Tall-Wraith monoid

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#### Algebra
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# Tall--Wraith monoids
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## Idea

Given an [[algebraic theory]] $V$, a $V$-algebra is a model of $V$ in the category $Set$.  A _Tall--Wraith $V$-monoid_ is _the kind of thing that acts on $V$-algebras_.

## Definition

+-- {: .num_definition #twmonoid}
###### Definition
Let $V$ be an [[algebraic theory]] and let $V Alg$ be the category of models of this theory in $Set$.  Then a **Tall--Wraith $V$-monoid** is a [[monoid|monoid object]] in the category of co-$V$-objects in $V Alg$.
=--

To see why these are _what acts on $V$-algebras_ one needs to understand what a co-$V$-object in $V Alg$ actually is.
A co-$V$-object in some category $D$ is a [[representable functor|representable covariant functor]] from $D$ to $V Alg$.  To give a particular $D$-object, $d$, the structure of a co-$V$-object is to give a lift of the $Set$-valued $Hom$-functor $D(d,-)$ to $V Alg$.
Thus a co-$V$-object in $V Alg$ is a representable covariant functor from $V Alg$ to itself.

One can therefore consider composition of such representable covariant functors.
The main result of this can be simply stated:

+-- {: .num_proposition #repcomp}
###### Proposition
The composition of representable covariant functors $V Alg \to V Alg$ is again representable.
=--

This is a basic result in general algebra, and is not stated here in its full generality (although see [below](/nlab/show/Tall-Wraith+monoid#general) for some reasonably general constructions).

An almost corollary of this is that the category of representable covariant functors from $V Alg$ to itself is monoidal (the "almost" refers to the fact that you have to show that the identity functor is representable, but this is not hard).

Thus for two co-$V$-algebra objects in $V$, say $R_1$ and $R_2$, there is a product $R_1 \odot R_2$ and a natural isomorphism

$$
Hom_V(R_1 \odot R_2,A) \cong Hom_V(R_1, Hom_V(R_2,A))
$$

for any $V$-algebra, $A$.

A **Tall--Wraith $V$-monoid** is thus a triple $(P,\mu,\eta)$ with $\mu : P \odot P \to P$ and $\eta : I \to P$ (where $I$ is the free $V$-algebra on one element --- this represents the identity functor), satisfying the obvious coherence diagrams.
An action of $P$ on a $V$-algebra, say $A$, is then a morphism $\rho : P \odot A \to A$ again satisfying certain coherence diagrams.

Ah, but I have not told you what $P \odot A$ is!
At the moment, one can take the "product" of two co-$V$-algebra objects in $V Alg$ but now I want to take the product of a co-$V$-algebra object with a $V$-algebra.
How do I do this?
I do this by observing that a $V$-algebra is a _co-$Set$-algebra object in $V Alg$_!
That's a complicated way of saying that $V$ represents a covariant functor $V Alg \to Set$.
Precomposing this with the functor represented by $P$ yields again a covariant functor $V Alg \to Set$.
This is again representable and we write its representing object $P \odot A$.

As an aside, we note a consequence.  As we've seen, the category of co-$V$-algebra objects in $V$ is a monoidal category, with the tensor product $\odot$.  Now we're seeing this monoidal category [[action|acts]] on the category of $V$-algebras.   Indeed, it acts on the categories of $V$-algebra and co-$V$-algebra objects in a reasonably arbitrary base category.

One postscript to this is that although the category of co-$V$-algebra objects in $V Alg$ is not a variety of algebras, for a specific Tall--Wraith $V$-monoid $P$, the category of $P$-modules _is_ a variety of algebras.

## Examples

* If $V$ is the theory of [[commutative ring|commutative unital rings]], a $V$-algebra is a commutative unital ring, a co-$V$-algebra object in $V$ is a [[biring]] and the corresponding sort of Tall--Wraith $V$-monoid is called, in Tall and Wraith's original paper, a *biring triple*.

* If $V$ is the theory of [[commutative algebra|commutative associative algebras]] over a [[field]] $k$, then a $V$-algebra is a commutative associative algebra over $k$, and the corresponding sort of Tall--Wraith $V$-monoid is called a [[plethory]].

* If $V$ is the theory of [[abelian group|abelian groups]], than a $V$-algebra is an abelian group, and the corresponding sort of Tall--Wraith $V$-monoid is a [[ring]].

  To understand the last example, we need to think about _co-abelian group objects in the category of abelian groups_.  Abstractly, such a thing is an abelian group object [[internalization|internal to]] $AbGp^{op}$ (though this picture gets the morphisms the wrong way around; in full abstraction then the category of co-abelian group objects in $AbGrp$ is the _opposite_ category of the category of abelian group objects in $AbGrp^{op}$). Concretely, such a thing is an abelian group $A$ together with group homomorphisms

  $$
  \begin{aligned}
  \mu &: A \to A \coprod A, \\
  \epsilon &: A \to I \\
  \iota &: A \to A
  \end{aligned}
  $$

  where $I$ is the initial object in the category of abelian groups.  These homomorphisms must satisfy certain laws: just the abelian group axioms written out diagrammatically, with all the arrows turned around.

  In fact, $I = \{0\}$.  Thus $\epsilon$ is forced to be the map that sends everything to $0$: we have no choice here.

  We also have that $A \coprod A = A \oplus A$.
That means that for $a \in A$, $\mu(a) = (a_1,a_2)$ for some $a_1, a_2 \in A$.  Now, one of the laws says that $\epsilon$ is a counit for $\mu$. This means that $(\epsilon \oplus 1) \mu = 1$ and similarly for $1 \oplus \epsilon$. Thus $a_1 = a_2 = a$ and $\mu$ is the diagonal map.  So, we have no choice here either.

  The diagram for $\iota$ (representing the inverse map) is a little more complicated.  As $I$ is the initial object in $AbGrp$, there is a unique morphism $I \to A$ (inclusion of the zero).  Composing this with $\epsilon$ yields a morphism $A \to A$ which maps every element to the zero in $A$.  Using $\mu$ and $\iota$ we can construct another morphism $A \to A$ as

  $$
  A \overset{\mu}\rightarrow A \coprod A \overset{1 \coprod \iota}\rightarrow A \coprod A \overset{\Delta^c}\rightarrow A
  $$

  where $\Delta^c$ is the co-diagonal.  The relations for abelian groups say that this morphism must be the same as the zero morphism $A \to A$.  Using the fact that $A \coprod A \cong A \oplus A$ and that $\mu$ is the diagonal, this says that $a + \iota(a) = 0$.  Hence, by the uniqueness of inverses for abelian groups, $\iota(a) = -a$.

  Thus _if_ $(A, \mu, \iota, \epsilon)$ is a co-abelian group object in $AbGrp$ then $\mu$ is the diagonal, $\iota$ the inverse from abelian groups, and $\epsilon$ the zero morphism.

  However, that is still not quite the same as saying that $(A, \mu, \iota, \epsilon)$ is a co-abelian group object in $AbGrp$.  Certainly, $(A, \mu, \epsilon)$ is a co-commutative co-monoid object in $AbGrp$ since $\mu$ is the diagonal, which is automatically co-commutative and co-associative, and $\epsilon$ the zero map, which is the co-unit for the diagonal.  What remains is to fit $\iota$ into the structure.

  The first issue is that $\iota$ is not _automatically_ a morphism in $AbGrp$.  That is to say, when defining an algebraic theory then the operations are defined on the underlying objects.  It is a consequence of the relations of abelian groups that the operations lift to morphisms of abelian groups (algebraic theories where this happens for all operations are sometimes called _commutative_).  Thus $\iota$ is a morphism of abelian groups and so $(A, \mu, \iota, \epsilon)$ is a co-commutative co-monoid with an extra unary co-operation.  In fact, it is an involution from the relations for abelian groups.

  The final relation is that $\iota$ is the inverse for $\mu$.  The relation that $\iota$ is the inverse for addition (let us write it as, say, $\alpha$) is that

  $$
  A \overset{\Delta}\rightarrow A \times A \overset{1 \times \iota}\rightarrow A \times A \overset{\alpha}\rightarrow A
  $$

  is the zero map $A \to I \to A$.  This is precisely the relation that $\iota$ is the inverse for $\mu$ since we have the following identifications: $\mu = \Delta$, $A \coprod A = A \times A$, and $\Delta^c = \alpha$.  Also, $\epsilon = 0$ and $\eta : I \to A$ is the initial morphism in $AbGrp$.

  Thus the fact that $\iota$ is the inverse for the diagonal+zero co-monoidal structure is due to the fact that $\iota$ is the inverse for $(\alpha,\eta)$ and $\alpha : A \oplus A \to A$ is the co-diagonal in $AbGrp$ and $\eta : I \to A$ is the unit.

  It is part of the general theory that the category of co-$V$-objects in $V$ is monoidal (though not, in general, symmetric). For details on this see _The Hunting of the Hopf Ring_, referred to below.  This monoidal structure for abelian groups turns out to be the tensor product.

  Thus a Tall--Wraith monoid for abelian groups is actually an ordinary monoid in the category of abelian groups: in other words, a [[ring]]!

## General results and constructions {#general}

We now recapitulate the discussion above in a slightly more general context.

### Bialgebras over a monad

For now our context is that of [[monads]] $T$ on [[Set]], although all of what follows can be generalized considerably, for example to [[enriched category theory]] replacing $Set$ by a suitable [[cosmos]] $\mathbf{V}$.

Notation: the category of $T$-[[Eilenberg-Moore category|algebras]] is denoted $Set^T$, with forgetful functor $U: Set^T \to Set$ and free functor $F: Set \to Set^T$, whose composite is the monad $T = U F$, and whose [[counit]] is denoted $\epsilon: F U \to 1_{Set^T}$.

