold|smooth]] [[submanifold]].

If in addition $X$ is equipped with a [[Riemannian metric]] and $G$ acts by [[isometries]] then the [[submanifold]] $X^G$ is a totally geodesic submanifold.

=--

(e.g. [Ziller 13, theorem 3.5.2](equivariant+differential+topology#Ziller13))

## References

* [[eom]], _[Totally-geodesic manifold](https://www.encyclopediaofmath.org/index.php/Totally-geodesic_manifold)_

[[!redirects totally geodesic submanifolds]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[∞-connected (∞,1)-topos]]

$$
  (\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \to \infty Grpd
$$

is called **strongly connected** if $\Pi$ preserves [[finite limit|finite]] [[(∞,1)-limit]]s.

Similarly for an $n$-connected $(n,1)$-topos.

For $n = 1$ this yields the notion of [[totally connected topos]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]], [[locally ∞-connected site]]

* [[connected site]]

* [[local site]]

* [[cohesive site]]

[[!redirects totally ∞-connected (∞,1)-topos]]
[[!redirects totally ∞-connected (∞,1)-toposes]]
[[!redirects totally ∞-connected (∞,1)-topoi]]

[[!redirects totally infinity-connected (infinity,1)-toposes]]
[[!redirects totally infintiy-connected (infinity,1)-topoi]]

[[!redirects totally connected (∞,1)-topos]]


totally infinity-connected site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *totally $\infty$-connected site* is a [[site]] satisfying sufficient conditions to make the [[(∞,1)-sheaf (∞,1)-topos]] over it a [[totally ∞-connected (∞,1)-topos]].

## Definition

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

Let $C$ be a locally and globally [[∞-connected site]]; we say it is a **strongly $\infty$-connected site** if it is also a [[cofiltered (∞,1)-category]].

=--

## Properties

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

If $C$ is a totally $\infty$-connected site, then the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over it is a [[totally ∞-connected (∞,1)-topos]].

=--

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

We need to check that the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves [[finite limit|finite]] [[(∞,1)-limit]]s.

By the discussion at [[∞-connected site]] we have that $\Pi$ is given by the [[(∞,1)-colimit]] [[(∞,1)-functor]] $\lim_\to : Func(C^{op}, \infty Grpd) \to \infty Grpd$. On the [[opposite (∞,1)-category|opposite]] and therefore [[filtered (∞,1)-category]] $C^{op}$ these preserve finite [[(∞,1)-limit]]s.

=--

## Examples

* [[CartSp]], [[ThCartSp]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / **totally ∞-connected site**

* [[local site]], [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects totally ∞-connected site]]

[[!redirects totally ∞-connected sites]]
[[!redirects totally infinity-connected sites]]


totally ordered abelian group

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

#Contents#
* table of contents
{:toc}

## Idea

A totally ordered abelian group is an [[ordered abelian group]] whose order forms a [[total order]].

## Definition

The following definition is from [[Peter Freyd]]:

A __totally ordered abelian group__ is an [[pseudolattice ordered abelian group]] $G$ such that for all elements $a$ in $G$, $a \leq 0$ or $-a \leq 0$.

In a totally ordered abelian group, the [[join]] is usually called the __maximum__, while the [[meet]] is usually called the __minimum__

## Examples

The [[integers]], the [[rational numbers]], and the [[real numbers]] are totally ordered abelian groups.

## Related concepts

* [[abelian group]]

* [[ordered abelian group]]

* [[total order]]

* [[pseudolattice ordered abelian group]]

* [[totally ordered ring]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))


totally ordered ring

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

\tableofcontents

## Idea

A totally ordered ring is an [[ordered ring]] whose order forms a [[total order]].

## Definition

This definition is adapted from [[Peter Freyd]]'s definition of a [[totally ordered abelian group]]:

A __totally ordered ring__ is an [[pseudolattice ordered ring]] $R$ such that for all elements $a$ in $R$, $a \leq 0$ or $-a \leq 0$.

In a totally ordered ring, the [[join]] is usually called the __maximum__, while the [[meet]] is usually called the __minimum__

If the relation $\leq$ is only a [[preorder]], then the [[pseudolattice preordered ring]] $R$ is said to be a **totally preordered ring**.

## Examples

The [[integers]], the [[rational numbers]], and the [[real numbers]] are totally ordered rings.

