Frölicher spaces and Isbell envelopes

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# Frölicher Spaces and Isbell Envelopes

[[Frölicher spaces]] are examples of [[generalized smooth space|generalised smooth spaces]].  The category of Frölicher spaces is also closely related to the concept of the [[Isbell envelope]] of a category.

+-- {: mynumdef #RCat}
###### Definition
Let $\mathcal{R}$ denote the category with one object and morphism set $C^\infty(\mathbb{R},\mathbb{R})$.
=--

There is a close relationship between Frölicher spaces and the subcategory of $E(\mathcal{R})$, the Isbell envelope of $\mathcal{R}$, of those objects satisfying Isbell duality.

+-- {: .num_prop #IsbellIsFroelicher}
###### Proposition
An object of $E(\mathcal{R})$ that satisfies Isbell duality is a Frölicher space.
=--

+-- {: myproof}
###### Proof
Let $X = (C,F)$ be a generalised $\mathcal{R}$-object satisfying Isbell duality.
From the page about the [[Isbell envelope]], $X$ is concrete.
Let $|X|$ denote the set of constant elements of $C$.
By concreteness, $C$ injects into $Set(\mathbb{R},|X|)$ and $F$ injects into $Set(|X|,\mathbb{R})$.
For clarity, we shall distinguish between an element of $C$ and its image in $Set(\mathbb{R},|X|)$ writing $\alpha$ for the former and $|\alpha|$ for the latter.
We shall do similarly for elements of $F$.

Suppose that $\beta : \mathbb{R} \to |X|$ is such that $|\phi| \circ \beta \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $\phi \in F$.
Then the map $\phi \mapsto |\phi|\circ \beta$ is a natural transformation $F \to C^\infty(\mathbb{R},\mathbb{R})$ (it obvious commutes with the left $C^\infty(\mathbb{R},\mathbb{R})$-actions).
Hence it is an element of $C$.
That is, there is an element $\alpha$ of $C$ such that $\alpha(\phi) = |\phi| \circ \beta$ for all $\phi \in F$.

Let us compare $|\alpha|$ with $\beta$.
Let us put $\delta_t : \mathbb{R} \to \mathbb{R}$ as the constant function at $t$ then for all $\phi \in F$,
\[
\phi \circ (|\alpha|(t)) = \phi \circ \alpha \circ \delta_t = \alpha(\phi) \circ \delta_t = |\phi| \circ \beta \circ \delta_t = |\phi|(\beta(t)) = \phi \circ (\beta(t))
\]
But if $|\alpha|$ and $\beta$ differ, then they differ at some $t \in \mathbb{R}$.
Now for $x \ne y \in |X|$, as $X$ satisfies Isbell duality, there must be some $\phi \in F$ such that $\phi \circ x \ne \phi \circ y$.
Hence $|\alpha| = \beta$ and so $\beta$ is in the image of $C$ in $Set(\mathbb{R}, |X|)$.

For the other part, let $\theta : |X| \to \mathbb{R}$ be such that $\theta \circ |\alpha| \in C^\infty(\mathbb{R},\mathbb{R})$ for all $\alpha \in C$.
Then, exactly as above, we define a natural transformation $C \to C^\infty(\mathbb{R},\mathbb{R})$ by $\alpha \mapsto \theta \circ |\alpha|$.
This corresponds to some $\psi \in F$ and it remains to compare $|\psi|$ with $\theta$.
This is simpler since $x \in |X|$ is an element of $C$ and so $|\psi|(x) = \psi(x) = \theta \circ |x| = \theta(x)$.
=--

However, not all Frölicher spaces can be obtained in this manner.
The simplest example is the following:
\[
(\{0,1\},{\{0,1\}} ^{\mathbb{R}},{\Delta}_{\mathbb{R}})
\]
where this is taken to mean that all curves are smooth and only the constant functionals are smooth.
The problem here is that there are far more curves than the functionals warrant.
Put another way, the functionals cannot distinguish between the points of the set.

All Frölicher spaces satisfy half of the requirements for Isbell duality: the functions are always the natural transformations of the curves.

+-- {: .num_prop #FroFSat}
###### Proposition
The object of $E(\mathcal{R})$ corresponding to a Frölicher space is $F$-saturated.
=--

+-- {: myproof}
###### Proof
Let $(X,C,F)$ be a Frölicher space.
Recall from the page about the [[Isbell envelope]] that $F$-saturated means that $F$ is precisely the set of $C^\infty(\mathbb{R}, \mathbb{R})$-homs $C \to C^\infty(\mathbb{R},\mathbb{R})$.

We know that every element of $F$ gives a map $C \to C^\infty(\mathbb{R}, \mathbb{R})$ which commutes with the right actions.
As $C$ contains the constant functions at the points of $X$, this assignment is injective.

Let $\phi : C \to C^\infty(\mathbb{R},\mathbb{R})$ commute with the right actions.
Define $|\phi| : X \to \mathbb{R}$ by $|\phi|(x) = \phi(\delta_x)$ where $\delta_x \in C$ is the constant function at $x$.
Then for $\alpha \in C$, $|\phi| \circ \alpha (t) = |\phi|(\alpha(t)) = |\phi|(\alpha \circ \delta_t) = \phi(\alpha \circ \delta_t) = \phi(\alpha) \circ \delta_t = \phi(\alpha)(t)$.
Hence $|\phi| \circ \alpha \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $\alpha \in C$ and so $|\phi|$ is in $F$.

Hence $(X,C,F)$ is $F$-saturated.
=--

The other half is more complicated.
Using concreteness one can see that the essence depends on comparing the set $X$ with the set of _constant_ natural transformations $F \to C^\infty(\mathbb{R}, \mathbb{R})$.
That is, those natural transformations $\gamma$ with the property that $\gamma(\phi)$ is a constant function in $C^\infty(\mathbb{R}, \mathbb{R})$ for all $\phi \in F$.

+-- {: .num_lemma #NatTransFSet}
###### Lemma
A constant natural transformation $F \to C^\infty(\mathbb{R}, \mathbb{R})$ is a function $\alpha : F \to \mathbb{R}$ with the property that $\alpha(\phi \circ \psi) = \phi(\alpha(\psi))$.
=--

Let us write $|F|$ for this set.

Now every point of $X$ defines a constant natural transformation via evaluation but the map $X \to |F|$ need be neither injective nor surjective.
However, we can determine conditions on when it is injective or surjective.
Injectivity is related to a fairly simple condition (as indicated by the example earlier).

+-- {: mynumdef #FroHausdorff}
###### Definition
A Frölicher space is said to be _[[Hausdorff space|Hausdorff]]_ if the smooth functions separate points.
=--

+-- {: .num_prop #FroHausCSat}
###### Proposition
A Frölicher space $(X,C,F)$ is Hausdorff if and only if the map $X \to |F|$ is injective.
=--

+-- {: myproof}
###### Proof
The key point here is that an element of $|F|$ is completely determined by its effect on functions in $F$.
Thus $x, y \in X$ are such that they define the same element of $|F|$ if and only if $\phi(x) = \phi(y)$ for all $\phi \in F$.
This means that the smooth functions do not separate $x$ and $y$.
Hence $(X,C,F)$ is Hausdorff if and only if $X \to |F|$ is injective.
=--

It is simple to construct non-Hausdorff Frölicher spaces.
Indeed, the example earlier was one.

Surjectivity is more complicated.
As currently stated, not even very simple Frölicher spaces satisfy the surjectivity condition.