For each $T$-algebra $R$, there is an [[adjunction|adjoint pair]] of functors

$$- \cdot R \dashv \hom(R, -): Set^T \to Set$$

with associated monad $\hom(R, - \cdot R)$. The functor $- \cdot R: Set \to Set^T$ takes a set $X$ to the $T$-algebra $X \cdot R$, an $X$-indexed [[coproduct]] of copies of $R$ in $Set^T$. We define a $T$-*bialgebra* to be a $T$-algebra $R$ equipped with a morphism of monads $\phi: T \to \hom(R, -\cdot R)$. By the adjunction, the datum $\phi$ is equivalent to a left $T$-algebra structure

$$\alpha: T \circ \hom(R, -) \to \hom(R, -)$$

on $\hom(R, -): Set^T \to Set$, thus giving a lifting denoted (by abuse of language) $\hom(R, -): Set^T \to Set^T$. This datum is also equivalent to a right $T$-algebra (aka right $T$-module) structure

$$\beta: W T \to W$$

where $W = W_R \coloneqq - \cdot R: Set \to Set^T$. A $T$-*bialgebra map* is a $T$-algebra map $f: R \to S$ such that the induced map $W_f: W_R \to W_S$ is a morphism of right $T$-modules.

+-- {: .num_example}
###### Example
A good case to keep in mind is that of [[birings]], which are $T$-bialgebras for the [[Lawvere theory]] $T$ of [[commutative rings]]. The monad morphism $T \to \hom(R, -\cdot R)$ has components $T X \to \hom(R, X \cdot R)$ for each set $X$. Here $X \cdot R$ is an $X$-indexed coproduct of copies of $R$, where coproduct in the category of commutative rings $Set^T$ is given by tensor product. Thus, for example, $2 \cdot R$ is the ring $R \otimes R$. The component $T(2) \to \hom(R, 2\cdot R)$ therefore "interprets" each element $\theta \in T(2)$, i.e., each binary operation in the Lawvere theory, as a binary *co*-operation $R \to R \otimes R$. This applies in particular to the elements $m, a \in T(2)$ which abstractly represent multiplication and addition (seen as natural operations on the category of commutative rings).
=--

### Monoidal product on bialgebras over a monad

We thus have several perspectives on what a $T$-bialgebra is:

* A $T$-algebra $R$ equipped with a monad morphism $T \to \hom(R, -\cdot R)$,

* A $T$-algebra equipped with a compatible $T$-coalgebra structure (actually the same as the preceding item, but in different words),

* A $T$-algebra $R$ for which $\hom(R, -)$ is provided with a left $T$-algebra structure,

* A $T$-algebra $R$ for which $-\cdot R$ is provided with a right $T$-algebra/module structure.

The following proposition gives two more useful descriptions:

+-- {: .num_prop}
###### Proposition
Let $Ladj(Set^T, Set^T)$ ($Radj(Set^T, Set^T)$) be the category of left (right) adjoint functors $\Psi: Set^T \to Set^T$. The functor $T$-$BiAlg \to Ladj(Set^T, Set^T)$ that takes $(R, \phi)$ to the right $T$-module $(W_R, \beta)$ is an equivalence. Or, what is the same, the functor $T$-$BiAlg \to Radj(Set^T, Set^T)^{op}$, taking $(R, \phi)$ to the left $T$-algebra $\hom(R, -), \alpha)$, is an equivalence.
=--

+-- {: .proof}
###### Sketch of proof
The main thing to check is that the functor $R \mapsto \hom(R, -)$ to $Radj(Set^T, Set^T)^{op}$ is [[essentially surjective functor|essentially surjective]]. The essential point is that $\Phi$ has a left adjoint iff $U \Phi$ has a left adjoint iff $U \Phi: Set^T \to Set$ is [[representable functor|representable]]: $U \Phi \cong \hom(R, -)$ for some $T$-algebra $R$ (in which case the lift $\Phi$ of $\hom(R, -)$ through $U$ is tantamount to a $T$-algebra structure on $\hom(R, -)$). The only (mildly) tricky part is that $\Phi$ has a left adjoint if $U\Phi$ has a left adjoint $W = W_R$. To define the left adjoint $\Psi$ of $\Phi$ objectwise, we take any $T$-algebra $S$ with its canonical presentation

$$F U F U S \stackrel{\overset{\epsilon F U S}{\to}}{\underset{F U \epsilon}{\to}} F U S \stackrel{\epsilon}{\to} S$$

which is a [[coequalizer]] diagram. A left adjoint $\Psi$ must preserve this coequalizer, and we must have $\Psi F \cong W$ since both sides are left adjoint to $U \Phi$. Thus we define $\Psi (S)$ to be a coequalizer

$$W(T U S) \stackrel{\overset{\beta U S}{\to}}{\underset{W U\epsilon S}{\to}} W(U S) \to \Psi(S)$$

where $\beta: W T \to T$ is the $T$-module structure coming from the monad morphism $\phi: T \to \hom(R, -\cdot R)$. This objectwise definition of $\Psi$ easily extends to morphisms by [[universal property|universality]] and provides a left adjoint to $\Phi$. Remaining details are left to the reader.
=--

One import of this proposition is that left adjoint endofunctors on $Set^T$ compose, i.e., endofunctor composition gives a monoidal structure on $Ladj(Set^T, Set^T)$, and this monoidal structure transports across the categorical equivalence of the proposition to give a monoidal structure on $T$-$BiAlg$. The resultant monoidal product on $T$-bialgebras is denoted $\odot$.

A second import of this proposition is that the canonical functor $T\text{-}Bialg \to Ladj(Set^T, Set^T)$ induces a functor which is reasonably denoted

$$\odot: T\text{-}Bialg \times Set^T \to Set^T,$$

realizing an [[actegory]] structure over the monoidal category $T\text{-}Bialg$.

A direct construction of the monoidal product $\odot$ can be extracted by following the proof of the proposition. If $R, S$ are $T$-bialgebras, then the underlying $T$-algebra of $S \odot R$ (corresponding to composition of $\hom(S, -) \circ \hom(R, -)$ of right adjoints $Set^T \to Set^T$) is computed as a reflexive coequalizer in $Set^T$:

$$T U S \cdot R \stackrel{\overset{\epsilon U S \cdot R}{\to}}{\underset{\beta U S}{\to}} U S \cdot R \to S \odot R.$$

Here $\epsilon U S$ is the same as the $T$-algebra structure $T U S \to U S$ on $S$. Whereas $\beta X: T X \cdot R \to X \cdot R$ is a component of the $T$-module structure $W_R T \to W_R$; it is mated by the $- \cdot R \dashv \hom(R, -)$ adjunction to the component of the coalgebra structure $\phi X: T X \to \hom(R, X \cdot R)$.

To extract the $T$-coalgebra structure on $S \odot R$, let us observe generally that if $F: C \to D$ is a left adjoint, then for any category $E$ there is an induced left adjoint $[1_E, F]: [E, C] \to [E, D]$ and similarly an induced left adjoint $Ladj(E, C) \to Ladj(E, D)$. Applying this to the case where $C = D = E = Set^T$ and where $F$ is the left adjoint to the lift $\hom(R, -): Set^T \to Set^T$, we find that

$$- \odot R: Ladj(Set^T, Set^T) \to Ladj(Set^T, Set^T)$$

is a left adjoint, and in particular preserves $Y$-indexed copowers $Y \cdot S$. In other words, for each $X$ we have canonical isomorphisms

$$X \cdot (S \odot R) \cong (X \cdot S) \odot R, \qquad T X \cdot (S \odot R) \cong (T X \cdot S) \odot R$$

so that the desired right $T$-module structure is given componentwise by a composite

$$\beta (S \odot R)_X \coloneqq \left(T X \cdot (S \odot R) \cong (T X \cdot S) \odot R \stackrel{(\beta S)_X \odot R}{\to} (X \cdot S) \odot R \cong X \cdot (S \odot R)\right).$$

### Tall-Wraith monoids relative to a monad

A *Tall-Wraith monoid* over $T$ is of course a monoid in the monoidal category $(T\text{-}BiAlg, \odot)$. We note that the unit in his monoidal category is the free $T$-algebra $F(1)$, equipped with its canonical lift $id: Set^T \to Set^T$. That is, the $T$-coalgebra structure on $F(1)$ is given tautologously by

$$T(X) \cong U F(X) \cong Set^T(F(1), F(X)) \cong Set^T(F(1), X \cdot F(1)).$$

So, multiplication on a Tall-Wraith monoid is a bialgebra map $m: R \odot R \to R$ and the unit is a bialgebra map $u: F(1) \to R$. Such a monoid is tantamount precisely to a monoid in $Ladj(Set^T, Set^T)$, i.e., to a *left adjoint monad* on $Set^T$. In particular, for a Tall-Wraith monoid $R$, one has a category $R Alg$ of algebras over that monad, giving a monadic functor $Alg_R \to Set^T$.

Now, recall that left adjoint monads are canonically [[mate|mated]] to right adjoint comonads $C$, in such a way that the category of algebras over the monad is equivalent to the category of coalgebras over the comonad. In short, Tall-Wraith monoids over $T$ are essentially the same thing as functors

$$G: C \to Set^T$$

which are simultaneously *monadic and comonadic*: the comonadicity means $G$ is a left adjoint and has a left adjoint $F$, so that the monad $G F: Set^T \to Set^T$ resides in $Ladj(Set^T, Set^T)$, and such monads are tantamount to Tall-Wraith monoids.

+-- {: .num_example}
###### Example
In the important example where $T$ is the theory of commutative rings and $\Lambda$ is the bialgebra $\mathbb{Z}[x_1, x_2, \ldots]$, equipped with a Tall-Wraith multiplication $\Lambda \odot \Lambda \to \Lambda$ given by [[plethysm]] (a decategorified product that arises by viewing $\Lambda$ as the Grothendieck ring of the category of $Ab$-valued [[species]] together with its substitution or plethystic product), the category $Alg_\Lambda$ may be identified with the category of [[lambda-rings]]. In this case the monad $\Lambda \odot -$ has right adjoint given by $\hom(\Lambda, -)$. The *right* adjoint $CRing \to \Lambda Ring$ to the forgetful functor is the big Witt functor, often denoted $W$.
=--

## References

* D. Tall, [[Gavin Wraith|G. Wraith]], _Representable functors and operations on rings_,  Proc. London Math. Soc. (3), 1970, 619--643, [MR265348](http://www.ams.org/mathscinet-getitem?mr=265348), [doi](http://dx.doi.org/10.1112/plms/s3-20.4.619)

* [[James Borger]], B. Wieland, _Plethystic algebra_, Adv. Math. __194__ (2005), no. 2, 246–283, [doi](http://dx.doi.org/10.1016/j.aim.2004.06.006), [pdf](http://wwwmaths.anu.edu.au/~borger/papers/03/lambda.pdf), [MR2006i:13044](http://www.ams.org/mathscinet-getitem?mr=2139914)

* [[Andrew Stacey|A. Stacey]] and S. Whitehouse, _The Hunting of the Hopf Ring_,   Homology, Homotopy and Applications __11__(2), 2009, 75--132, [online](http://intlpress.com/HHA/v11/n2/a6), [arXiv/0711.3722](http://arxiv.org/abs/0711.3722).