## Related concepts

* [[ring]]

* [[ordered ring]]

* [[total order]]

* [[pseudolattice ordered ring]]

* [[totally ordered abelian group]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

## External links

* Wikipedia, _Ordered ring_

[[!redirects totally ordered ring]]
[[!redirects totally ordered rings]]

[[!redirects totally preordered ring]]
[[!redirects totally preordered rings]]


totally ordered set > history

< [[totally ordered set]]
[[!redirects totally ordered set -- history]]


totally positive matrix

A matrix $A$ over the field of [[real numbers]] is __totally positive__ (resp. __totally nonnegative__) if every minor (= [[determinant]] of any submatrix) is a positive (resp. nonnegative) real number. Total positivity implies a number of remarkable properties; for example all eigenvalues are distinct and positive. George Lusztig discovered that this classical total positivity is a related to more general __total positivity__ phenomena in the theory of [[Lie groups]] and [[quantum groups]].

### Literature and related entries

* [[George Lusztig]], _Total positivity in reductive groups_, In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics __123__; Birkhäuser 1994 [doi](https://doi.org/10.1007/978-1-4612-0261-5_20)
* [[cluster algebra]]
* many lecture notes listed at a minicourse [page](http://www.math.lsa.umich.edu/~fomin/eidma.html) on total positivity
* M. Skandera, _Introductory notes on total positivity_, [ps](http://www.math.lsa.umich.edu/~fomin/565/intp.ps), June 2003.
* Yuji Kodama, Lauren Williams, _KP solitons and total positivity for the Grassmannian_, [arXiv:1106.0023](https://arxiv.org/abs/1106.0023)

category: algebra
[[!redirects total positivity]]


* David Mumford, _Towards An Enumerative Geometry of the Moduli Space of Curves_, Progress in Math., Vol. II (1983), pp. 271-328

category: reference

[[!redirects Towards an enumerative geometry of the moduli space of curves]]
[[!redirects Towards an enumerative geometry of the moduli space of curves.]]


Towards Higher Categories

This entry is to record the book collection

* [[John Baez]] and [[Peter May]] (eds.),

  **[[John Baez:Towards Higher Categories|Towards Higher Categories]]**

  The IMA volumes in Mathematics and its applications,

  Springer (2010)

  [doi:10.1007/978-1-4419-1524-5](https://link.springer.com/book/10.1007/978-1-4419-1524-5)

on [[higher category theory]].

## Contents

On [[2-category theory]]:

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

On the history of Australian higher category theory

* {#Street09} [[Ross Street]], _An Australian conspectus of higher categories_, talk at Institute for Mathematics and its Applications Summer Program: *$n$-Categories: Foundations and Applications* at the University of Minnesota (Minneapolis, 7–18 June 2004) [[pdf](http://www.math.uchicago.edu/~may/IMA/Street.pdf), [[Street-Conspectus.pdf:file]]]

(...)

category: reference

[[!redirects Approaching Higher Category Theory]]


[[!redirects Eric Finster, Towards Higher Universal Algebra in Type Theory]]
## Contents

* table of contents
{:toc}

## Idea

While [[homotopy type theory]] formalizes [[homotopy theory]], it is not a priori clear -- and in fact is or was an open problem -- how to formalize general homotopy-[[coherence|coherent]] [[structures]] of [[higher algebra]]/[[higher category theory]]: Since these typically involve an _infinite_ hierarchy of [[coherence]]-conditions, these cannot be [[axiom|axiomatized]] directly, but one needs some scheme that generates them. This turned out to be subtle.

[[Eric Finster]] had previously considered another variant of [[type theory]], called _[[opetopic type theory]]_ which natively talks about [[infinity-categories]] and their higher [[coherences]] by type-theoretically formalizing the [[structure]] of [[opetopic sets]]. In new work [Finster 18](#Finster2018) he gives something like an implementation of aspects of [[opetopic type theory]] _within_ [[homotopy type theory]] and provides evidence that this  yields a tool to solve the general problem of [[coherences]] of [[higher algebra]]/[[higher category theory]] within [[homotopy type theory]].

Polynomials serve as a notion of "*higher signature*". Following ideas from the categorical approach to universal algebra, we are going to encode the relations or axioms of our structure using a monadic multiplication on P.

## Eric's talk

In Eric's Agda formalisation and talk, a polynomial has its parameter sorts encoded with the parameter. We will follow a suggestion by Mike to have a $\SortOf$ function instead. This should make some definitions less cluttered.