+-- {: .num_lemma #PlaneNonSurj}
###### Lemma
The Frölicher space defined by the usual structure on $\mathbb{R}^2$ does not satisfy the surjectivity condition.
=--

+-- {: myproof}
###### Proof
We need to construct a map $\alpha : F \to C^\infty(\mathbb{R}, \mathbb{R})$ such that $\alpha(\psi \circ \phi) = \psi(\alpha(\phi))$ but which does not correspond to a point in $\mathbb{R}^2$.

A non-zero point $p \in \mathbb{R}^2$ defines an element $\pi_p$ of $F$ by composing orthogonal projection $\mathbb{R}^2 \to \langle p \rangle$ with the map $\langle p \rangle \to \mathbb{R}$ defined by $p \mapsto 1$.
If $p, q \in \mathbb{R}^2$ are not collinear then the only situation in which $\psi \circ \pi_p = \phi \circ \pi_q$ is if $\psi$ and $\phi$ are constant functions.
To see this, let $p'$ be orthogonal to $p$ and $q'$ orthogonal to $q$ be such that the orthogonal projection of $p'$ to the line spanned by $q$ is $q$, and similarly $q'$ maps to $p$.
Then $p'$ and $q'$ span $\mathbb{R}^2$ and
\[
\begin{aligned}
\psi \circ \pi_p(\lambda p' + \mu q') &= \psi \circ \pi_p(\mu q') &&= \psi(\mu) \\
\phi \circ \pi_q(\lambda p' + \mu q') &= \phi \circ \pi_q(\lambda p') &&= \phi(\lambda)
\end{aligned}
\]
As this holds for all $\lambda, \mu$ we see that $\psi$ and $\phi$ are constant functions.

Moreover, the functions $\pi_p$ are "initial" in $F$ in the sense that if $\pi_p = \psi \circ \phi$ for some $\phi \in F$ then $\phi$ is of the form $\theta \circ \pi_p$.
We thus conclude that in defining a map $\alpha : F \to \mathbb{R}$ such that $\alpha(\psi \circ \phi) = \psi(\alpha(\phi))$ we have free choice on the values $\alpha(\pi_p)$.
However, the maps $F \to C^\infty(\mathbb{R}, \mathbb{R})$ which come from evaluation do not have this free choice: their value on $\pi_p$ is completely determined by the values on $\pi_x$ and $\pi_y$.
=--

However, all is not lost.
The set of functions in a Frölicher space has much more structure than simply composition by functions from $C^\infty(\mathbb{R}, \mathbb{R})$.

+-- {: .num_lemma #FroFunAlg}
###### Lemma
The set $F$ of functions in a Frölicher space is a commutative $\mathbb{R}$-algebra.
=--

+-- {: myproof}
###### Proof
Let $(X,C,F)$ be a Frölicher space.
Let $\phi, \psi, \theta \in F$.
Then $\phi \circ \alpha$, $\psi \circ \alpha$, and $\theta \circ \alpha$ lie in $C^\infty(\mathbb{R}, \mathbb{R})$.
Thus as $C^\infty(\mathbb{R}, \mathbb{R})$ is a ring,
\[
\phi \circ \alpha + (\theta \circ \alpha) \cdot (\psi \circ \alpha) = (\phi + \theta \cdot \psi) \circ \alpha
\]
is in $C^\infty(\mathbb{R}, \mathbb{R})$.
As this holds for all $\alpha \in C$, $\phi + \theta \cdot \psi \in F$.
It is commutative because $C^\infty(\mathbb{R}, \mathbb{R})$ is commutative.
Finally we note that there is an obvious ring homomorphism $\mathbb{R} \to F$ sending $\lambda$ to the function $x \mapsto \lambda$.
=--

This suggests that we should consider a Frölicher space not as a pair of functors $\mathcal{R}, \mathcal{R}^{op} \to Set$ but as a pair of functors $\mathcal{R} \to Set$ and $\mathcal{R}^{op} \to Alg$.

There is yet more structure on $F$.
Not only can we compose element of $F$ with elements of $C^\infty(\mathbb{R}, \mathbb{R})$ but if $\phi \in F$ is a particular element then we can compose $\phi$ with an element of $C^\infty(\im \phi, \mathbb{R})$.
This suggests that we ought to enlarge the category $\mathcal{R}$ so that its objects are the power set $\mathcal{P}(\mathbb{R})$.

(An alternative to this extension is to insist that the natural transformations be continuous with respect to a topology on $F$ compatible with the compact-open topology on $C^\infty(\mathbb{R}, \mathbb{R})$.)

With these two augmentations, a constant natural transformation from $F(A) \to C^\infty(-,A)$, with $A \subseteq \mathbb{R}$, defines an algebra homomorphism $F(\mathbb{R}) \to \mathbb{R}$ with the property that $\alpha(\psi) \in \im \psi$ for all $\psi \in F(\mathbb{R})$.

+-- {: .num_lemma #LemPairEval}
###### Lemma
Such a natural transformation, $\alpha$, has the property that for any pair $\phi, \psi \in F(\mathbb{R})$ there is a point $x \in X$ such that $\alpha(\psi) = \psi(x)$ and $\alpha(\phi) = \phi(x)$.
=--

+-- {: myproof}
###### Proof
Consider the function
\[
\theta = (\psi - \alpha(\psi))^2 + (\phi - \alpha(\phi))^2.
\]
As $\alpha$ is an algebra homomorphism, $\alpha(\theta) = 0$.
Since $\alpha(\theta) \in \im \theta$, there is thus some $x \in X$ such that $\theta(x) = 0$.
For this $x$ we therefore have that $\psi(x) = \alpha(\psi)$ and $\phi(x) = \alpha(\phi)$.
=--

This clearly extends to any finite family.
Indeed, from the proof of this result we deduce that the family
\[
\mathcal{A} := \big\{\{x : \phi(x) = \alpha(\phi)\} : \phi \in F(\mathbb{R})\big\}
\]
is directed and thus defines a filter on $X$.
If our natural transformation $\alpha$ is _not_ represented by a point on $X$ then this filter will have empty intersection.

Now consider the subfamily of $F(\mathbb{R})$ consisting of those functions $\phi$ with the property that $\im \phi \subseteq [0,1]$ and $\alpha(\phi) = 1$.
It is clear that if we restrict to this family then we still get all of $\mathcal{A}$.
We can order this family; there are two equivalent (but not isomorphic) orderings:

1. $\phi \leq \psi$ if $\phi(x) \leq \psi(x)$ for all $x \in X$, or
2. $\phi \leq \psi$ if $\phi^{-1}((0,1]) \subseteq \psi^{-1}((0,1])$.

This family is directed (downwards) since $\phi \psi \leq \phi$ (and $\psi$).
In any _reasonable_ topology on $F$ then this net converges to the zero function: on any compact subset of $X$ then we have $(\phi) \to 0$ uniformly.
Therefore if we simply add the condition that our natural transformation $\alpha$ be continuous (something we might have been ready to do anyway) we see that it must be represented by an element of $X$ since otherwise we have $\alpha(0) = \lim \alpha(\phi) = 1 \notin \im 0$.

Thus $X$ is the set of _continuous_ algebra homomorphisms $F \to \mathbb{R}$ and we finally see the relationship between Hausdorff Frölicher spaces as objects in the Isbell envelope of $\mathbf{R}$ satisfying Isbell duality.