An old and long query-discussion has been archived starting [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=291&Focus=26302#Comment_26302).

[[!redirects Tall-Wraith monoid]]
[[!redirects Tall-Wraith monoids]]
[[!redirects Tall–Wraith monoid]]
[[!redirects Tall–Wraith monoids]]
[[!redirects Tall--Wraith monoid]]
[[!redirects Tall--Wraith monoids]]


Tamagawa number

#Contents#
* table of contents
{:toc}

## Idea

Let $G$ be a [[connected]] [[semisimple]] [[algebraic group]] over the [[field]] of [[rational numbers]] $\mathbf{Q}$ or more generally over a [[global field]] $K$.  The **Tamagawa measure** of $G$ is a canonical normalization of the [[Haar measure]] on $G$.  (Recall that the latter is well-defined only up to scalar multiplication.)

The **Tamagawa number** of $G$ is essentially the Tamagawa measure of the [[quotient]] [[coset space]] $G(K)\backslash G(\mathbf{A}_K)$, where $\mathbf{A}$ is the ring of [[adeles]] of $K$.

This is [[conjecture|conjectured]] and known in many cases to be a [[rational number]], the quotient $Pic/Sha$ of the [[order of a group|order]] of the [[Picard group]] divided by that of the [[Tate-Shafarevich group]]. ([MO comment](http://mathoverflow.net/q/44184/381))

+-- {: .num_remark}
###### Remark
For $G= GL_1 = \mathbb{G}_m$ the [[multiplicative group]] then the quotient $G(\mathbf{Q})\backslash G(\mathbf{A})$ is the [[idele class group]].

=--

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

The further [[stacky quotient]] of $G(\mathbf{Q})\backslash G(\mathbf{A})$ by $G(\mathbb{A}_{\mathbb{Z}})$ is [[analogy|analogous]], under the [[function field analogy]] and in view of the [[Weil uniformization theorem]], with an incarnation of the [[moduli stack of bundles|moduli stack of G-principal bundles]] over a [[curve]]. A relation of the Tamagawa numbers to the properties of this stack is the content of the [[Weil conjecture on Tamagawa numbers]].)

=--

## Related concepts

* [[Weil conjecture on Tamagawa numbers]]

* [[Birch and Swinnerton-Dyer conjecture]]

## References

* Wikipedia, _[Tamagawa numbers](http://en.wikipedia.org/wiki/Tamagawa_number#Tamagawa_numbers)_, _[Weil conjecture on Tamagawa numbers](http://en.wikipedia.org/wiki/Weil_conjecture_on_Tamagawa_numbers)_

* {#Bloch80} [[Spencer Bloch]], _A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture_, Invent. Math. 58, 65-76 (1980) ([[BlochTamagawa.pdf:file]])

* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincare Duality_, Lecture notes ([web](http://www.math.harvard.edu/~lurie/283.html))

Relation to [[Yang-Mills theory]] and the [[moduli space of connections]]:

* Aravind Asok, Brent Doran, Frances Kirwan, _Yang-Mills theory and Tamagawa numbers_ ([arXiv:0801.4733](http://arxiv.org/abs/arXiv:0801.4733))

Some other relation to [[quantum mechanics]] and maybe to the [[Witten genus]] (?) is claimed in

* M. A. Olshanetsky, _Quantum-mechanical calculations in the algebraic group theory_, Comm. Math. Phys. Volume 132, Number 2 (1990), 441-459. ([Euclid](http://projecteuclid.org/euclid.cmp/1104201160))

[[!redirects Tamagawa numbers]]
[[!redirects Tamagawa measure]]
[[!redirects Tamagawa measures]]


Tamar Janelidze

* [Institute page](http://www.mth.uct.ac.za/dr-tamar-janelidze-gray)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=151223)

## Selected writings

On non-abelian generalizations of aspects of [[homological algebra]] (such as the [[diagram chasing lemmas - contents|diagram chasing lemmas]]) in [[semi-abelian categories]], [[homological categories]]:

* {#Janelidze09} [[Tamar Janelidze]], *Foundations of relative non-abelian homological algebra*, 2009 ([pdf](https://open.uct.ac.za/bitstream/item/4777/thesis_sci_2009_janelidze_t.pdf?sequence=1), [[Janelidze_NonabelianHomologicalAlgebra.pdf:file]] [hdl:11427/4891](http://hdl.handle.net/11427/4891))

category: people


Tamara von Glehn

[[!redirects Tamara van Glehn]]

Tamara von Glehn works on [[type theory]], [[category theory]] and the [[relation between category theory and type theory]].

* [webpage](https://www.dpmms.cam.ac.uk/people/tlv22/)

## Selected writings

On [[polynomial functors]] and [[two-sided fibrations]] in the [[categorical semantics of dependent types]]:

* {#vonGlehn2015} [[Tamara von Glehn]], *Polynomials and models of type theory*, PhD thesis (2015) [[doi10.17863/CAM.16245](https://doi.org/10.17863/CAM.16245)]

category: people


Tamaz Kandelaki

* [webpage](http://www.rmi.ge/~kandel/)

category: people


Tambara functor

#Contents#
* table of contents
{:toc}

## Idea

**Tambara functors** are algebraic structures similar to [[Mackey functors]], but with multiplicative norm maps as well as additive transfer maps, and a rule governing their interaction. They were introduced by Tambara, as TNR-functors, to encode the relationship between Trace (additive transfer), Norm (multiplicative transfer) and Restriction maps in the representation theory and cohomology theory of finite groups ([Tam93](#Tam93)).

>In the $G$-equivariant context for a finite group $G$, the role of abelian groups in non-equivariant algebra is played by Mackey functors. The category of Mackey functors is a closed symmetric monoidal category with symmetric monoidal product, the box product. In addition to the expected generalization of commutative rings to simply commutative monoids for the box product, there is a poset of generalizations of the notion of commutative rings to the $G$-equivariant context: the incomplete Tambara functors. These interpolate between [[Green functors]], the ordinary commutative monoids for the box product, and Tambara functors. The distinguishing feature for [incomplete] Tambara functors is the presence of certain multiplicative transfer maps, called norm maps. For a [[Green functor]], we have no norm maps; for a Tambara functor, we have norm maps for any pair of subgroups $H \subset K$ of $G$. ([Hill 17](#Hill17))

>A Mackey functor is represented by a "$G$-equivariant Eilenberg–MacLane spectrum", ... a Tambara functor is represented by a commutative "$G$-equivariant Eilenberg-MacLane ring spectrum" ([AB15](#AB15))

## Examples

Burnside rings, representation rings, zeroth stable homotopy group of a genuine equivariant $E_{\infty}$-ring.

>the homotopy category of Eilenberg MacLane commutative ring spectra is equivalent to the category of Tambara functors. ([Ull13](#Ull13))

Some other examples are related to [[Witt-Burnside functors]], Witt rings in the sense of Dress and Siebeneicher.

## Related concepts

* [[Mackey functor]]
* [[polynomial functor]]
* [[Witt vectors]]

##References

* [[Neil Strickland]], _Tambara Functors_, [arXiv:1205.2516](http://arxiv.org/abs/1205.2516)
* [[Morten Brun]], _Witt vectors and Tambara functors_, [arXiv:math/0304495](http://arxiv.org/abs/math/0304495)
* {#Tam93} [[Daisuke Tambara]], _On multiplicative transfer_, Comm. Algebra 21 (1993), no. 4, 1393–1420 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/tambara.pdf)).
* {#AB15} [[Vigleik Angeltveit]] and [[Anna Marie Bohmann]], _Graded Tambara Functors_, ([arXiv:1504.00668](http://arxiv.org/abs/1504.00668))
[[!redirects Tambara functors]]
* {#Ull13} [[John Ullman]], _Tambara Functors and Commutative Ring Spectra_, ([arXiv:1304.4912](http://arxiv.org/abs/1304.4912))
* [[Michael Hill]], _Derived Equivariant Algebraic Geometry_, ([lecture and notes](https://www.msri.org/workshops/689/schedules/18236))
* Kristen Mazur, _On the Structure of Mackey Functors and Tambara Functors_, ([thesis](http://sites.lafayette.edu/mazurk/files/2013/07/Mazur-Thesis-4292013.pdf))
* {#Hill17} [[Michael Hill]], _On the Andre-Quillen homology of Tambara functors_, ([arXiv:1701.06219](https://arxiv.org/abs/1701.06219))
* [[Andrew Blumberg]], [[Michael Hill]], _Incomplete Tambara functors_, ([arXiv:1603.03292](https://arxiv.org/abs/1603.03292))

[[!redirects Tambara functors]]


Tambara module

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## Idea

A **Tambara module** is a [[profunctor]] endowed with additional [[mathematical structure|structure]] that makes it interact nicely with the [[action]] of a [[monoidal category]].   In other words, Tambara modules generalize profunctors from [[categories]] to [[actegories]].   Tambara modules are used in the theory of [[optics (in computer science)]].

## Definition

\begin{definition}
Let $(\mathbf M, i, \odot)$ be a [[monoidal category]], let $\mathbf C$ and $\mathbf D$ be two [[actegory|left $\mathbf M$-actegories]] (in other terminology, left $\mathbf M$-modules). We denote $\mathbf M$ actions by $(-)\cdot (-)$.

A (left) **Tambara module** is a [[profunctor]] $P \,\colon\, \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ equipped with a [[indexed set|family]] of [[morphisms]] called (left) *strength*
$$
   s_{a,b,m} \,\colon\, P(a,b) \longrightarrow P(m \cdot a, m \cdot b)
$$
which is [[natural transformation|natural]] in $a$ and $b$ and [[dinatural transformation|dinatural]] in $m$, and satisfies two [[coherence laws]]:

1. $s_{a,b,i} = P(\rho_a, \rho_b^{-1})$, where $\rho$ comes with the [[actegory|module structures]] of $\mathbf C$ and $\mathbf D$.