Let $I : \mathcal{U}$ be a type of sorts. A polynomial over $I$ consists of the following data:

 * A family of operations $\Op : I \to \mathcal{U}$
 * For each operation, a family of parameters $\Param_i : (f : \Op i) \to \mathcal{U}$
 * A function $\SortOf : (f : \Op i) \to \Param f \to I$

We define the type of operations of a polynomial $P$ as:

$$ \Ops \equiv \sum_{i : I} \Op_P i$$

A polynomial $P : \Poly I$ generates a type of trees:

The inductive famiy $\Tr_P : I \to \mathcal{U}$ has constructors

 * $\lf : (i : I) \to \Tr_P i$
 * $\nd : (i : I) \to (f : \Op_P i) \to (\phi : (p : \Param f) \to \Tr_P (\SortOf(f,p))) \to \Tr_P i$

For a tree $w : \Tr_P i$ we define its type of leaves by induction on $w$.

$$ \Leaf_i : (w : Tr_P i) \to I \to \mathcal{U} $$

* $\Leaf( (\lf i), j) \equiv (i =_I j)$
* $\Leaf( \nd (f, \phi), j) \equiv \sum_{p : \Param f} \Leaf(\phi(p), j)$

and its type of nodes:

$$ \Node_i : (w : \Tr_P i) \to \Ops \to \mathcal{U} $$

* $\Nodes(\lf i, jg) \equiv \mathbf{0}$
* $\Nodes(\nd(f,\phi), jg) \equiv ((i,f) = jg) + \sum_{p : \Param f} \Node(\phi(p), jg)$

Let $w : \Tr_i P$ be a tree and $f : Op_P i$ an operation. A frame from $w$ to $f$ is an equivalence:
$$\Frame(w,f) \equiv \sum_{j:I} \Leaf(w,j) \simeq \Param_i(f)$$

i.e. the type of leaves of the tree is equivalent to the type of parameters of the operator.

A polynomial relation for $P$ is a [[type family]]:

$$ R :{i : I} \to (f : \Op i) \to (w : \Tr i) (\alpha : \Frame(w,f)) \to \mathcal{U}$$

Let $P : \Poly I$ and let $R$ be a relation on $P$. The slice of $P$ by $R$, denoted $P // R$, is the polynomial with sorts $\Ops$ defined as follows:

TODO: Continue writing this out. Essentially following [this coq file](https://github.com/Alizter/HoTT/blob/higher-alg/contrib/higher-alg/Polynomial3.v)

## Definitions and Examples

Here we collect the definitions and ideas that were given in Eric's talk. We can also use the Agda formalisation for reference too.

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

Fix a [[type]] $I$ of sorts. A **polynomial** over $I$, $\Poly I$, is the data of

 * A [[type family|family]] of operations $\Op : I \to \mathcal{U}$
 * For each operation, for each $i : I$, a family of sorted parameters

	$$\Param_i : (f : \Op i) \to  I \to \mathcal{U}$$

=--

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

 * For $i : I$, an element $f: \Op i$ represents an operation whose output sort is $i$.
 * For $f : \Op i$ and $j : I$, and element $p : \Param_i(f, j)$ represents an input parameter of sort $j$.

 * The $\Op i$ and $\Param_i (f, j)$ are not truncated at set level. So operations and parameters can have higher homotopy.

 * In [[HoTT book]] notation, we can write the previous definition as:

$$\Poly I \equiv \sum_{\Op : I \to \mathcal{U}}\prod_{i : I} \Param_i$$

$$\Param_i \equiv \prod_{f : \Op i}I \to \mathcal{U}$$
=--

+-- {: .num_remark}
###### Example
We start with the simplest non-trivial example, when $I \equiv \mathbf{1}$ the [[unit type]]. Looking at $\Poly \mathbf{1}$ we clearly have $\Op : \mathbf{1} \to \mathcal{U}\simeq \mathcal{U}$. So $\Op$ is simply a type. Next we have $\Param : (f : \Op) \to \mathbf{1} \to \mathcal{U}\simeq \Op \to \mathcal{U}$ so $\Param$ is just a family of types indexed over the type $\Op$.

Now if $\Op \equiv \Fin_3$ then there would be 3 operations. Take one of these operations and call it $o_1$, $\Param$ would then have to chose what the type of parameters should be, let's choose $\Fin_2$. This would mean that $o_1$ would be an operation ($o_1 : \Op$) with two parameters $p : \Fin_2 \equiv \Param o_1$. Note $o_1$ is very much like the operator in an abstract syntax tree in this case.