[[!redirects Frölicher spaces and Isbell envelopes]]
[[!redirects Frolicher spaces and Isbell envelopes]]
[[!redirects Froelicher spaces and Isbell envelopes]]


Frölicher spectral sequence

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#Contents#
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## Idea

The _Frölicher spectral sequence_ is the [[spectral sequence of a double complex]] of the [[Dolbeault cohomology|Dolbeault]] [[double complex]] $(\Omega^{\bullet,\bullet}, \partial, \bar \partial)$ of a [[complex analytic space]] $X$. It converges to the complex [[de Rham cohomology]] and hence may be thought of as interpolating between that and [[Dolbeault cohomology]]. In particular on a [[compact topological space|compact]] [[Kähler manifold]] $X$ the Frölicher spectral sequence degenerates on the $E_1$-page and exhibits the [[Hodge filtration]] (see [here](Hodge%20structure#HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure)) on its complex de Rham cohomology. Therefore the higher differentials of the Frölicher spectral sequence may be thought of as witnessing [[obstructions]] to $X$ admitting a Kähler structure.

Write $\Omega^\bullet$ for the [[holomorphic de Rham complex]].
The Frölicher spectral sequence has the form

$$
  E_1^{p,1}
  =
  (H^q(X,\Omega^p), \partial)
   \Rightarrow
  H^{p+q}(X, \Omega^\bullet)
$$

with $E_\infty^{p,q} = Gr^p H^{p+q}(X)$ the [[associated graded]] of the [[Hodge filtration]]

$$
  F^p H^k(X,\mathbb{C})
  \coloneqq
  im\left(
    H^k(X, \Omega^{\bullet \geq p})
   \to
    H^k(X, \Omega^\bullet)
  \right)
  \,.
$$

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

If $X$ is [[compact topological space|compact]] and carries the structure of a [[Kähler manifold]], then the Frölicher spectral sequence degenerates at $E_1$.

=--

Traditionally this was proven ([Frölicher 55](#Frolicher55)) by way of  [[harmonic differential forms]].  In ([Deligne-Illusie 87](#DeligneIllusie87)) this is proven instead by reduction to [[positive characteristic]].

## Related concepts

* [[Hodge-de Rham spectral sequence]]

## References

Original articles include

* {#Frolicher55} [[Alfred Frölicher]], _Relations between the cohomology groups of Dolbeault and topological invariants_, Proceedings of the National Academy of Sciences 41: 641–644 (1955)

* {#DeligneIllusie87} [[Pierre Deligne]], [[Luc Illusie]] _Relèvements modulo $p^2$ et decomposition du complexe de de Rham_, Invent. Math. 89 (1987), no. 2, 247-270.

Further developments include

* Laura Bigalke, Sönke Rollenske, _The Frölicher spectral sequence can be arbitrarily non degenerate_, Math. Ann., 341(3), 623--628, 2008 ([arXiv:0709.0481](http://arxiv.org/abs/0709.0481))

A textbook account is in

* {#Voisin02} [[Claire Voisin]], section 8.3.3 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#Voisin} [[Claire Voisin]], _Hodge theory and the topology of compact Kähler and complex projective manifolds_ ([pdf](http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf))

[[!redirects Frolicher spectral sequence]]


Frölicher–Nijenhuis bracket

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## Definition

Suppose $M$ is a [[smooth manifold]].
Recall (e.g. from the article *[[Nijenhuis–Richardson bracket]]*) that any [[differential form|differential $(k+1)$-form]] $K\in\Omega^{k+1}(M,T M)$ valued in the [[tangent bundle]] of $M$ gives rise to a [[graded derivation|graded]] [[derivation]] $\iota_K$ of degree $k$ on the [[de Rham algebra]] of [[differential forms]] on $M$: on [[differential 1-form|1-forms]] we have $\iota_K \omega=\omega\circ K$ and on higher forms we extend using the [[Leibniz rule]].

Concretely,
$$
  \iota_K \omega(X_1,\ldots,X_{k+l})
  \;=\;
  \frac
    {1}
    {(k+1)!(l-1)!}
  \sum_\sigma (-1)^\sigma
  \omega\big(
    K(Y_{\sigma(1)} ,\, \ldots ,\, Y_{\sigma(k+1)})
    ,\,
    Y_{\sigma(k+2)}
    ,\,\ldots
  \big)
  \,,
$$
where the [[sum]] is over all [[permutations]] $\sigma \in$ [[symmetric group|$Sym(k+l)$]] and where $(-1)^\sigma$ denotes the [[signature of a permutation|sign of the permutation]].

[[Cartan's magic formula]]
$$L_X=[\iota_X,d]$$
makes it natural to define the [[Lie derivative]] with respect to $K\in\Omega^k(M,\, T M)$:
$$L_K=[\iota_K,d].$$

The map $L$ defines an [[injective]] [[homomorphism]] of [[graded vector spaces]] from $\Omega(M,TM)$ to graded derivations of $\Omega(M)$.  Its image comprises precisely those derivations $D$ such that $[D,d]=0$ and is closed under the commutator operation.  Transferring the bracket to its source yields the __Frölicher–Nijenhuis bracket__:
$$L_{[K,L]} = [L_K,L_L]$$
for a uniquely defined $[K,L]\in\Omega^{k+l}(M,TM)$.

## Classification of graded derivations of differential forms

Taken together, the [[Frölicher–Nijenhuis bracket]] and the [[Nijenhuis–Richardson bracket]] allows us to fully classify the [[graded derivations]] of the [[de Rham algebra]] of [[differential forms]] on $M$:

A graded dervation $D$ of degree $k$ on $\Omega(M)$ has a unique presentation of the form
$$D=L_K + \iota_L,$$
where $K\in\Omega^k(M,TM)$, $L\in\Omega^{k+1}(M,TM)$.

We have $L=0$ if and only if $[D,d]=0$
and $K=0$ if and only if $D$ vanishes on 0-forms.

Finally, the graded commutator of graded derivations can be expressed in terms of the [[Frölicher–Nijenhuis bracket]] (for $K$) and the [[Nijenhuis–Richardson bracket]] (for $L$):
$$[L_K,L_L]=L_{[K,L]},$$
$$[\iota_K,\iota_L]=\iota_{[K,L]^\wedge},$$
$$[L_K,\iota_L]=\iota_{[K,L]}-(-1)^{k l}L_{\iota_L K},$$
$$[\iota_K,L_L]=L_{\iota_L K}+(-1)^k \iota_{[L,K]}.$$

## Explicit formula

Given a smooth manifold $M$ and differential forms $P\in\Omega^k(M,TM)$, $Q\in\Omega^l(M,TM)$
valued in the [[tangent bundle]] $TM$ of $M$,
their __Frölicher–Nijenhuis bracket__ is a [[differential form]]
$$[P,Q]\in\Omega^{k+l}(M,TM)$$
defined by the formula
$$
  \begin{array}{l}
  [P,Q](X_1,\ldots,X_{k+l})
  \;\coloneqq\;
  \frac{1}{k! l!}
  \sum_\sigma (-1)^\sigma
  \Big(
    &
    \big[
      P(Y_{\sigma(1)},\ldots, Y_{\sigma(k)}),
      \,
      Q(Y_{\sigma(k+1)},\ldots,Y_{\sigma(k+l)})
    \big]
    \\
    &
    -l
    \,
    Q\big(
      [P(Y_{\sigma(1)},\ldots,Y_{\sigma(k)})
      ,\,
      Y_{\sigma(k+1)}]
      ,\,Y_{\sigma(k+2)},
      \ldots
    \big)
    \\
    &
    +(-1)^{k l} k
    \,
    P\big(
      [Q(Y_{\sigma(1)},\ldots,Y_{\sigma(k)})
      ,\,
      Y_{\sigma(k+1)}], Y_{\sigma(k+2)},\ldots\big)
    \\
    &
    + \frac{1}{2} (-1)^{k-1}  k l
    \,
    Q\big(
      P([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)},
      \ldots),Y_{\sigma(k+2)},\ldots
    \big)
    \\
    &
    + \frac{1}{2} (-1)^{(k-1)l} k l
    \,
    P\big(
      Q([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)},\ldots),Y_{\sigma(k+2)}, \ldots
    \big)
  \Big)
  \,,
  \end{array}
$$
and $(-1)^\sigma$ is the [[signature of a permutation|sign of the permutation]] $\sigma$.