2. $s_{a,b,m \odot n} = P(\mu^{-1}_{m,n,a}, \mu_{m,n,b}) \circ s_{a,b,m} \circ s_{a,b,n}$ where $\mu$ comes with the [[actegory|module structures]] of $\mathbf C$ and $\mathbf D$.

\end{definition}

This is same thing as (left) **strong profunctors**.

A right Tambara module is defined in the exact way except $\mathbf C$ and $\mathbf D$ are assumed to be right $\mathbf M$-actegories and everything is correspondigly 'done on the other side'.

\begin{definition}
  Suppose now $\mathbf C$ and $\mathbf D$ have both left and right $\mathbf M$-actegories structures.
  A **Tambara bimodule** is a profunctor $P : \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ equipped with compatible left and right Tambara module structures $s$ and $s'$, i.e. such that they satisfy
  $$
    s'_{m \cdot a, m \cdot b, n} \circ s_{a,b,m} = s_{a \cdot n, b \cdot n, m} \circ s'_{a,b,n}
  $$
\end{definition}

\begin{definition}
  Let $P, Q: \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ be (left) Tambara modules.
  A **morphism of Tambara modules** $\alpha : P \to Q$ is a natural transformation $P \Rightarrow Q$ that commutes with $P$ and $Q$'s strenghts:
$$
\alpha_{a,b} \circ s^P_{a,b,m} = s^Q_{a,b,m} \circ \alpha_{m \cdot a,m \cdot b}
$$
\end{definition}

\begin{remark}
  Tambara modules are named after [[Daisuke Tambara]] who introduced them in [Tamb06](#Tamb06) to prove that for a [[enriched category|$\mathbf V$-enriched]] category $\mathbf A$, $Z(\mathbf A^{op}, \mathbf V) \cong \mathbf{Tamb}(\mathbf A, \mathbf A)$.
\end{remark}

\begin{remark}
  Both [Tamb06](#Tamb06) and [PS07](#PS07) define Tambara module to mean Tambara bimodule. In the optical literature, however, left Tambara modules are called simply modules and they are the one used. This doesn't pose any problem when $\mathbf M$ is *symmetric* monoidal since it's evident how, in that case, any left module structure can be made into a right module structure, so that any left/right module is automatically a bimodule.
\end{remark}

## References

* {#Tamb06} [[Daisuke Tambara]], _Distributors on a tensor category_, Hokkaido mathematical journal, 35(2):379–425, 2006.

* [[Bartosz Milewski]], _Profunctor optics: the categorical view_ ([blog post](https://bartoszmilewski.com/2017/07/07/profunctor-optics-the-categorical-view/))

* {#Rom20} [[Mario Román]], _Profunctor optics and traversals_, 2020 ([arXiv](https://arxiv.org/abs/2001.08045))

* {#PS07} [[Craig Pastro]], [[Ross Street]], _Doubles for monoidal categories_, 2007 ([arXiv](https://arxiv.org/abs/0711.1859))

[[!redirects Tambara modules]]
[[!redirects Tambara bimodule]]
[[!redirects Tambara bimodules]]
[[!redirects strong profunctor]]
[[!redirects strong profunctors]]


Tambara-Yamagami category

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## Idea

A **Tambara-Yamagami** (TY) category is a [[fusion category]] that may be regarded as a non-trivial extension of a category of $G$-[[graded vector spaces]], for $G$ a [[finite group]]. By nontrivial extension we mean that, even though it is constructed from a category of $G$-graded vector spaces, it itself cannot be presented as such.

In 2-dimensional [[quantum field theory]] (QFT), a TY category is thought to encode as a *symmetry category* what is nowadays known as a [[non-invertible symmetry]], where the [[objects]] of the category correspond to line [[defects]].

## Definition

A Tambara-Yamagami category $TY(G)$ is defined as follows.

Starting from a [[finite group]] $G$, consider the set $I=G\coprod \{m\}$. We construct $TY(G)$ by defining the simple objects as $\{U_g\}_{g\in G}\coprod \{U_m\}$. These have fusion rules:

* $U_{g}\otimes U_{g'}=U_{gg'}$
* $U_{g}\otimes U_m = U_m$
* $U_m\otimes U_g = U_m$
* $U_m\otimes U_m = \sum_{g\in G} U_g$

Unless stated otherwise, we assume the ground field of $TY(G)$ to be $\mathbb{C}$. It has been shown that the above fusion rules also admit categorification over the reals (see [Plavnik & Sanford & Sconce 2023](#PlavnikSanfordSconce2023)).

## Properties

It was shown by [Tambara & Yamagami (1998)] (#Tambara1998tensor) that these categories are classified by pairs $(\chi,\tau)$ for $\chi:G\times G\to \mathbb{C}^*$ a bicharacter, and $\tau$ a square root of $\frac{1}{|G|}$.

+-- {: .num_remark}
###### Remark
Since $\chi$ is symmetric and nondegenerate, $TY(G)$ exists only if $G$ is abelian. $TY(G)$ is not necessarily unique for a choice of $G$.
=--

+-- {: .num_prop}
###### Proposition
$TY(G)$ is a [unitary fusion category](unitary+fusion+category).
=--

This follows from ([Galindo & Hong & Rowell 2013, theorem 5.20](#GalindoHongRowell2013)).

+-- {: .num_prop}
###### Proposition
$TY(G)$ is a braided fusion category if and only if $G\cong\mathbb{Z}_{2}^{n}$.
=--

The above is due to ([Siehler 2000, theorem 1.2(1)](#Siehler2000)).

## Examples

* $TY(\mathbb{Z}_{2})$ is realised by two distinct unitary fusion categories (distinguished by the Frobenius-Schur indicator $\varkappa_{m}\in\{+1,-1\}$ of $m$), each of which admit $4$ distinct braidings. All $8$ of these unitary braided fusion categories (UBFCs) are [modular](modular+tensor+category). In particular, the $4$ UBFCs with $\varkappa_{m}=+1$ are also known as the Ising categories (which describe [$SU(2)$-anyons](su2-anyon) at level $k=2$).

* As worked out in [Tambara & Yamagami (1998)](#Tambara1998tensor), for $TY(\mathbb{Z}_2\times\mathbb{Z}_2)$, there are two possible choices of roots ($\tau=\pm\frac{1}{2}$), and two choices of classes of bicharacters. For $\tau=\frac{1}{2}$ and trivial bicharacter, $TY$ is the [[category of representations]] of the [[dihedral group]] $D_8$. For $\tau=-\frac{1}{2}$ and trivial character, this is the representation category of the [[quaternion group]] $Q_8$. For $\tau=\frac{1}{2}$ and nontrivial character, this is the representation category of the Kac-Paljutkin 8-dimensional [[Hopf algebra]]. The remaining choice does not correspond to the category of representations of a Hopf algebra (see also [[Tannaka duality]]).

## References

* {#PlavnikSanfordSconce2023} Julia Plavnik, Sean Sanford, Dalton Sconce, *Tambara-Yamagami Categories over the Reals: The Non-Split Case*, preprint (2023) [[arXiv:2303.17843](https://arxiv.org/abs/2303.17843)]

* {#GalindoHongRowell2013} Cesar Galindo, Seung-Moon Hong, [[Eric Rowell]], *Generalized and Quasi-Localizations of Braid Group Representations*, International Mathematics Research Notices **2013(3)** (2013) 693-731 [[doi:10.1093/imrn/rnr269](https://doi.org/10.1093/imrn/rnr269)]

* {#Siehler2000} Jacob Siehler, *Braided Near-Group Categories*, preprint (2000) [[arXiv:0011037](https://arxiv.org/abs/math/0011037)]

* {#AnyonWiki} AnyonWiki, [List of small multiplicity-free fusion rings](http://www.thphys.nuim.ie/AnyonWiki/index.php/List_of_small_multiplicity-free_fusion_rings)

### General

Original articles:

* {#Tambara1998tensor} [[Daisuke Tambara]], [[Shigeru Yamagami]], *Tensor categories with fusion rules of self-duality for finite abelian groups*, Journal of Algebra **209**  (1998)  692-707 [[doi:10.1006/jabr.1998.7558](https://doi.org/10.1006/jabr.1998.7558)]

[[!redirects Tambara-Yamagami categories]]


tame topology

**Tame topology** is the name for the largely programmatic quest for a refoundation of [[topology]] and [[geometry]] that avoids 'pathological' objects like space-filling curves or counter-intuitive results like the [[Banch-Tarski theorem|Banach-Tarski paradox]] that occur in the traditional approach. The advantages of such a program that has among others been proposed by [[A. Grothendieck]] and [[W. Lawvere]] are philosophical by reconciling concepts and intuitions on space as well as technical by excluding pathological spaces.

The notion of [[o-minimal structure]] has been proposed as a reasonable candidate for an axiomatic approach to this hoped-for tame topology ( _topologie modérée_).

* Michel Coste, _An Introduction to O-Minimal Geometry_ , Lecture notes Pisa 1999. ([pdf](http://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf))

* [[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM vol. 1488 (1991).

* [[F. W. Lawvere]], _Left and right adjoint operations on spaces and data types_ , Theoretical Computer Science **316** (2004) pp.105-111.

* [[Lou van den Dries]], _Exponential rings, exponential polynomials and exponential functions_ , Pacific Journal of Mathematics **113** no.1 (1984) pp.51–66. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.pjm/1102709376))

* [[Lou van den Dries]], _A Generalization of the Tarski-Seidenberg Theorem and Some Nondefinability Results_ , Bull. Am. Math. Soc. **15** (1986) pp.189-193. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.bams/1183553469))

* [[Lou van den Dries]], _Tame topology and O-minimal structures_, London Math. Soc. Lecture Notes Series __248__, Cambridge U. Press 1998.

* [[Alexandre Grothendieck]], _Esquisse d'un Programme_, section 5. English translation available in Geometric Galois Actions I (edited by L. Schneps and P. Lochak), LMS Lecture Notes Ser. 242, CUP 1997.