When the type of sorts $I$ gets more structure, the objects we discuss get complicated quickly. Even letting $\Param o_1$ be some arbitrary type gives us an interesting object.

=--

A polynomial $P : \Poly I$ generates an associated type of trees.

+-- {: .num_defn}
###### Definition
The type of **trees** associated to a polynomial $P : \Poly I$ is an [[inductive family]] $\Tr P : I \to \mathcal{U}$ that has constructors

 * $\lf : (i : I) \to \Tr(P, i)$
 * $\nd : (i : I) \to (f : \Op(P,i)) \to (\phi : (j : I) \to (p : \Param_i (f, j)) \to \Tr(P, j)) \to \Tr(P, i)$

=--

For a tree $w : \Tr (P, i)$, we will need its type of leaves and type of nodes

+-- {: .num_defn}
###### Definition
The type of **leaves** of a tree $P : \Poly I$ is given by:

 * $\Leaf : (i : I) \to (w : \Tr i) \to I \to \mathcal{U}$
 * $\Leaf_i((\lf i), j) \equiv (i = j)$
 * $\displaystyle\Leaf_i ( \nd(f, \phi),j) \equiv \sum_{k : I} \sum_{p : \Param (f, k)} \Leaf_i ((\phi (k, p)), j)$

=--

+-- {: .num_defn}
###### Definition
The type of **nodes** of a tree $P : \Poly I$ is given by:

 * $\Node : (i : I) \to (w : \Tr i) \to (j : I) \to \Op j \to \Type$
 * $\Node_i ((\lf i), j, g) \equiv \mathbf{0}$
 * $\displaystyle \Node_i (\nd(f,\phi),j,g) \equiv ((i,f)=(j,g)) + \sum_{k : I} \sum_{p : \Param(f,k)} \Node_i ( \phi( j, p), j, g) $

=--

+-- {: .num_defn}
###### Definition
Let $P : \Poly I$ be a polynomial $w : \Tr(P,i)$ a tree and $f : \Op(P,i)$ an operation. A **frame** from $w$ to $f$ is a [[family]] of [[equivalences]]
$$\Frame(w,f):(j:I) \to \Leaf_i(w,j) \simeq \Param(P,f,j)$$
=--

+-- {: .num_defn}
###### Definition
A *polynomial relation* for $P$ is a [[type family]]:

$$R : (i:I) (f:\Op i)(w : \Tr i)(\alpha : \Frame(w,f) \to \mathcal{U}$$

=--

## References
 {#Link}

* {#Finster2018} [[Eric Finster]], _Towards Higher Universal Algebra in Type Theory_, [Homotopy Type Theory Electronic Seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html) 2018 ([compact pdf slides](/nlab/files/Finster-2018-HoTTEST-compact.pdf), [original pdf slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Finster-2018-12-6-HoTTEST.pdf), [recording](https://www.youtube.com/watch?v=hlCVHVtAlqQ))

* [[Agda]] code at: [github.com/ericfinster/higher-alg](https://github.com/ericfinster/higher-alg)

* [Notes on a definition of higher structure](http://ericfinster.github.io/files/huadtt.pdf).

The idea of [[opetopic sets]] that [Finster 18](#Finster2018) is inspired by goes back to

* [[John Baez]], [[James Dolan]], _Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes_, [arXiv:q-alg/9702014](https://arxiv.org/abs/q-alg/9702014)

category: reference


tower

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

#Contents#
* table of contents
{:toc}

## Idea

A _tower_ is a [[diagram]] of the shape of the [[poset]] of [[natural numbers]]

$$
  \cdots \to X_2 \to X_1 \to X_0
  \,.
$$

A [[limit]] over a diagram of this form is called a [[sequential limit]]/[[directed limit]]. If all connecting morphisms $X_{i+1}\to X_i$ are epimorphism then we sometimes say that the tower is a [[cofiltration]] (though the same term may be sometimes applied more generally for more general index categories,
or more restrictively when there is an object in the ambient category such that
all $X_i$ are its quotients).

Different contexts lead to different notions of morphism of towers, so it is important to consider what category of towers is appropriate for any given use of these objects. In addition to [[Postnikov towers]], and related uses in decomposing homotopy types, towers also occur as a simple type of [[pro-object]] in a category.  In that situation the morphisms considered between towers are usually pro-morphisms.