## Applications

The [[Nijenhuis tensor]] of an [[almost complex structure]] $J\in\Omega^1(M,TM)$ is $[J,J]$.
The explicit formula yields
$$
  [J,J](X,Y)
  \;=\;
  2\big(
    [J X, J Y] - [X,Y] - J[X, J Y] - J[J X,Y]
  \big).
$$

## Related concepts

* [[Schouten–Nijenhuis bracket]]

* [[Nijenhuis–Richardson bracket]]

* [[Nijenhuis tensor]]

## References

The original definition, with an explicit formula is in Section 6 of

* [[Albert Nijenhuis]], _Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II_, [[Indagationes Mathematicae]] (Proceedings) 58: (1955), 398-403.  [doi][Nijenhuis]

[Nijenhuis]: http://dx.doi.org/10.1016/s1385-7258(55)50055-2

Further development:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part I.  Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359.  [doi:10.1016/s1385-7258(56)50046-7][1] and [doi:10.1016/s1385-7258(56)50047-9][2].

[1]: https://doi.org/10.1016/s1385-7258(56)50046-7
[2]: https://doi.org/10.1016/s1385-7258(56)50047-9

Refinements for [[almost complex structures]]:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part II.  Almost-complex structures_, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429.  [doi:10.1016/s1385-7258(58)50057-2][3] and [doi:10.1016/s1385-7258(58)50058-4][4].

[3]: https://doi.org/10.1016/s1385-7258(58)50057-2
[4]: https://doi.org/10.1016/s1385-7258(58)50058-4

Discussion as a [[natural operation]]:

* {#KolarMichorSlovak} [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], p. 175 of: _[[Natural operations in differential geometry]]_, Springer (1993) [[webpage](https://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3), [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

An article in [[Encyclopedia of Mathematics]]:

* [[Peter W. Michor]], _Frölicher-Nijenhuis bracket_, [HTML](https://encyclopediaofmath.org/wiki/Frölicher-Nijenhuis_bracket), [PDF](https://www.mat.univie.ac.at/~michor/fn-rev.pdf).

A textbook account: Chapter 16 of

* [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008).  [PDF](https://www.mat.univie.ac.at/~michor/dgbook.pdf).

[[!redirects Frölicher–Nijenhuis brackets]]
[[!redirects Frölicher-Nijenhuis brackets]]
[[!redirects Frölicher-Nijenhuis bracket]]


Fubini theorem

* In [[analysis]] the _Fubini theorem_ is a classical [[theorem]] about interchangeability of operations of [[integration]].

* In [[enriched category theory]] the term _Fubini theorem_  is suggestively re-used to refer to a similar interchangeability of [[ends]]/[[coends]], see [there](end#Fubini) for more.

category: disambiguation


Fubini-Study metric

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## Idea

There is a unique (up to a scalar) [[hermitian metric]] on  [[complex projective space]], the _[[Fubini-Study metric]]_.

All analytic subvarieties of a complex projective space are in fact [[algebraic variety|algebraic subvarieties]] and they inherit the [[Kähler manifold structure]] from the projective space.

Examples include [[complex tori]] $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, [[K3-surfaces]], compact [[Calabi-Yau manifolds]], quadrics, products of projective spaces and so on.

## Related concepta

* [[symplectic formulation of quantum mechanics]]

## References

* Wikipedia, _[Fubini-Study metric](https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric)_

[[!redirects Fubini-Study metric]]


Fuchsian equation

## Idea

A **Fuchsian (differential) equation** is a linear homogeneous ordinary differential equation with analytic coefficients in the complex domain whose singular points, including at infinity, are all [[regular singular point]]s. In other words, it is the 1-dimensional case of the theory of meromorphic differential equations with only regular singular points. [[Riemann-Hilbert problem|Hilbert's 21st problem]] is concerned with finding a Fuchsian equation with prescribed points of singularities and prescribed [[monodromies]]. The corresponding [[connection]] is also called Fuchsian.

## References

* M.V. Fedoryuk, _[Fuchsian equation](http://eom.springer.de/F/f041880.htm)_, Springer online Enc. of Math.

*  J.L. Fuchs, J. Reine Angew. Math. __66__, 121–160 (1866); __68__, 354–385 (1868)

* A. Haefliger, _Local theory of meromorphic connections in dimension one (Fuchs theory)_,  pages 129-149 of "Algebraic D-modules", A. Borel, ed.

* P. [[Deligne]], _Équations différentielles à points singuliers réguliers_, Lect. Notes in Math. __163__, Springer-Verlag (1970)

* M. Yoshida, _Fuchsian differential equations_, Aspects of Math. Eil, Vieweg, Braunschweig 1987.

* Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida, _From Gauss to Painlevé, A modern theory of special functions_, 184 pp.

[[!redirects Fuchsian differential equation]]
[[!redirects Fuchsian connection]]

category: analysis


Fuchsian group

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Idea

A [[discrete group|discrete]] [[subgroup]] of the [[real numbers|real]] [[projective linear group]] $PSL(2,\mathbb{R})$.

## Related concepts

* [[Riemann surface]]

* [[Kleinian group]]

## References

* Wikipedia, _[Fuchsian group](http://en.wikipedia.org/wiki/Fuchsian_group)_

[[!redirects Fuchsian groups]]


Fukaya category

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###Context###
#### Symplectic geometry
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[[!include symplectic geometry - contents]]
=--
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
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#### Stable Homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

The _Fukaya category_ of a [[symplectic manifold]] $X$ is an
[[A-∞ category]] having [[Lagrangian submanifold]]s of $X$ as objects.
When two Lagrangian submanifolds $L_1$ and $L_2$ of $X$ meet
transversally, their [[hom-object|hom-space]] in the Fukaya category can be roughly
defined as the free vector space generated by the intersection points
$x\in L_1\cap L_2$; one of the main difficulties in giving a rigorous
definition of the Fukaya category in general relies precisely in the
problem of correctly defining the hom-spaces for nontransversal
intersections. As one could expect, the same difficulty carries on to the
definition of the multilinear operations in the Fukaya category: when
Lagrangians $L_1, L_2,\dots,L_{k+1}$ intersect transversally one has
a clear geometric intuition of the multiplication
$$
m_k\colon Hom(L_1,L_2)\otimes\cdots\otimes Hom(L_k,L_{k+1})\to
Hom(L_1,L_{k+1})
$$
in terms of counting pseudo-holomorphic disks into $X$ whose boundaries
lie on the given Lagrangian submanifolds, but when intersections are
nontransverse, the definition of $m_k$ becomes more evasive.

## In string theory

### A-Model topological string

In [[string theory]], the Fukaya category of a [[symplectic manifold]] $X$ represents the category of [[D-branes]] in the [[A-model]] -- the [[A-branes]] -- with [[target space]] $X$. For [[Landau-Ginzburg models]], the category of [[A-branes]] is described by [[Fukaya-Seidel categories]].

The assignment that sends a [[symplectic manifold]] to its Fukaya category extends to a [[functor]] on a variant of the [[symplectic category]] with [[Lagrangian correspondences]] as morphisms. This is supposed to be the [[FQFT]] incarnation of [[Donaldson theory]]. See at _[[Lagrangian correspondences and category-valued TFT]]_ for more on this and see at _[[homological mirror symmetry]]_.