* Yuliy Baryshnikov, [[Robert Ghrist]], _Euler integration over definable functions_ ([pdf](http://www.math.upenn.edu/~ghrist/preprints/definable.pdf))

* Jean Philippe Rolin,


Tamiaki Yoneya

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tamiaki_Yoneya)

## related $n$Lab entries

* [[Nambu mechanics]]

category: people


Tammo tom Dieck

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Tammo_tom_Dieck)

## Selected writings

On [[equivariant bundles]]:

* [[Tammo tom Dieck]], _Faserbündel mit Gruppenoperation_, Arch. Math 20, 136–143 (1969) ([doi:10.1007/BF01899003](https://doi.org/10.1007/BF01899003))

On [[equivariant bordism homology theory]]:

* [[Tammo tom Dieck]], _Kobordismentheorie und Transformationsgruppen_, Aaarhus University  Preprint Series 1968/69, No. 30 ([gbooks](https://books.google.de/books/about/Kobordismen_Theorie_und_Transformationsg.html?id=1ohGAAAAYAAJ&redir_esc=y))

* [[Tammo tom Dieck]], _Bordism of $G$-manifolds and integrality theorems_, Topology Volume 9, Issue 4, November 1970, Pages 345-358 (doi:10.1016/0040-9383(70)90058-3)

* {#tomDieck75} [[Tammo tom Dieck]], _Orbittypen und äquivariante Homologie II_, Arch. Math. (Basel) 26 (1975), no. 6, 650–662 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tomdieck2.pdf))

On [[cobordism theory]]:

* [[Theodor Bröcker]], [[Tammo tom Dieck]], *Kobordismentheorie*, Lecture Notes in Mathematics **178**, Springer (1970) [[ISBN:9783540053415](https://link.springer.com/book/9783540053415)]

On [[classifying spaces]] via [[geometric realization of simplicial topological spaces]]:

* {#tomDieck74} [[Tammo tom Dieck]], *On the homotopy type of classifying spaces*, Manuscripta Math 11, 41–49 (1974) ([doi:10.1007/BF01189090](https://doi.org/10.1007/BF01189090))

On [[homotopy theory]]:

* {#Homotopietheorie} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

On [[equivariant homotopy theory]] and [[equivariant cohomology]]:

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979

* [[Tammo tom Dieck]], _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

On [[representation theory]] of [[compact Lie groups]]:

* [[Tammo tom Dieck]], [[Theodor Bröcker]], *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

* {#tomDieck09} [[Tammo tom Dieck]], _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

On [[algebraic topology]]:

* [[Tammo tom Dieck]], _Topologie_, De Gruyter 2000 ([doi:10.1515/9783110802542](https://doi.org/10.1515/9783110802542))

* {#tomDieck2008} [[Tammo tom Dieck]],  _Algebraic topology_. European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf))

## Related $n$Lab entries

* [[algebraic topology]]

* [[representation theory]]

* [[equivariant cohomotopy]]

* [[Burnside ring]]

* [[equivariant Hopf degree theorem]]

* [[equivariant stable homotopy theory]]

* [[tom Dieck splitting]]

* [[equivariant bordism homology theory]]

* [[CW-approximation]]

category: people

[[!redirects tom Diek]]


Tamás Matolcsi

* [webpage](http://web.cs.elte.hu/~matolcsi/)

## related $n$Lab entries

* [[mathematical physics]]

category: people

[[!redirects Tamas Matolcsi]]


Tamás Szamuely

* [webpage](https://pagine.dm.unipi.it/tamas/)

## Selected writings

On [[Galois cohomology]]:

* {#GilleSzamuely06} [[Philippe Gille]], [[Tamás Szamuely]], *Central Simple Algebras and Galois Cohomology*, Cambridge University Press  2006 ([doi:10.1017/CBO9780511607219](https://doi.org/10.1017/CBO9780511607219), [pdf](http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf))

On [[Galois groups]] and [[fundamental groups]] of [[schemes]]:

* _Galois Groups and Fundamental Groups_, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, 2009.

category:people


tangent

+-- {: .rightHandSide}
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

\begin{imagefromfile}
    "file_name": "DoCarmo-Tangents.jpg",
    "float": "right",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 10,
        "right": 0,
        "left": 15
    },
    "caption": "From [Do Carmo 1976](#DoCarmo76)"
\end{imagefromfile}

**In the [[differential geometry of curves and surfaces]]**, *the tangent* at a point on a [[smooth function|smooth]] [[curve]] in the [[Euclidean plane]] is the straight [[line]] through that point whose [[slope of a line|slope]] is that of the curve at that point. Unless the curve is actually itself "straight" in a vicinity of the point, the tangent is the unique straight line which in an [[open neighbourhood]] of the point on the curve has exactly that one point in common with the curve. In any case, the tangent to a curve may be thought of as the best approximation by a straight line of that curve at the given point.

\begin{imagefromfile}
    "file_name": "Mathnet-TangentFunction.jpg",
    "float": "left",
    "width": 210,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 10,
        "right": 15,
        "left": 0
    },
    "caption": "From [math.net](#Mathnet)"
\end{imagefromfile}

**In [[trigonometry]]** one considers such tangents in the [[Euclidean plane]] for the case that the [[curve]] is a unit [[circle]]. Here the tangent to a point on the circle may also be understood as the unique straight line through that point which makes a [[right angle]] with the "radial" line through that point and the center of the circle.

The [[length]] of the segment of a tangent to a unit circle between its point of tangency and its intersection with any other radial line is, as a [[function]] of the [[angle]] between the two radial lines, known as the *[[tangent function]]*, traditionally denoted "$tan(-)$".

\begin{imagefromfile}
    "file_name": "TangentSpaceToSphere.png",
    "float": "right",
    "width": 220,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 10,
        "right": 0,
        "left": 15
    },
    "caption": "From [Hatcher](vector+bundle#Hatcher)"
\end{imagefromfile}

**More generally in [[differential geometry]]**, for a [[smooth manifold]] [[embedding of smooth manifolds|embedded]] inside a [[Euclidean space]], *a tangent line* through a point on the manifold is a straight line in the ambient Euclidean space through that point which is the "best approximation" of the manifold by a straight line "at that point and in that [[direction of a vector|direction]]".

This is made precise by the notion of *[[tangent vectors]]*, which may be understood as [[infinitesimal]] tangent lines to a smooth manifold and are as such independent of a choice of [[embedding of smooth manifolds|embedding]]. The collection of tangent vectors to a manifold is thus known as its *[[tangent bundle]]*.

More generally but analogously there are higher dimensional *tangent [[planes]]* and tangent [[hypersurfaces]] which infinitesimally are given by *tangent [[multivectors]]*.

## Related concepts

* [[slope of a line]], [[direction of a vector]]

tangents...

* in [[differential geometry]]: _[[tangent vector]]_

* in [[trigonometry]]: _[[tangent function]]_

## References

Most texts on [[differential geometry]] discuss tangents, for instance

* {#DoCarmo76} Manfredo P. Do Carmo, §1-5 in: *Differential Geometry of Curves and Surfaces*, Prentice-Hall (1976) [[pdf](http://www2.ing.unipi.it/griff/files/dC.pdf)]

* Victor A. Toponogov, §1.3 in: *Differential Geometry of Curves and Surfaces --- A Concise Guide*, Springer (2006) [[doi:10.1007/b137116](https://doi.org/10.1007/b137116)]

See also:

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

* {#Mathnet} math.net, *[tangent](https://www.math.net/tangent)*

category: disambiguation

[[!redirects tangents]]


tangent (infinity,1)-category

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## Idea

For $K$ a [[locally presentable (∞,1)-category]] whose objects we think of as [[space]]s of sorts, its **tangent $(\infty,1)$-category**

$$
  (T_{K^{op}})^{op} \to K
$$

is an [[(∞,1)-category]] over $K$, whose objects may be thought of as spaces that are **[[infinitesimal space|infinitesimal]] thickenings** of those of $K$.

More concretely, the tangent $(\infty,1)$-category $T_C \to C$ for $C = K^{op}$ is the fiberwise [[stabilization]] of the [[codomain fibration]] $Func(\Delta[1], C) \to C$.

This generalizes -- as discussed at [[deformation theory]] -- the classical example of the [[bifibration]] [[Mod]] $\to$ [[CRing]] of the category of all [[module]]s over the cateory [[CRing]] of all commutative rings:

the fiber of the tangent $(\infty,1)$-category $T_C$ over an object $A \in C$ may be thought of as the $(\infty,1)$-category of **square-0-extensions** $A \oplus N$ of $A$, for $N$ a [[module]] over $A$. Dually, in $K = C^{op}$ we may think of these as being infinitesimal neighbourhoods of 0-sections of [[vector bundle]]s -- or rather of [[quasicoherent sheaves]] -- over whatever [[space]] $A$ is regarded to be the algebra of functions on.

A remarkable amount of information about the geometry of these spaces/objects in $K$ is encoded in the fiber of the tangent $(\infty,1)$-category over them. Notably the [[left adjoint|left]] [[adjoint (∞,1)-functor]]

$$
  \Omega : C \to T_C
$$

to the domain projection $dom : T_C \to C$ turns out to send each $A$ to its [[cotangent complex]] $\Omega(A)$, to be thought of as the module of [[Kähler differentials]] on the space that $A$ is functions on.

A [[category theory|1-categorical]] approximation to the notion of tangent $(\infty,1)$-category is that of [[tangent category]].

## Definition

Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]].

+-- {: .num_defn #FiberwiseStabilization}
###### Definition
**(fiberwise stabilization)**

For $\mathcal{C}' \to \mathcal{C}$ a [[model structure for quasi-categories|categorical fibration]], [[generalized the|the]] **fiberwise stabilization** $Stab(\mathcal{C}' \to \mathcal{C})$ is -- roughly -- the fibration universal with the property that for each $A \in C$ its [[fiber]] over $A$ is the [[stabilization]] $Stab(\mathcal{C}'_A)$ of the fiber $\mathcal{C}'_A$ over $A$.

=--

This is ([Lurie, section 1.1](#Lurie)) formulated in view of ([Lurie, remark 1.1.8](#Lurie)). There $Stab(\mathcal{C}' \to \mathcal{C})$ is called the _stable envelope_ .

+-- {: .num_defn #TangentCategory}
###### Definition
**(tangent $(\infty,1)$-category)**

[[generalized the|The]] **tangent $(\infty,1)$-category** $T_{\mathcal{C}} \to \mathcal{C}$ is [[generalized the|the]] _fiberwise stabilization_ of the [[codomain fibration]]  $cod : \mathcal{C}^{\Delta^1} \to \mathcal{C}$:

$$
  (T_{\mathcal{C}} \stackrel{p}{\to} \mathcal{C}) := Stab(Func(\Delta[1], \mathcal{C}) \stackrel{cod}{\to} \mathcal{C}  )
  \,.
$$

=--

This is [[Deformation Theory|DT, def 1.1.12]].