In [[homotopy theory]] in the presence of [[(infinity,1)-limits]]/[[(infinity,1)-colimits]], every tower is a [[tower of homotopy fibers]].

## The pro-category of towers
 {#TheProCategoryOfTowers}

### Definition

The tower diagram shape $\mathbb{N}_{\geq} = \{\cdots 3\to 2 \to 1 \to 0\}$ is evidently a small [[cofiltered category]]. As such it makes sense to consider _pro-morphisms_ between tower diagrams:

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

For any small category $\mathcal{C}$, the [[full subcategory]]

$$
  i
   \;\colon\;
  Tow_{pro}(\mathcal{C})
    \overset{}{\hookrightarrow}
  Pro(\mathcal{C})
    \hookrightarrow
  PSh(\mathcal{C})
$$

of the category $Pro(\mathcal{C})$ of [[pro-objects]] in $\mathcal{C}$ on those that have presentation by formal [[sequential limits]] (formal limits over tower diagrams) is the **pro-category of towers** $Tow_{pro}(\mathcal{C})$ in $\mathcal{C}$.

=--

(e.g. [Blanc 96, def. 2.5](#Blanc96))

For $X_\bullet \colon \mathbb{N}_{\geq} \longrightarrow \mathcal{C}$ a tower diagram, we write

$$
  \underset{\longleftarrow}{\lim}^f X_\bullet \in Tow_{pro}(\mathcal{C}) \hookrightarrow Pro(\mathcal{C})
$$

for its formal cofiltered limit, i.e.

$$
  \underset{\longleftarrow}{\lim}^f
   \;\colon\;
  Tow(\mathcal{C})
    \longrightarrow
  Top_{pro}(\mathcal{C})
  \,.
$$

### Properties

Notice that generally we have that morphisms between formal cofiltered limits (pro-objects) of shapes $\mathcal{K}_1$ and $\mathcal{K}_2$, respectively, are represented as cofiltered limits of systems of morphisms indeced on a single cofiltered category $\mathcal{K}$ equipped with [[final functors]] to $\mathcal{K}_1$ and $\mathcal{K}_2$, respectively. (See at _[[ind-object]]_ [this prop.](ind-object#MorphismsRepresentedByCofilteredSystemsOfMorphisms)).

The following says that in the case that both shapes are towers, then also $\mathcal{K}$ may always be chosen to be of tower shape:

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

For $X_\bullet$ and $Y_\bullet$ two tower diagrams in $\mathcal{C}$, then every morphism

$$
   \phi
     \;\colon\;
   \underset{\longleftarrow}{\lim}^f X_\bullet
     \longrightarrow
   \underset{\longleftarrow}{\lim}^f Y_\bullet
$$

between their formal cofiltered limits (every pro-morphism between the diagrams) in $Tow_{pro}(\mathcal{C}) \hookrightarrow Pro(\mathcal{C})$ is represented by component morphisms

$$
  \phi_{k} \;\colon\; X_{h(k)} \longrightarrow Y_k
$$

making [[commuting diagrams]]

$$
  \array{
     X_{h(k+1)} &\overset{\phi_{k+1}}{\longrightarrow}& Y_{k+1}
     \\
     \downarrow
       &&
     \downarrow
     \\
     X_{h(k)}
       &\underset{\phi_k}{\longrightarrow}&
     Y_k
  }
$$

i.e. by a [[natural transformation]] $\phi_\bullet$ between

$$
  \mathbb{N}_{\geq} \overset{h}{\longrightarrow} \mathbb{N}_{\geq} \overset{X_\bullet}{\longrightarrow} \mathcal{C}
$$

and

$$
  \mathbb{N}_{\geq} \overset{Y_\bullet}{\longrightarrow} \mathcal{C}
$$

as

$$
  \phi \simeq \underset{\longleftarrow}{\lim}^f (\phi_\bullet)
  \,.
$$

=--

(e.g. [Blanc 96, p. 6](#Blanc96), [Libman, p.4](#Libman))

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

The pro-category of towers $Tow_{pro}(\mathcal{C})$ (def. \ref{ProCategoryOfTowers}) has all [[finite limits]] and [[finite colimits]]. Moreover these are presented by the degreewise finite limits of any representing diagram of towers, via prop. \ref{ProMorphismsBetweenTowerDiagrams}.