### Yukawa couplings in intersecting D-brane models

In [[intersecting D-brane models]] Yukawa couplings are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by  [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).

\begin{center}
<img src="https://ncatlab.org/nlab/files/YukawaFukaya.jpg" width="600">
\end{center}

> table grabbed from [Marchesano 03](#Marchesano03)

## Related concepts

* [[Weinstein symplectic category]]

* [[intersecting D-brane model]], [[Yukawa coupling]]

## References

Fukaya categories have first been introduced in

* [[Kenji Fukaya]], _Morse homotopy,
   $A_\infty$-category, and Floer homologies_. Proceedings of GARC Workshop
   on Geometry and Topology '93 (Seoul, 1993). ([link](http://www.math.kyoto-u.ac.jp/~fukaya/mfikki.pdf))

The definitive reference is the book

* Fukaya-Oh-Ohta-Ono, _Lagrangian intersection Floer theory - anomaly and obstruction_

See also

* [[Paul Seidel]], _Fukaya categories and Picard-Lefschetz theory_.

A beginners introduction is given by Denis Auroux, see [link](http://arxiv.org/abs/1301.7056), a text (available as ArXiv  1301.7056), based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Université de Nantes in June 2011.

Discussion of the relation to [[Lagrangian cobordism]] is in

* [[David Nadler]], [[Hiro Tannaka]], _A stable infinity-category of Lagrangian cobordisms_ ([arXiv:1109.4835](http://arxiv.org/abs/1109.4835))
  {#NadlerTannaka11}

Relation to [[Yukawa couplings]] in [[intersecting D-brane models]]:

* {#Marchesano03} [[Fernando Marchesano]], section 7.5 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

[[!redirects Fukaya categories]]


Fukaya-Seidel category

In [[string theory]], Fukaya--Seidel categories are [[categories]] of [[D-brane|D-branes]] for the [[A-models]] of [[Landau-Ginzburg model|Landau–Ginzburg models]].

See [[Fukaya category]].

## References

* [[Paul Seidel]], _Fukaya categories and Picard--Lefschetz theory_.

[[!redirects Fukaya-Seidel category]]
[[!redirects Fukaya-Seidel categories]]
[[!redirects Fukaya–Seidel category]]
[[!redirects Fukaya–Seidel categories]]
[[!redirects Fukaya--Seidel category]]
[[!redirects Fukaya--Seidel categories]]


full > history

< [[full]]
[[!redirects full -- history]]


full and faithful functor

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#### Category theory
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# Contents
* table of contents
{: toc}

## Definition

A **full and faithful functor** is a [[functor]] which is both [[full functor|full]] and [[faithful functor|faithful]].  That is, a [[functor]] $F\colon C \to D$ from a [[category]] $C$ to a category $D$ is called _full and faithful_ if for each pair of [[objects]] $x, y \in C$, the [[function]]
$$F\colon C(x, y) \to D(F(x), F(y))$$
between [[hom sets]] is [[bijective]].  "Full and faithful" is sometimes shortened to "fully faithful" or "ff."   See also [[full subcategory]].

\begin{remark}\label{NeedForNaturality}
It is not sufficient for there simply to exist _some_ isomorphism between $C(x, y)$ and $D(F(x), F(y))$. For instance, consider the category comprising a [[parallel pair]] $f, g : x \rightrightarrows y$ and the [[identity-on-objects]] [[endofunctor]] $F$ sending $f \mapsto f$ and $g \mapsto f$. We have $C(x, y) \cong C(F(x), F(y)) = C(x, y)$, but this functor is not fully faithful.
\end{remark}

## Properties

* Together with [[bo functor|bijective-on-objects functors]], fully faithful  functors form an [[bo-ff factorization system|orthogonal factorization system]] on $Cat$; see [[bo-ff factorization system]]. More invariantly, pair them with [[essentially surjective functor]]s to get a bicategorial factorization system.

* In particular, fully faithful functors are stable under [[pullback]].

* A [[fully faithful functor]] (hence a [[full subcategory]] inclusion) [[reflected limit|reflects]] all [[limits]] and [[colimits]], also all [[isomorphisms]] (is a [[conservative functor]]).  This is evident from inspection of the defining [[universal property]].
 {#FullSubcategoryInclusionsReflectCoLimits}

* Fully faithful functors are closed under [[pushouts]] in [[Cat]].  For ordinary categories this was proven by [Fritch and Latch](#FL); for [[enriched categories]] it is proven in [Stanculescu, Prop. 3.1](#Stanculescu), and for [[(∞,1)-categories]] it is proven in [Simspon, Cor. 16.6.2](#Simpson).

* Fully faithful functors $F : C \to D$ can be characterized as those functors for which the following square is a pullback, where the vertical maps are source and target, and the horizontal maps are induced by $F$
$$
  \array{
    C^{[1]} &\to& D^{[1]}
    \\
    \downarrow && \downarrow
    \\
    C \times C &\to& D \times D
  }
$$

* The bijections exhibiting full faithfulness of $F$ form a natural isomorphism, by functoriality of $F$ and of pre- and postcomposition.

* Let $I L \dashv R$ be an [[adjunction]]. If $I$ is [[fully faithful]], then $L \dashv R I$. In this case, the two adjunctions induce the same [[monad]]. This is Proposition 1.1 of [DFH75](#DFH75). (For a converse, see [[dominant functor]].)

## Generalizations

* For [[(∞,1)-category|(∞,1)-categories]] the corresponding notion of fully faithful functor is described at [[fully faithful (∞,1)-functor]].  This is part of a bigger pattern at work here which is indicated at [[stuff, structure, property]] and [[k-surjective functor]].

* Inside a [[2-category]] there is a "representable" notion of [[ff morphism]].

* There is also a notion of fully faithful functor in [[enriched category theory]], which in general is stronger than being an ff morphism in the 2-category of enriched categories.  But it can be expressed internally in any [[proarrow equipment]].

## Related concepts

[[!include properties of functors -- contents]]

## References

* [[Francis Borceux]], Section 1.5 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* {#DFH75} [[Aristide Deleanu]], [[Armin Frei]], [[Peter Hilton]], _Idempotent triples and completion_, Mathematische Zeitschrift **143** (1975) 91-104 [[doi:10.1007/BF01173053](https://doi.org/10.1007/BF01173053)]

[[!redirects full and faithful functor]]
[[!redirects full and faithful functors]]
[[!redirects fully faithful functor]]
[[!redirects fully faithful functors]]

[[!redirects full and faithful]]
[[!redirects fully faithful]]
[[!redirects fully faithfully]]


full field algebra

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

Where a [[vertex operator algebra]] encodes one chiral half of a [[2d CFT]], a _full field algebra_ combines two vertex operator algebras to produce a genuine 2d CFT defined on the [[complex plane]] (i.e. on [[genus]] 0, notice that to define it on all genera one needs still more information, see at _[[FRS-theorem]]_).

## Related concepts

* [[chiral algebra]]

## References

* {#HuangKong05} [[Yi-Zhi Huang]], [[Liang Kong]], _Full field algebras_, Commun. Math. Phys. **272** (2007) 345-396 [[arXiv:0511328](http://arxiv.org/abs/math/0511328), [doi:10.1007/s00220-007-0224-4](https://doi.org/10.1007/s00220-007-0224-4)]

* {#Kong06} [[Liang Kong]], _Full field algebras, operads and tensor categories_, Adv. Math.213:271-340, 2007 [[arXiv:0603065](http://arxiv.org/abs/math/0603065), [doi:10.1016/j.aim.2006.12.007](https://doi.org/10.1016/j.aim.2006.12.007)]

Survey and review:

* {#Kong11} [[Liang Kong]], *Conformal field theory and a new geometry*, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Proceedings of Symposia in Pure Mathematics **83**, AMS (2011) [[arXiv:1107.3649](http://arxiv.org/abs/1107.3649), [ISBN:978-0-8218-5195-1](http://bookstore.ams.org/PSPUM-83)]

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp 11 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]


full functor

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# Contents
* table of contents
{: toc}

## Definition

A [[functor]] $F\colon C \to D$ from a [[category]] $C$ to a category $D$ is called _full_ if for each pair of [[objects]] $x, y \in C$, the [[function]]
$$F\colon C(x,y) \to D(F(x), F(y))$$
between [[hom sets]] is [[surjective]].