For a maybe more explicit definition see below at _[Tangent ∞-topos -- General](#TangentToposGeneral)_.

Explicitly, the tangent $\infty$-category is given as follows.

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

Given a presentable [[(∞,1)-category]] $\mathcal{C}$, the [[(∞,1)-functor]]

$$
  \chi_{cod} \colon \mathcal{C}^{op} \to (\infty,1)Cat
$$

which classifies the [[codomain fibration]] $cod \colon \mathcal{C}^{\Delta^1} \to \mathcal{C}$ under the [[(∞,1)-Grothendieck construction]] factors through the wide non-full inclusion

$$
  (\infty,1)Cat^R \to (\infty,1)Cat
$$

of [[(∞,1)-functors]] which are [[right adjoint|right]] [[adjoint (∞,1)-functors]].  For these the further (now full) inclusion

$$
  i \colon (\infty,1)StabCat^R \hookrightarrow (\infty,1)Cat^R
$$

of the [[stable (∞,1)-categories]] has a [[right adjoint|right]] [[adjoint (∞,1)-functor]]

$$
  (i \dashv Stab)
$$

given by [[stabilization]]. (Note that this is not a functor on all of $(\infty,1)Cat$, where instead the obstructions to functoriality are given by [[Goodwillie calculus]].)

So the classifying map of the codomain fibration factors through this and hence we can postcompose with the [[stabilization]] functor to obtain

$$
  i \circ Stab i \chi_{cod} \colon \mathcal{C}^{op} \to (\infty,1)Cat
  \,.
$$

This sends an object $c \in \mathcal{C}$ to the [[stabilization]] of the [[slice (∞,1)-category]] over $c$:

$$
  Stab \circ \chi_{cod} \colon c \mapsto Stab(\mathcal{C}_{/c})
  \,.
$$

Again by the [[(∞,1)-Grothendieck construction]] this classifies a [[Cartesian fibration]] over $\mathcal{C}$ and this now is the tangent $(\infty,1)$-category projection

$$
  \array{
    T_{\mathcal{C}}
    \\
    \downarrow^{\mathrlap{p}}
    \\
    \mathcal{C}
  }
  \,.
$$

=--

This is the first part of the proof of [[Deformation Theory|DT. prop. 1.1.9]].

## Properties

### Presentability and limits
 {#PresentabilityAndLimits}

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

The tangent $(\infty,1)$-category $T_C$ of the [[locally presentable (∞,1)-category]] $C$ is itself a locally presentable $(\infty,1)$-category.

In particular, it admits all [[(∞,1)-limit]]s and [[(∞,1)-colimit]]s.

=--

This is ([Lurie, prop. 1.1.13](#Lurie)).

Moreover:

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

A diagram in the tangent $(\infty,1)$-category $T_{\mathcal{C}}$ is an [[(∞,1)-limit|(∞,1)-(co-)limit]] precisely if

1. it is a [[relative (infinity,1)-limit|relative (∞,1)-(co-)limit]] with respect to the projection $p \colon T_{\mathcal{C}} \to \mathcal{C}$;

1. its image under this projection is an [[(∞,1)-limit|(∞,1)-(co-)limit]] in $\mathcal{C}$.

=--

([Lurie, HigherAlgebra, prop. 7.3.1.12](#LurieHigherAlgebra))

### Relation to modules
 {#RelationToModules}

We discuss how the tangent $(\infty,1)$-category construction indeed generalizes the equivalence between the [[tangent category]] over [[CRing]] and the category [[Mod]] of all [[modules]] over commutative rings.

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

Let $\mathcal{O}^\otimes$ be a [[coherent (∞,1)-operad]] and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a [[stable (∞,1)-category|stable]] $\mathcal{O}$-[[monoidal (∞,1)-category]].

Let

$$
  A \in Alg_\mathcal{O}(\mathcal{C})
$$

be an $\mathcal{O}$-[[algebra over an (infinity,1)-operad|algebra]] in $\mathcal{C}$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$ is canonically equivalent to $Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C}))$

$$
  Stab( Alg_\mathcal{O}(\mathcal{C})/A)
  \simeq
  Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C}))
  \,.
$$

=--

This is ([Lurie, theorem 1.5.14](#Lurie)).

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

Let $\mathcal{O}^\otimes$ be a [[coherent (∞,1)-operad]] and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a [[presentable (∞,1)-category|presentable]] [[stable (∞,1)-category|stable]] $\mathcal{O}$-[[monoidal (∞,1)-category]]. Then there is a canonical [[equivalence of (∞,1)-categories|equivalence]]

$$
  \phi : T_{Alg_\mathcal{O}(\mathcal{C})}
    \stackrel{\simeq}{\to}
  Alg_\mathcal{O}(\mathcal{C})
   \times_{Func(\mathcal{O}, Alg_\mathcal{O}(\mathcal{C}))}
  Func_\mathcal{O}(\mathcal{O}, Mod^\mathcal{O}(\mathcal{C}))
$$

of presentble [[Cartesian fibration|fibrations]] over $Alg_\mathcal{O}(\mathcal{C})$.

=--

This is ([Lurie, theorem, 1.5.19](#Lurie)).

In words this says that under the given assumptions, objects of $T_{\mathcal{C}}$ may be identified with pairs

$$
  (A, N)
$$

where

* $A$ is an $\mathcal{O}$-[[∞-algebra over an (∞,1)-operad|algebra]] in $\mathcal{C}$;

* $N$ is an $A$-[[∞-module over an ∞-algebra over an (∞,1)-operad|module]].

### Cotangent complex
 {#CotangentComplex}

From its definition as the fiberwise stabilization of the [[codomain fibration]] $cod : Func(\Delta[1], C) \to C$ the tangent $(\infty,1)$-category $p : T_C \to C$ inherits a second $(\infty,1)$-functor to $C$, coming from the _domain_ evaluation

$$
  dom : T_C \to C
  \,.
$$

+-- {: .num_prop #CotangentComplexAdjunction}
###### Definition/Proposition
**(cotangent complex)**

The domain evaluation $dom : T_C \to C$ admits a [[left adjoint|left]] [[adjoint (∞,1)-functor]]

$$
  (\Omega \dashv dom) : T_C \stackrel{\overset{\Omega}{\leftarrow}}{\underset{dom}{\to}}
  C
$$

that is also a [[section]] of $p : T_C \to C$ in that

$$
  (C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C) \simeq Id_C
$$

and which hence exhibits $C$ as a [[retract]] of $T_C$.

This $\Omega$ is the **cotangent complex $(\infty,1)$-functor** : for $A \in C$ the object $\Omega(A)$ is the [[cotangent complex]] of $A$.

=--

This is ([Lurie, def. 1.2.2, remark 1.2.3](#Lurie)).

In more detail this adjunction is the composite

$$
  (\Omega \dashv dom)
  \;\colon\;
  T_C
  \stackrel{\overset{\Sigma^\infty_C}{\leftarrow}}{\underset{\Omega^\infty_{C}}{\to}}
  C^{\Delta^1}
  \stackrel{\overset{const}{\leftarrow}}{\underset{dom}{\to}}
  C
  \colon
  \Omega
  \,,
$$

where $(\Sigma^\infty_C \dashv \Omega^\infty_C)$ is the fiberwise [[stabilization]] relative adjunction, def. \ref{FiberwiseStabilization}.

### Tangent $\infty$-topos of an $\infty$-topos
 {#TangentTopos}

We discuss how the tangent $\infty$-category of an [[(∞,1)-topos]] is itself an [[(∞,1)-topos]] over the tangent $\infty$-category of the original [[base (∞,1)-topos]].

In terms of [[Omega-spectrum]] [[spectrum objects]] this is due to ([Joyal 08](#Joyal08)) joint with [[Georg Biedermann]]. In terms of [[excisive functors]] this is due to observations by [[Georg Biedermann]], [[Charles Rezk]] and [[Jacob Lurie]], see at <a href="n-excisive+(∞,1)-functor#nExcisiveApproximation">$n$-Excisive functor -- Properties -- $n$-Excisive reflection</a<.

#### General
 {#TangentToposGeneral}

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

Let $seq$ be the [[diagram]] category as follows:

$$
  seq
  \coloneqq
  \left\{
    \array{
    && \vdots && \vdots
    \\
    && \downarrow &&
    \\
    \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast
    \\
    &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}}
    \\
    &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast
    \\
    && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}}
    \\
    && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots
    \\
     && && && \downarrow
     \\
     && && && \vdots
    }
  \right\}_{n \in \mathbb{Z}}
  \,.
$$

=--

([Joyal 08, section 35.5](#Joyal08))

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

Given an [[(∞,1)-topos]] $\mathbf{H}$, an [[(∞,1)-functor]]

$$
  X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}
$$

is equivalently

1. a choice of [[object]] $B \in \mathbf{H}$ (the image of $\ast \in seq$]);

1. a sequence of objects $\{X_n\} \in \mathbf{H}_{/B}$ in the [[slice (∞,1)-topos]] over $B$;

1. a sequence of [[morphisms]] $X_n \longrightarrow \Omega_B X_{n+1}$ from $X_n$ into the [[loop space object]] of $X_{n+1}$ in the slice.

This is a [[prespectrum object]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/B}$.

A [[natural transformation]]
$f \;\colon \;X_\bullet \to Y_\bullet$ between two such functors with components

$$
  \left\{
  \array{
    X_n &\stackrel{f_n}{\longrightarrow}& Y_n
    \\
    \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}}
    \\
    B_1 &\stackrel{f_b}{\longrightarrow}& B_2
  }
  \right\}
$$

is equivalently a morphism of base objects $f_b \;\colon\; B_1 \longrightarrow B_2$ in $\mathbf{H}$ together with morphisms $X_n \longrightarrow f_b^\ast Y_n$ into the [[(∞,1)-pullback]] of the components of $Y_\bullet$  along $f_b$.

Therefore the [[(∞,1)-presheaf (∞,1)-topos]]

$$
  \mathbf{H}^{seq}
  \coloneqq
  Func(seq, \mathbf{H})
$$

is the [[codomain fibration]] of $\mathbf{H}$ with "fiberwise pre-stabilization".