=--

([Blanc 96, pop. 2.7](#Blanc96))

## Examples

* [[Postnikov tower]]

* [[Whitehead tower]]

* [[chromatic tower]]

## Related concepts

* [[filtered object in an (infinity,1)-category]]

## References

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

* {#Libman} [[Assaf Libman]], _Tower techniques for cofacial resolutions_,  Hopf archive ([pdf](http://hopf.math.purdue.edu/Libman/towers.pdf))

[[!redirects tower diagram]]
[[!redirects towers]]
[[!redirects tower diagrams]]

[[!redirects cotower]]
[[!redirects cotowers]]

[[!redirects co-tower]]
[[!redirects co-towers]]

[[!redirects cotower diagram]]
[[!redirects cotower diagrams]]

[[!redirects co-tower diagram]]
[[!redirects co-tower diagrams]]

[[!redirects pro-category of towers]]
[[!redirects pro-categories of towers]]


tower of homotopy fibers

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

#Contents#
* table of contents
{:toc}

## Definition

Given a [[homotopy theory]], i.e. an [[(infinity,1)-category]], then a _tower of homotopy fibers_ or _tower of fibrations_ or similar is a [[tower]] [[diagram]] of the form

$$
  \array{
    \vdots
    \\
    {}^{\mathllap{hofib(f_2)}}\downarrow
    \\
    X_2 &\stackrel{f_2}{\longrightarrow}& A_2
    \\
    {}^{\mathllap{hofib(f_1)}}\downarrow
    \\
    X_1 &\stackrel{f_1}{\longrightarrow}& A_1
    \\
    {}^{\mathllap{hofib(f_0)}}\downarrow
    \\
    X &\stackrel{f_0}{\longrightarrow}& A_0
  }
$$

where each hook is a [[homotopy fiber sequence]].

## Properties

The [[long exact sequences of homotopy groups]] for each of the hooks in the tower combine to yield an [[exact couple]]. The corresponding [[spectral sequence of an exact couple]] is a means to (approximately) compute the [[homotopy groups]] of the base object $X$ of the tower

## Examples

* [[Postnikov tower]]

* [[Adams tower]]

## Related concepts

* [[filtered object in an (infinity,1)-category]]

  * [[filtered chain complex]]

  * [[filtered spectrum]]

* [[spectral sequence of a tower of fibrations]]

## References

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], chapter IX of _Homotopy limits, completions and localization_, Springer 1972

* [[Paul Goerss]], [[Rick Jardine]], chapter VI of _[[Simplicial homotopy theory]]_, 1996

[[!redirects towers of homotopy fibers]]

[[!redirects tower of fibrations]]
[[!redirects towers of fibrations]]

[[!redirects fibrational tower]]
[[!redirects fibrational towers]]


tphyahoo

### Compactness for locales

this is a scratch space for where I was working on clarifying

https://ncatlab.org/nlab/show/compact+space#compactness_for_locales

In another context, the definition 2.2 also works for [[locales]], since it refers only to the [[frame]] of open sets.  Here is an equivalent way to phrase it that is often convenient for locale theory.

Firstly, note that given any open cover, adding to it the closure of all unions of finite opens defines a [[direction]] on any [[open cover]].  This directed open cover is useful for locales.

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

A space is compact iff every directed open cover of it has the entire space as one of its opens.

=--

+-- {: .proof}
###### Proof
For the "only if" case: Let $\mathcal{U}$ be a directed open cover of compact $X$, with the direction defined by unioning. By the open-cover definition of compactness, $X$ is the union of finitely many opens $U_i$ of $\mathcal{U}$. By directedness, these $U_i$ have an upper bound in $\mathcal{U}$. Since the only upper bound of opens unioning to $X$ is $X$, it follows that $X$ belongs to $\mathcal{U}$.