More abstractly, we may say a functor is full if it is [[k-surjective functor|1-surjective]] -- or, in simple terms, 'surjective on morphisms between given objects'.  (Note that a functor may be full without being surjective on morphisms, overall, since $F$ is allowed to not hit morphisms between objects that are not in the image of $F$.)

Fullness is most important for functors which are also [[faithful functor|faithful]], and full and faithful functors are often called _[[fully faithful functor|fully faithful]]_.  For ordinary functors this may sound odd, because there is no real sense in which "full" modifies "faithful."  However, in some contexts (such as for morphisms in a general [[2-category]]), there is a good notion of "full-and-faithful" or "fully faithful," but the right notion of "full" alone is not so clear.  "Fully faithful" is also sometimes abbreviated to "ff"; see also [[bo-ff factorization system]].

A [[subcategory]] is called a _[[full subcategory]]_ if its inclusion functor (which is automatically faithful) is also full, and any full and faithful functor exhibits an equivalence of its domain with a full subcategory of its codomain.

## Examples

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

Given a pair of [[adjoint functors]]

$$
  \mathcal{C}
    \underoverset
     {\underset{R}{\longrightarrow}}
     {\overset{L}{\longleftarrow}}
     { \phantom{AA} \bot \phantom{AA} }
  \mathcal{D}
$$

* the [[right adjoint]] $R$ is a [[full functor]] precisely if the component of the [[unit of an adjunction|counit]] over every object $x$ is a [[split monomorphism]] $L R x \stackrel{}{\to} x $;

* the [[left adjoint]] $L$ is a [[full functor]] precisely if the component of the [[unit of an adjunction|unit]] over every object $x$ is a [[split epimorphism]] $x \to R L x $.

=--

For [[proof]] see [this Prop](adjoint+functor#FullyFaithfulAndInvertibleAdjoints) at _[[adjoint functor]]_.

## Related concepts

[[!include properties of functors -- contents]]

[[!redirects full functor]]
[[!redirects full functors]]

[[!redirects full]]


full image

# Full images of functors #
* tic
{: toc}

## Idea ##

The full image of a [[functor]] $F\colon C \to D$ is a version of its [[image of a functor|image]] that gets its [[objects]] from the functor\'s [[source]] $C$ but its [[morphisms]] from the functor\'s [[target]] $D$. It is the object through which the [[(bo, ff) factorization system|(bo, ff) factorization]] of a functor factors.

You may think of it as (up to [[equivalence of categories|equivalence]]) the [[full subcategory]] of $D$ whose objects lie in the literal image of $F$.

We may call it the $1$-image of the functor, because it reduces (again, up to equivalence) to the ordinary [[image]] for a functor between $0$-[[0-category|categories]].

## Definition ##

Let $C$ and $D$ be [[categories]], and let $F\colon C \to D$ be a [[functor]].  Then the __full image__ of $F$ is the category $\overline{im} F$ with:

*  as objects, the objects of $C$;
*  as morphisms from $x$ to $y$, the morphisms in $D$ from $F(x)$ to $F(y)$.

If $C$ is a [[subcategory]] of $D$, then the full image is the [[full subcategory]] of $D$ whose objects belong to $C$.

The full image should be taken as equipped with a functor to $D$, which acts as $F$ on objects and the identity on morphisms.  This functor is [[fully faithful functor|fully faithful]], so $\overline{im} F$ is always equivalent to a full subcategory of $D$.

From in internal point of view, if $codisc(S)$ is the category with object set $S$ and a unique arrow between any ordered pair of objects (that is, $Mor(codisc(S)) = S\times S$), the full image can be defined as a pullback:
$$
\begin{matrix}
  \overline{im} F& \to & D \\
  \downarrow&&\, \downarrow \\
  codisc(Obj(C))&\underset{codisc(F_0)}{\to} & codisc(Obj(D))
\end{matrix}
$$
in the category [[Cat]]. Here $F_0$ is the object component of $F$ and $codisc(F_0)$ is the obvious functor. This determines $\overline{im} F$ up to canonical isomorphism as a [[strict category]] (or other [[internal category]]).

## Full images of forgetful functors ##

Let $F$ be interpreted as a [[forgetful functor]], so that the objects of $C$ are thought of as objects of $D$ with some [[stuff, structure, property]].  Then the full image of $F$ consists of objects of $D$ with only a property: specifically the property that they are capable of taking the stuff or structure of being an object of $C$.

For example, any [[inhabited set]] is capable of taking the structure of a [[group]] (at least, assuming the [[axiom of choice]]).  So the full image of the forgetful functor from [[Grp]] to [[Set]] is equivalent to the category $Set \setminus \{\empty\}$ of inhabited sets.

[[!redirects full images]]


full sub-2-category

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#### Notions of subcategory
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[[!include notions of subcategory]]
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#Contents#
* table of contents
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## Definition

A [[2-functor]] $F \,\colon\, C \to D$ exhibits the [[2-category]] $C$ as a **full sub-2-category** of $D$ if for all [[object]]s $c_1,c_2 \in C$ the component [[functor]] $F_{c_1, c_2}$ is an [[equivalence of categories]]

$$
  F_{c_1, c_2}
  \;\colon\;
  C(c_1,c_2)
  \xrightarrow{\;\; \simeq \;\;}
  D\big(
    F(c_1), F(c_2)
  \big)
  \,,
$$

hence if $F$ is a **2-fully-faithful 2-functor**.

## Properties

$C$ and $D$ can be considered as (1-)categories by forgetting their 2-morphisms, and $F$ can be considered as a (1-)functor via [[decategorification]]. As a result, every full sub-2-category is also a full subcategory.

If $D$ is a [[(2,1)-category]] a full sub-2-category is equivalently a [[full sub-(∞,1)-category]].

## References

* Math Overflow, "When is a full sub-2-category not a full subcategory?", [web](https://math.stackexchange.com/q/4821592/425071)

## Related concepts

[[!include properties of functors -- contents]]

[[!redirects full sub-2-categories]]

[[!redirects full and faithful 2-functor]]
[[!redirects full and faithful 2-functors]]

[[!redirects fully faithful 2-functor]]
[[!redirects fully faithful 2-functors]]

[[!redirects sub-2-category]]


full subcategory

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## Idea

A [[subcategory]]

$$
  \mathcal{C}
  \overset{\iota}{\hookrightarrow}
  \mathcal{D}
$$

of a given [[category]] $\mathcal{D}$ is meant to be _full_, if it includes "some [[objects]] but _all_ the [[morphisms]] between these objects".

This means _at least_ that $\iota$ is a [[fully faithful functor]]. In fact, that is the most one may demand while respecting the [[principle of equivalence]] of [[category theory]] and hence constitutes an invariant definition of _full subcategory_ (Def. \ref{InvariantDefinition} below).

However, a [[fully faithful functor]] need not be an [[injective function]] on [[objects]] (it is so only up to [[equivalence of categories]]). If one insists on defining a _[[subcategory]]_ inclusion to involve an [[injective function]] on the sets of objects and morphisms, this condition must be added to the condition of the inclusion functor being fully faithful. This leads to a non-invariant definition, discussed [below](#NonInvariantDefinition).