A genuine [[spectrum object]] is a [[prespectrum object]] for which all the structure maps $X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}$ are [[equivalence in an (∞,1)-category|equivalences]]. The [[full sub-(∞,1)-category]]

$$
  T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}
$$

on the genuine [[spectrum objects]] is therefore the "fiberwise [[stabilization]]" of the self-indexing, hence the tangent $(\infty,1)$-category.

=--

+-- {: .num_lemma #SpectrificationLemma}
###### Lemma
**(spectrification is left exact reflective)**

The inclusion of [[spectrum objects]] into $\mathbf{H}^{seq}$  is [[left exact (infinity,1)-functor|left]] [[reflective sub-(infinity,1)-category|reflective]], hence it has a [[left adjoint]] [[(∞,1)-functor]] $L$ -- [[spectrification]] --  which preserves [[finite (∞,1)-limits]].

$$
  T \mathbf{H}
  \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow}
  \mathbf{H}^{seq}
  \,.
$$

=--

([Joyal 08, section 35.1](#Joyal08))

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

Forming degreewise [[loop space objects]] constitutes an [[(∞,1)-functor]] $\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}$ and by definition of $seq$ this comes with a [[natural transformation]] out of the identity

$$
  \theta \;\colon\; id \longrightarrow \Omega
  \,.
$$

This in turn is compatible with $\Omega$ in that

$$
  \theta \circ \Omega \simeq \Omega \circ \theta
  \;\colon\;
  \rho \longrightarrow \rho \circ \rho = \rho^2
  \,.
$$

Consider then a sufficiently deep [[transfinite composition]] $\rho^{tf}$. By the [[small object argument]] available in the [[presentable (∞,1)-category]] $\mathbf{H}$ this stabilizes, and hence provides a [[reflective sub-(infinity,1)-category|reflection]] $L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}$.

Since [[transfinite composition]] is a [[filtered (∞,1)-colimit]] and since in an [[(∞,1)-topos]] these commute with [[finite (∞,1)-limits]], it follows that spectrum objects are an [[exact (∞,1)-functor|left exact]] [[reflective sub-(∞,1)-category]].

=--

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

For $\mathbf{H}$ an [[(∞,1)-topos]] over the [[base (∞,1)-topos]] $\infty Grpd$, its [[tangent (∞,1)-category]] $T \mathbf{H}$ is an [[(∞,1)-topos]] over the base $T \infty Grpd$ (and hence in particular also over $\infty Grpd$ itself).

=--

([Joyal 08, section 35.5](#Joyal08))

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

By the the spectrification lemma \ref{SpectrificationLemma}
$T \mathbf{H}$ has a [[geometric embedding]] into the [[(∞,1)-presheaf (∞,1)-topos]] $\mathbf{H}^{seq}$, and this implies that it is an [[(∞,1)-topos]] (by the discussion there).

Moreover, since both [[adjoint (∞,1)-functor]] in the [[global section geometric morphism]] $\mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd$ preserve [[finite (∞,1)-limits]] they preserve [[spectrum objects]] and hence their immediate [[(∞,1)-presheaf]] prolongation immediately restricts to the inclusion of spectrum objects

$$
  \array{
    T \mathbf{H} &\stackrel{\overset{T \Delta}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}}&
   T \infty Grpd
   \\
   \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
   \\
   \mathbf{H}
    &
    \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}
    &
   \infty Grpd
  }
  \,.
$$

=--

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

This statement also follows from the general theory of [[excisive functors]], and in this form it is due to [[Charles Rezk]]. See at _[n-Excisive functor -- Properties -- n-Excisive reflection](n-excisive+%28?%2C1%29-functor#nExcisiveApproximation)_ for the above fact and its generalization to "[[Goodwillie calculus|Goodwillie]] [[jet bundles]]".

=--

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

We may think of the tangent $\infty$-topos $T \mathbf{H}$ as being an [[extension]] of $\mathbf{H}$ by its [[stabilization]] $Stab(\mathbf{H}) \simeq T_\ast \mathbf{H}$:

$$
  \array{
    Stab(\mathbf{H}) &\stackrel{\overset{Stab(\Delta)}{\leftarrow}}{\underset{Stab(\Gamma)}{\longrightarrow}}&
    Sp
    \\
    \downarrow && \downarrow
    \\
    T\mathbf{H} &\stackrel{\overset{T\Delta}{\leftarrow}}{\underset{T\Gamma}{\longrightarrow}}&
    T \infty Grpd
    \\
    \downarrow^{\mathrlap{base}} && \downarrow^{\mathrlap{base}}
    \\
    \mathbf{H} &\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}&
    \infty Grpd
  }
  \,.
$$

Crucial for the internal interpretation in [[homotopy type theory]] is that the [[homotopy types]] in $T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ are [[stable homotopy types]].

=--

#### As the classifying $\infty$-topos for a universal stable object

One may understand $T \mathbf{H}$ as the result of
adjoining to $\mathbf{H}$ a universal "stable object"

$$
  T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ]
  \,.
$$

For details see at _[[excisive (∞,1)-functor]]_ the section _[Characterization via a generic stable object](excisive+%28%E2%88%9E%2C1%29-functor#CharacterizationViaAGenericStableObject)_.

#### Cohesive tangent $\infty$-topos of a cohesive $\infty$-topos

Assume that $\mathbf{H}$ is a [[cohesive (∞,1)-topos]] over [[∞Grpd]],
in that there is an [[adjoint quadruple]]

$$
  \mathbf{H}
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
$$

with $Disc, coDisc$ being [[full and faithful (∞,1)-functors]] and $\Pi$ preserving finite [[(∞,1)-products]].

Since [[(∞,1)-limits]] and [[(∞,1)-colimits]] in an [[(∞,1)-presheaf (∞,1)-topos]] are computed objectwise, this [[adjoint quadruple]] immediately prolongs to $\mathbf{H}^{seq}$

$$
  \mathbf{H}^{seq}
    \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  \infty Grpd^{seq}
  \,.
$$

Moreover, all three [[right adjoints]] preserves the [[(∞,1)-pullbacks]] involved in the characterization of [[spectrum objects]] and hence restrict to $T \mathbf{H}$

$$
  T\mathbf{H}
    \stackrel{}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  T\infty Grpd
  \,.
$$

But then we have a further [[left adjoint]] given as the composite

$$
  T\mathbf{H}
  \hookrightarrow
  \mathbf{H}^{seq}
   \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\underset{Disc^{seq}}{\leftarrow}}
  \infty Grpd^{seq}
   \stackrel{\overset{L}{\longrightarrow}}{\underset{}{\leftarrow}}
  T \infty Grpd
  \,.
$$

Again since $L$ is a [[left exact (∞,1)-functor]] this composite $L \Pi$ preserves finite [[(∞,1)-products]].

So it follows in conclusion that if $\mathbf{H}$ is a [[cohesive (∞,1)-topos]] then its tangent $(\infty,1)$-category $T \mathbf{H}$ is itself a [[cohesive (∞,1)-topos]] over the tangent $(\infty,1)$-category  $T \infty Grpd$ of the [[base (∞,1)-topos]], which is an [[extension]] of the cohesion of the $\infty$-topos $\mathbf{H}$ over $\infty Grpd$ by the cohesion of the stable $\infty$-category $Stab(\mathbf{H})$ over $Stab(\infty Grpd) \simeq Spec$:

$$
  \array{
    Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd)
    \\
    \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}}
  &&
  {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}}
  \\
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
$$

For more on this see at _[[tangent cohesive (∞,1)-topos]]_.

## Examples

### Of $E_\infty$-rings

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

Let $E_\infty$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $A \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$

$$
  Stab(E_\infty/A) \simeq A Mod(Spec)
$$

is equivalent to the category of $A$-[[module spectra]].

=--

([Lurie, cor. 1.5.15](#Lurie)).

### Of an $\infty$-topos
 {#ExamplesTangentOfAnInfinityTopos}

We discuss here aspects of the tangent $\infty$-categories of [[(∞,1)-toposes]].

First consider the [[base (∞,1)-topos]] $\mathbf{H} = $ [[∞Grpd]].

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

For each [[∞-groupoid]]/[[homotopy type]] $X \in \infty Grpd$.
there is a [[natural equivalence|natural]] [[equivalence of (∞,1)-categories]]

$$
  \infty Grpd_{/X}
  \simeq
   Func(X, \infty Grpd)
$$

between the [[slice (∞,1)-category]] of [[∞Grpd]] over $X$ and the [[(∞,1)-functor (∞,1)-category]] of maps $X \to \infty Grpd$.

=--

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

By the [[(∞,1)-Grothendieck construction]].

=--

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

For each [[∞-groupoid]]/[[homotopy type]] $X \in \infty Grpd$.
there is a [[natural equivalence|natural]] [[equivalence of (∞,1)-categories]]

$$
  T_X (\infty Grpd) \simeq Func(X, Spec)
$$

between the fiber of the tangent (∞,1)-category of [[∞Grpd]] over $X$, def. \ref{TangentCategory}, and the [[(∞,1)-category]] of [[parameterized spectra]] over $X$.

=--

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

Applying remark \ref{SliceOfinfinityGroupoidsIsFunctorCategory} in remark \ref{TangentCatByExplicitFiberwiseStabilization} yields that

$$
 T_X(\infty Grpd) \simeq Stab(Func(X,\infty Grpd))
  \,.
$$

The statement then follows with the "[stable Giraud theorem](stable%20%28infinity,1%29-category#StabGiraud)".

=--

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

This means that the tangent $(\infty,1)$-category $T(\infty Grpd)$
is equivalently what in ([Joyal 08, section 30.34](#Joyal08))
is denoted $D(Kan, X)$ in the case that $X = Spec$ is the [[(∞,1)-category of spectra]].

=--

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

The tangent $(\infty,1)$-category $T (\infty Grpd)$ is itself
an [[(∞,1)-topos]].

=--

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

With the above equivalence this is ([Joyal 08, section 35.5, 35.6](#Joyal08) (with [[Georg Biedermann]])).