For the "if" case: given an open cover $\mathcal{U}$ of $X$, let $\mathcal{U}'$ be all the unions of finitely many opens of $\mathcal{U}$. $\mathcal{U}'$ is an open cover of $X$ since $\mathcal{U}$ is. Taking the upper bound for a direction as unioning, clearly $\mathcal{U}'$ is directed. So, by hypothesis, $X$ belongs to $\mathcal{U}'$. So $X$ is a union of finitely many opens of $\mathcal{U}$. This shows $X$ is compact according to Definition \ref{hb}.
=--


trace

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

# Traces
* table of contents
{: toc}

## Idea

Given an [[endomorphism]] $f:A\to A$ in a [[traced monoidal category]] $\mathscr{C}$ there is a natural notation of a _trace_ $\tr(f):1\to 1$, where $1$ is the tensor unit. When $\mathscr{C}$ is a $\mathbb{C}$-linear category for which the tensor unit (e.g. a fusion categories), we can canonically identity $\tr(f)$ with a complex number. That is, the trace of $f$ is the unique scalar $\lambda$ such that

$$\tr(f)=\lambda \cdot \text{id}_A.$$

In the case that $f$ is a [[linear map | linear]] endomorphism of a [[finite number| finite]] [[dimension | dimensional]] [[vector space]], we recover the usual notion of trace from [[linear algebra]].

## Definition

The idea of the trace operation is easily seen in [[string diagram]] notation: essentially one takes an endomorphism
$f: A\xrightarrow{}A$, and "closes the loop".

$$
 \array{
   1
   \\
   \;\;\;\downarrow^{tr(f)}
   \\
   1
 }
  \;\;\;
  :=
  \;\;\;
  \array{
     & 1
     \\
     & \downarrow
     \\
     a^* &\otimes& a
     \\
     \downarrow^{\mathrlap{Id_{a^*}}} && \;\;\downarrow^f
     \\
     a^* &\otimes& a
     \\
     & \downarrow^{\mathrlap{b_{a^*, a}}}
     \\
     a &\otimes& a^*
     \\
     & \downarrow
     \\
     & 1
  }
$$

This definition makes sense in any [[braided monoidal category]], but often in non-symmetric cases one wants instead a slightly modified version which requires the extra structure of a [[balanced monoidal category|balancing]].

The trace of the identity $1_a:a \to a$ is called the **[[dimension]]** or [[Euler characteristic]] of $a$.

## Examples

* [[Vect|$C = Vect_k$]] with its standard monoidal structure ([[tensor product of vector spaces]]): in this case $tr(f)$ is the usual trace of a linear map. Indeed, suppose $f$ is an [[endomorphism]] on a [[finite dimensional vector space|finite dimensional $k$-vector space]] $V$. Then, the trace $tr(f) \colon k\to k$ is given by the following composition.
\[k\xrightarrow{\eta} V\otimes V^\ast\xrightarrow{f\otimes id}V\otimes V^\ast\xrightarrow{\varepsilon} k\]
Here $\varepsilon$ is the standard [[evaluation]]-pairing. The map $\eta$ is obtained from the [[dual linear map]] of $\varepsilon$ by applying the following isomorphisms, see ([Dold & Puppe 1978](#DoldPuppe1978)).
\[k\cong k^\ast,\qquad (V\otimes V^\ast)^\ast\cong(V^\ast)^\ast\otimes V^\ast\cong V\otimes V^\ast\]
One can then compute that
\[\eta(\lambda)=\sum_{i=1}^n\lambda (e_i\otimes e_i^\ast),\]
where $\{e_1,\ldots,e_n\}$ is a [[linear basis]] of $V$ and $\{e_1^\ast,\ldots,e_n^\ast\}$ is the [dual basis](dual+vector+space#dual_bases). Now, if $f$ is given in this basis by a matrix $(a_ij)$, then
\[(f\otimes id)(\eta(\lambda))=\lambda\sum\begin{pmatrix}a_{1i}\\\vdots\\a_{ni}\end{pmatrix}\otimes e_i^\ast.\]
Finally, $(tr(f))(\lambda)$ is obtained by applying the pairing on this tensor: $(tr(f))(\lambda)=\lambda\sum a_{ii}$.

* $C = SuperVect = (Vect_{\mathbb{Z}_2}, \otimes, b)$, the category of $\mathbb{Z}_2$-graded vector spaces with the _non_trivial symmetric braiding which is $-1$ on two odd graded vector spaces: in this case the above is the **[[supertrace]]** on supervectorspaces, $str(V) = tr(V_{even}) - tr(V_odd)$.

* $C = Span(Top^{op})$: here the trace is the [[co-span co-trace]] which can be seen as describing the gluing of in/out boundaries of [[cobordism|cobordisms]].

* $C = Span(Grpd)$: this reproduces the notion of trace of a linear map within the interpretation of spans of groupoids as linear maps in the context of [[groupoidification]] and [[geometric function theory]], made explicit at [[span trace]].