See also the discussion at _[[subcategory]]_ ([here](subcategory#VariantsInAccordWithThePrincipleOfEquivalence)).

## Definition

### Non-invariant definition
 {#NonInvariantDefinition}

If one accepts the notion of [[subcategory]] without any qualification (as discussed there), then:

A [[subcategory]] $S$ of a category $C$ is a **full subcategory** if for any $x$ and $y$ in $S$, every morphism $f : x \to y$ in $C$ is also in $S$ (that is, the inclusion [[functor]] $S \hookrightarrow C$ is [[full functor|full]]).

This inclusion functor is often called a __full embedding__ or a __full inclusion__.

Notice that to specify a full subcategory $S$ of $C$, it is enough to say which [[objects]] belong to $S$.  Then $S$ must consist of all [[morphism]]s whose [[source]] and [[target]] belong to $S$ (and no others).  One speaks of the _full subcategory on_ a given set of objects.

This means that equivalently we can say:

A [[subcategory]]-inclusion functor $F : S \to C$ exhibits $S$ as a **full subcategory** of $C$ precisely if it is a [[full and faithful functor]].
($S$ is the [[essential image]] of $F$).

Up to [[equivalence of categories]], every [[fully faithful functor]] is equivalent to a [[subcategory]]-inclusion in these sense of being an [[injection]] on the set of objects.

### Invariant definition
 {#InvariantDefinitionSection}

Therefore, the definition of _full subcategory_ which respects the [[principle of equivalence]] is simply this:

+-- {: .num_defn #InvariantDefinition}
###### Definition
**(invariant definition)**

A _full subcategory_-inclusion is a [[fully faithful functor]].

=--

## Properties

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

A [[fully faithful functor]] (hence a [[full subcategory]] inclusion) [[reflected limit|reflects]] all [[limits]] and [[colimits]].

=--

This is evident from inspection of the defining [[universal property]].

## Related concepts

* [[reflective subcategory]]

* [[coreflective subcategory]]

* [[fully faithful functor]]

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

* [[internal full subcategory]]

* [[full triangulated subcategory]]

[[!redirects full subcategories]]
[[!redirects full embedding]]
[[!redirects full embeddings]]
[[!redirects full imbedding]]
[[!redirects full imbeddings]]
[[!redirects full inclusion]]
[[!redirects full inclusions]]


full triangulated subcategory

## Definition

Recall that a [[triangulated category]] is given by a 1-category $C$ together with a suspension or translation endofunctor $X\mapsto X[1]$ and a distinguished family of diagrams of the form
$$
X \to Y \to Z\to X[1],
$$
called exact triangles, satisfying some axioms.

A triangulated category $B$ is a __full triangulated subcategory__ of a triangulated category $C$ if it is

* a [[full subcategory]] $B\subset C$ in 1-categorical sense

* closed under translation functor

* triangulated with respect to triangles in $C$

## Localization by a full triangulated subcategory

Given a full triangulated subcategory $B\subset C$, the class $\Sigma$ of morphisms $s$ which fit into a triangle in $C$ of the form
$$
X \overset{s}\to Y \to N\to X[1]
$$
where $N\in Ob(B)$
forms a (left and right) [[calculus of fractions]].
Hence we can form a localization (quotient) category $C[\Sigma^{-1}]$. This category equipped with induced translation functor and the exact triangles which are the images of the exact triangles in $C$ under the localization functor, is triangulated and usually denoted $C/B$. The localization functor is triangulated, sends all objects in $B$ into the $0$ object,  and is universal among all triangulated functors with this property.

## Literature

For example

* [[Dmitri Orlov]], _Triangulated categories of singularities and equivalences between Landau-Ginzburg models_, Sbornik: Mathematics __197__:12 (2006) 1827 [doi](https://doi.org/10.1070/SM2006v197n12ABEH003824)


fully dualizable object

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#### Duality
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=--
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#Contents#
* table of contents
{:toc}

## Idea

A [[dualizable object]] in a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}$ is called **fully dualizable** if the structure maps of the duality unit and counit each themselves have [[adjoints]], which have adjoints, and so on, up to level $(n-1)$.

## Properties

By the [[cobordism hypothesis]]-theorem, symmetric monoidal [[(∞,n)-functor]]s out of the [[(∞,n)-category of cobordisms]] are characterized by their value on the point, which is a fully dualizable object.

## Examples

* In the [[symmetric monoidal category]] [[Vect]] of [[vector space]]s (over some [[field]]), the fully dualizable objects are the finite-[[dimension]]al vector spaces.

* In the [[2-category]] of [[associative algebras]]  with [[bimodules]] between them as morphisms, over a [[commutative ring]], fully dualizable objects are [[separable algebras]] which are [[projective modules]] over the base ring (these conditions are given in [SchommerPries 11, Definition 3.70] (#SchommerPries11) and attributed to Lurie's paper cited below).

* In the [[symmetric monoidal (infinity,n)-category|symmetric monoidal 3-category]] of [[monoidal categories]] and [[bimodule categories]] between them, the fully dualizable objects are (or at least contain) the [[fusion categories]]. ([DSPS 13](#DSPS13)).

## Related concepts

* [[(∞,n)-category with adjoints]]

* [[(∞,n)-category with duals]]

[[!include finite objects -- table]]

## References

The definition appears around claim 2.3.19 of

* [[Jacob Lurie]], *[[On the Classification of Topological Field Theories]]*, Current Developments in Mathematics **2008** (2009) 129-280 [[arXiv:0905.0465](http://arxiv.org/abs/0905.0465), [doi:10.4310/CDM.2008.v2008.n1.a3](https://dx.doi.org/10.4310/CDM.2008.v2008.n1.a3), [euclid:cdm/1254748657](https://projecteuclid.org/euclid.cdm/1254748657)]

Review:

* [[Dominic Culver]], [[Mitchell Faulk]], *Duality Notes*, lecture notes for *West Coast Algebraic Topology Summer School* (2014) [[pdf](https://mathtube.org/sites/default/files/lecture-notes/Duality%20Notes.pdf), [[CulverFaulk-DualityNotes.pdf:file]] [mathtube](https://www.mathtube.org/lecture/notes/duality-notes)]

Detailed discussion in degree 2 and 3:

* {#SchommerPries11} [[Chris Schommer-Pries]], _The Classification of Two-Dimensional Extended Topological Field Theories_ ([arXiv:1112.1000](http://arxiv.org/abs/1112.1000))

* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))

[[!redirects fully dualizable objects]]


fully faithful (infinity,1)-functor

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

# Contents
* table of contents
{: toc}

## Idea

The generalization to the context of [[(∞,1)-category]]-theory of the notion of a [[full and faithful functor]] in ordinary [[category theory]].

## Definition

An [[(∞,1)-functor]] $F : C \to D$ is **full and faithful** if for all objects $x,y \in C$  it induces an equivalence on the [[hom-object in a quasi-category|hom-∞-groupoids]]

$$
  F_{x,y} : Hom_C(x,y) \stackrel{\simeq}{\to}
   Hom_D(F(x), F(y))
  \,.
$$

A full and faithful $(\infty,1)$-functor $F : C \to D$ exhibits $C$ as a [[full sub-(∞,1)-category]] of $D$ and one tends to write

$$
  F : C \hookrightarrow D
$$

to indicate this. This is definition 1.2.10 of [Lurie](#Lurie)

## Properties

+-- {: .num_lemma }
###### Lemma
For an $(\infty,1)$-functor $F : C \to D$, the following are equivalent:

* $F$ is fully faithful

* $Ho(F) : Ho(C) \to Ho(D)$ is a fully faithful functor of (1-)categories and the following square is a pullback in the $(\infty,1)$ category of $\infty$-categories:

$$
  \array{
    C &\to& D
    \\
    \downarrow && \downarrow
    \\
    Ho(C) &\to& Ho(D)
  }
$$

* The following square is a pullback in the $(\infty,1)$ category of $\infty$-categories:

$$
  \array{
    Core(C^{[1]}) &\to& Core(D^{[1]})
    \\
    \downarrow && \downarrow
    \\
    Core(C) \times Core(C) &\to& Core(D) \times Core(D)
  }
$$

* The following square is a pullback in the $(\infty,1)$ category of $\infty$-categories:

$$
  \array{
    C^{[1]} &\to& D^{[1]}
    \\
    \downarrow && \downarrow
    \\
    C \times C &\to& D \times D
  }
$$

=--

+-- {: .proof}
###### Proof
The equivalence of the first two points is basically remark 1.2.11.1 of [Lurie](#Lurie) that a fully faithful functor factors as an equivalence onto a full subcategory.