=--

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

The [[terminal object in an (∞,1)-category|terminal object]] in $T \infty Grpd$ should be the [[zero spectrum]] regarded as a parameterized spectrum over the point

$$
  0 \colon \ast \to Spec
  \,.
$$

=--

From this it follows that

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

The [[global elements]]/[[global sections]] functor (which forms the [[derived hom space|(∞,1)-categorical mapping space]] out of the terminal object)

$$
  \Gamma \coloneqq Hom(\ast, -)
  \;\colon\; T(\infty Grpd) \to \infty Grpd
$$

sends an $X$-[[parameterized spectrum]] to its base homotopy type $X$.

This functor has a [[left adjoint|left]] and [[right adjoint|right]] [[adjoint (∞,1)-functor]] both given by sending $X$ to the [[zero spectrum]] bundle over $X$.

So we have an infinite chain of [[adjoint (∞,1)-functors]]

$$
  (\cdots base \dashv 0 \dashv base \dashv 0 \dashv \cdots)
  \,.
$$

=--

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

The functor $0$ is a [[full and faithful (∞,1)-functor]]

$$
  0 \;\colon\; \infty Grpd \hookrightarrow T \infty Grpd
$$

and so the tangent $(\infty,1)$-category is [[cohesive (∞,1)-topos|cohesive]] over [[∞Grpd]], hence by prop. \ref{TangentOfInfinityGrpdIsTopos} $T(\infty Grpd)$ is a [[cohesive (∞,1)-topos]]:

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
  \;
  \colon
  \;
  T(\infty Grpd)
  \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}}
  \infty Grpd
  \,.
$$

Recalling that here $base = cod \circ \Omega^\infty$, we have one more adjunction, the [[cotangent complex]] adjunction due to prop. \ref{CotangentComplexAdjunction}

$$
  \infty Grpd
  \stackrel{
     \overset{\Omega}{\longrightarrow}
  }
  {\underset{dom\circ \Omega^{\infty}}{\leftarrow}}
  T(\infty Grpd)
  \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}}
  \infty Grpd
  \,.
$$

=--

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

For $\mathbf{H}$ a general [[(∞,1)-topos]] the above discussion goes through essentially verbatim. If $\mathbf{H}$ is itself [[cohesive (∞,1)-topos|cohesive]], then we end up with

$$
  \mathbf{H}
  \stackrel{\overset{\Omega}{\longrightarrow}}{\underset{dom}{\leftarrow}}
  T\mathbf{H}
  \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}}
  \mathbf{H}
  \stackrel{}{\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}}
  \infty Grpd
  \,.
$$

=--

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

For $\mathbf{H}$ a [[locally ∞-connected (∞,1)-topos]] (hence in particular for a [[cohesive (∞,1)-topos]]), there are canonical [[(∞,1)-functors]]

$$
  T \mathbf{H} \stackrel{\overset{T Disc}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}} T \infty Grpd
$$

and such that $T \Gamma$ covers the [[global section]] [[geometric morphism]] $\Gamma \;\colon\; \mathbf{H} \longrightarrow \infty Grpd$ in that it fits into a square

$$
  \array{
    T \mathbf{H}
    &\stackrel{}{\stackrel{}{\stackrel{\overset{T\Gamma}{\longrightarrow}}{}}}&
    T \infty Grpd
    \\
    {}^{\mathllap{0}}\uparrow\downarrow^{base}
    &&
    {}^{\mathllap{0}}\uparrow\downarrow^{base}
    \\
    \mathbf{H} &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{}}}& \infty Grpd
  }
$$

=--

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

By definition of [[stabilization]], $T \mathbf{H}$ is the [[(∞,1)-Grothendieck construction]] of

$$
  X \mapsto
  \underset{\leftarrow}{\lim}
  \left(
    \cdots \stackrel{\Omega}{\to} \mathbf{H}^{\ast/}_{/X} \stackrel{\Omega}{\to} \mathbf{H}^{\ast/}_{/X} \stackrel{\Omega}{\to} \cdots
  \right)
  \,.
$$

Since the [[loop space object]] [[(∞,1)-functor]] $\Omega$ is an [[(∞,1)-limit]] construction and since the [[right adjoint]] [[global section]] functor $\Gamma$ preserves all [[(∞,1)-limits]], there is a homotopy-[[commuting diagram]]

$$
  \array{
    \cdots &\stackrel{\Omega}{\to}& \mathbf{H}^{\ast/}_{/X} &\stackrel{\Omega}{\to}& \mathbf{H}^{\ast/}_{/X} &\stackrel{\Omega}{\to}& \cdots
    \\
    && \downarrow^{\mathrlap{\Gamma}} && \downarrow^{\mathrlap{\Gamma}}
    \\
    \cdots &\stackrel{\Omega}{\to}& \infty Grpd^{\ast/}_{/\Gamma(X)} &\stackrel{\Omega}{\to}& \infty Grpd^{\ast/}_{/\Gamma(X)} &\stackrel{\Omega}{\to}& \cdots
  }
$$

in [[(∞,1)Cat]]. This induces a natural morphism

$$
  Stab(\mathbf{H}_{/X})
  \longrightarrow
  Stab(\infty Grpd_{/\Gamma(X)})
$$

and hence a morphism

$$
  T \mathbf{H}
  \simeq
  \int_{X \in \mathbf{H}} Stab(\mathbf{H}_{/X}) \longrightarrow \int_{X \in \mathbf{H}} Stab(\infty Grpd_{\Gamma(X)})
  \,.
$$

The morphism in question is the postcomposition of this with pullback/restriction of the [[(∞,1)-Grothendieck construction]] along the [[reflective sub-(∞,1)-category|reflective inclusion]] (by assumption on $\mathbf{H}$) $Disc \;\colon\; \infty Grpd \longrightarrow \mathbf{H}$

$$
  T \mathbf{H}
  \simeq
  \int_{X \in \mathbf{H}} Stab(\mathbf{H}_{/X})
  \longrightarrow
  \int_{X \in \mathbf{H}} Stab(\infty Grpd_{/\Gamma(X)})
  \longrightarrow
\int_{S \in \infty Grpd} Stab(\infty Grpd_{/S})
  \simeq
  T \infty Grpd
  \,,
$$

where we used that by reflectivity $\Gamma \circ Disc \simeq id$.

=--

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

When $T \mathbf{H}$ is an $\infty$-topos it should carry another structure $\otimes$ of a [[symmetric monoidal (∞,1)-category]], induced by fiberwise [[smash product]] of [[spectrum objects]]....

=--

## Related concepts

* [[parameterized spectra]]

* [[model structure for parameterized spectra]]

* [[Joyal locus]]

* [[doubly closed monoidal (infinity,1)-category]]

* [[differentiable (∞,1)-category]]

* [[Goodwillie calculus]]

* [[jet (∞,1)-category]]

* [[tangent cohesion]]

* [[deformation theory]]

## References
 {#References}

Discussion of [[model category]] models is in

* {#Schwede97} [[Stefan Schwede]], section 3 of _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 104

The $\infty$-category theoretic definition and study of the notion of _tangent $(\infty,1)$-categories_ is from

* {#Lurie} [[Jacob Lurie]], _[[Deformation Theory]]_

and

* {#LurieHigherAlgebra} [[Jacob Lurie]], section 7.3 of _[[Higher Algebra]]_

The [[(infinity,1)-topos]] structure on tangent $(\infty,1)$-categories (cf. *[[Joyal loci]]*):

* {#Joyal08} [[André Joyal]], §35.5 of *Notes on Logoi* (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf), [[JoyalOnLogoi2008.pdf:file]]]

with expository emphasis in:

* [[Mathieu Anel]], [[André Joyal]], §4.2.3 in: *Topo-logie*, in *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) 155-257 [[doi:10.1017/9781108854429.007](https://doi.org/10.1017/9781108854429.007), [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf)]

Presentation by [[model categories]] is discussed in

* [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], _Tangent categories of algebras over operads_ ([arXiv:1612.02607](https://arxiv.org/abs/1612.02607))

* [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], _The abstract cotangent complex and Quillen cohomology of enriched categories_ ([arXiv:1612.02608](https://arxiv.org/abs/1612.02608))

* [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], _The tangent bundle of a model category_, Theory and Applications of Categories  Vol. 34, 2019, No. 33, pp 1039-1072, [journal web site](http://tac.mta.ca/tac/volumes/34/33/34-33abs.html)

* {#BraunackMayer19} [[Vincent Braunack-Mayer]], *Combinatorial parametrised spectra* (based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]]),
Algebr. Geom. Topol. **21** (2021) 801-891 [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]

Generalization to parameterized objects in any [[stable (∞,1)-category]] is discussed in:

* [[Marc Hoyois]], _Topoi of parametrized objects_, Theory and Applications of Categories, Vol. 34, 2019, No. 9, pp 243-248. ([arXiv:1611.02267](https://arxiv.org/abs/1611.02267), [tac:34-09](http://www.tac.mta.ca/tac/volumes/34/9/34-09abs.html))

A more general notion of a _tangent structure on an $(\infty, 1)$-category_ was attempted to reformulate the [[Goodwillie calculus]] in:

* {#BauerBurkeChing21} [[Kristine Bauer]], [[Matthew Burke]], [[Michael Ching]], _Tangent $\infty$-categories and Goodwillie calculus_ ([arXiv:2101.07819](https://arxiv.org/abs/2101.07819)).

Note that this is an extension to $(\infty, 1)$-categories of the concept of a [[tangent bundle category]]. In the following slides it is claimed that the tangent $(\infty, 1)$-category construction of this page is a particular case of their more general construction (see slide 3):

* [[Michael Ching]], _Tangent ∞-categories and Goodwillie calculus_ ([slides](http://www.math.muni.cz/~bourkej/29CHI/slides.pdf))

Two further examples of tangent structures on an $(\infty,1)$-category, on [[(∞,1)-Topos]] and its [[opposite (infinity,1)-category|opposite]], are given in:

* [[Michael Ching]], _Dual tangent structures for infinity-toposes_, ([arXiv:2101.08805](https://arxiv.org/abs/2101.08805))

[[!redirects tangent (infinity,1)-categories]]

[[!redirects tangent (∞,1)-category]]
[[!redirects tangent (∞,1)-categories]]
[[!redirects tangent ∞-category]]
[[!redirects tangent ∞-categories]]

[[!redirects tangent (∞,1)-topos]]
[[!redirects tangent (∞,1)-toposes]]
[[!redirects tangent (infinity,1)-topos]]
[[!redirects tangent (infinity,1)-toposes]]
[[!redirects tangent ∞-topos]]
[[!redirects tangent ∞-toposes]]