* In the [[symmetric monoidal (infinity,1)-category of spectra]], the trace on the identity on a [[suspension spectrum]] of a [[manifold]] $X$ is the [[Euler characteristic]] of $X$ (see there).

## Generalizations

### Vertical categorification

See [[trace of a category]].

### Horizontal categorification

See [[trace in a bicategory]].

### Partial trace

Let $V \in C$ be a dualizable object, and $W$ any object. From an endomorphism $f$ of $V \otimes W$, one can produce an endomorphism $tr_V(f)$ of $W$, by applying the duality data of $V$. In terms of string diagrams, this is "bending along" the strand representing $V$.

+--{: .query}
TO DO: Draw the diagram just described.
=--

#### Matrix representation

Suppose $V$, $W$ are finite-dimensional vector spaces over a field, with dimensions $m$ and $n$, respectively. For any space $A$ let $L(A)$ denote the space of linear operators on $A$. The __partial trace__ over $W$, Tr$_{W}$, is a mapping

$$
  T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).
$$

+-- {: .un_defn}
###### Definition
Let $e_{1}, \ldots, e_{m}$ and $f_{1}, \ldots, f_{n}$ be bases for $V$ and $W$ respectively.  Then $T$ has a matrix representation $\{a_{k l,i j}\}$ where $1 \le k,i \le m$ and $1 \le l,j \le n$ relative to the basis of the space $V \otimes W$ given by $e_{k} \otimes f_{l}$.  Consider the sum

$$
  b_{k,i} = \sum_{j=1}^{n}a_{k j,i j}
$$

for $k,i$ over $1, \ldots, m$.  This gives the matrix $b_{k,i}$.  The associated linear operator on $V$ is independent of the choice of bases and is defined as the partial trace.
=--

#### Example

Consider a quantum system, $\rho$, in the presence of an environment, $\rho_{env}$.  Consider what is known in [[quantum information theory]] as the CNOT gate:

$$
  U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.
$$

Suppose our system has the simple state ${|1\rangle}{\langle 1|}$ and the environment has the simple state ${|0\rangle}{\langle 0|}$.  Then $\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}$.  In the quantum operation formalism we have

$$
  T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}
          = \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|})
          = \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2}
          = {|1\rangle}{\langle 1|}
$$

where we inserted the normalization factor $\frac{1}{2}$.

## Related concepts

* [[relation between trace and determinant]]

* [[traced monoidal category]]

* [[Euler characteristic]]

* [[bicategorical trace]],

  * [[Reidemeister trace]]

* [[higher trace]]

* [[Dennis trace]], [[cyclotomic trace]]

* [[trace of horizontal to round chord diagrams]]

## References

The categorical notion of trace in a monoidal category is due to

* {#DoldPuppe1978} [[Albrecht Dold]],  and [[Dieter Puppe]], _Duality, trace, and transfer_ In Proceedings of the Inter-
national Conference on Geometric Topology (Warsaw, 1978), pages 81{102, Warsaw, 1980.
PWN.

and

* [[Max Kelly]] M. L. Laplaza, _Coherence for compact closed categories_  J. Pure Appl. Algebra, 19:193{213, 1980.

Surveys include

* [[Peter Selinger]], _A survey of graphical languages for monoidal categories_ ([pdf](http://www.mathstat.dal.ca/~selinger/papers/graphical.pdf)), Section 5

* [[Kate Ponto]], [[Mike Shulman]], _Traces in symmetric monoidal categories_ ([pdf](http://arxiv.org/pdf/1107.6032v1.pdf)).

Generalization of this to [[indexed monoidal categories]] is in

* [[Kate Ponto]], [[Mike Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))

and to [[bicategories]] in

* [[Kate Ponto]], [[Mike Shulman]], _Shadows and traces in bicategories_, JHRS, ([arXiv:0910.1306](https://arxiv.org/abs/0910.1306))

Further developments are in

* [[Andre Joyal]], [[Ross Street]], and [[Dominic Verity]], _Traced Monoidal Categories_

* [[David Ben-Zvi]], [[David Nadler]], _Nonlinear traces_ ([arXiv:1305.7175](http://arxiv.org/abs/1305.7175))
 {#BenZviNadler13}

For the notion of partial trace, particularly its application to [[quantum mechanics]], see:

* Nielsen and Chuang, _Quantum Computation and Quantum Information_

[[!redirects traces]]

[[!redirects partial trace]]
[[!redirects partial traces]]