The equivalence of the first and third points is proposition 3.9.6 of [Cisinski](#Cisinski). The idea is to exploit the fact that the hom-spaces of an $\infty$-category $X$ are the fibers of $X^{[1]} \to X \times X$ (and similarly for the map on cores).

The fourth point implies the third point since Core preserves pullbacks. The second point implies the fourth point by the following arugment.

$$
  \array{
    C^{[1]} &\to& D^{[1]} &\qquad \qquad& C^{[1]} &\to& D^{[1]}
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    Ho(C)^{[1]} &\to& Ho(D)^{[1]} && C \times C &\to& D \times D
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    Ho(C) \times Ho(C) &\to& Ho(D) \times Ho(D)
    &&     Ho(C) \times Ho(C) &\to& Ho(D) \times Ho(D)
  }
$$

Assume the second point. Since $Ho(C) \to Ho(D)$ is a fully faithful functor of 1-categories, the bottom square is a pullback (and a homotopy pullback). The top square is a pullback since $(-)^{[1]}$ preserves limits. Thus the outer square is a pullback.

On the right, we've seen the outer square is a pullback and the bottom square is a pullback since limits commute with limits. Thus, the upper square is a pullback.

=--

Every full and faithful $(\infty,1)$-functor is a [[monomorphism in an (∞,1)-category|monomorphism]] in [[(∞,1)Cat]], but being a full and faithful $(\infty,1)$-functor is a stronger condition. An $(\infty,1)$-functor $F$ is a monomorphism if and only if it induces a [[monomorphism in an (∞,1)-category|monomorphism]] on [[hom-spaces]] and every equivalence $F X \simeq F Y$ is in the effective image of $F$ (see [this MathOverflow question](https://mathoverflow.net/questions/345686/monomorphisms-in-mathcalc-at-infty/345700)).

An [[(∞,1)-functor]] which is both full and faithful as well as an [[essentially surjective (∞,1)-functor]] is an [[equivalence of (∞,1)-categories]].

## Related concepts

[[!include properties of functors -- contents]]

## References

This appears as definition 1.2.10 in

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_
* {#Cisinksi} [[Denis-Charles Cisinski]], "Higher Categories and Homotopical Algebra"

[[!redirects fully faithful (infinity,1)-functor]]
[[!redirects fully faithful (infinity,1)-functors]]
[[!redirects fully faithful (∞,1)-functor]]
[[!redirects fully faithful (∞,1)-functors]]
[[!redirects full and faithful (infinity,1)-functor]]
[[!redirects full and faithful (infinity,1)-functors]]
[[!redirects full and faithful (∞,1)-functor]]
[[!redirects full and faithful (∞,1)-functors]]


fully faithful (infinity,1)-functor > history

< [[fully faithful (infinity,1)-functor]]


fully faithful functor > history

< [[fully faithful functor]]
[[!redirects fully faithful functor -- history]]


fully faithful morphism

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

#Contents#
* table of contents
{:toc}

## Definition

Let $K$ be a [[2-category]].

A [[morphism]] $f:A\to B$ in $K$ is called **(representably) fully-faithful** (or sometimes just **ff**) if for all [[object]]s $X \in K$ , the [[functor]]
$$K(X,A) \to K(X,B)$$
is [[full and faithful functor|full and faithful]].

It is said to be **co-fully faithful** or **corepresentably fully faithful** if for all [[object]]s $X \in K$ , the [[functor]]
$$K(B,X) \to K(A,X)$$
is a full and faithful functor. Note that the shortened name "co-fully faithful" can be misleading, as a functor is corepresentably fully faithful if and only if it is representably fully faithful as a 1-cell in $Cat^{op}$ (rather than $Cat^{co}$.

## Remarks

* Fully faithful morphisms in a 2-category may also be called **1-monic**, and be said to make their source into a **1-subobject** of their target.  See [[subcategory]] for some discussion.

* Fully faithful morphisms are often the right class of a [[factorization system in a 2-category|factorization system]].  The left class in $Cat$ consists of [[essentially surjective functors]]; in a [[regular 2-category]] it consists of the [[eso morphisms]].

* Just as in a 1-category any [[equalizer]] is [[monic]], in a 2-category any [[inverter]] or [[equifier]] is fully faithful.

## Variations

This is not always the "right" notion of fully-faithfulness in a 2-category.  In particular, in [[enriched category theory]] this definition does not recapture the correct notion of enriched fully-faithfulness.  It is possible, however, to characterize $V$-fully-faithful functors 2-categorically; see [[codiscrete cofibration]]. In general, fully faithful morphisms should be defined with respect to a [[proarrow equipment]] (or some other context for [[formal category theory]]): in particular, this recovers $V$-fully-faithfulness. A 1-cell $f : A \to B$ in a proarrow equipment is **fully faithful** if the [[unit of an adjunction|unit of the adjunction]] $\eta \colon 1 \to f^* f_*$ is invertible.

## Examples

In the [[2-category]] [[Cat]] the full and faithful morphisms are precisely the [[full and faithful functor]]s; and the corepresentably fully faithful morphisms are precisely the [[absolutely dense functors]].

## Related pages

* [[subcategory]]
* [[faithful morphism]]
* [[conservative morphism]]
* [[pseudomonic morphism]]
* [[discrete morphism]]
* [[eso morphism]]

[[!redirects ff morphism]]
[[!redirects ff morphisms]]
[[!redirects fully faithful morphism]]
[[!redirects fully faithful morphisms]]
[[!redirects representably fully faithful]]
[[!redirects representably fully faithful morphism]]
[[!redirects representably fully faithful morphisms]]
[[!redirects 1-monic morphism]]
[[!redirects 1-monic morphisms]]
[[!redirects 1-subobject]]
[[!redirects 1-subobjects]]

[[!redirects co-fully faithful]]
[[!redirects co-fully faithful morphism]]
[[!redirects co-fully faithful morphisms]]
[[!redirects co-fully faithful functor]]
[[!redirects co-fully faithful functors]]
[[!redirects corepresentably fully faithful]]
[[!redirects corepresentably fully faithful morphism]]
[[!redirects corepresentably fully faithful morphisms]]


fully formal ETCS

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

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

## Introduction

This page addresses a frequently voiced but easily corrected misconception about [categorial](http://ncatlab.org/nlab/show/foundation+of+mathematics#CFM) approaches to [[foundations of mathematics]], namely that there is a logical circularity in using [[category theory]] to give an axiomatic [[set theory]] (such as [[ETCS]]), since categories themselves are [[sets]] (or collections) with extra [[structure]].

The most straightforward response is a formalist one: category theory and various elaborations like [[ETCS]] are [[first-order theories]], just as [[ZFC]] is. One doesn't need a theory of sets prior to a theory of categories, any more than one needs a theory of sets prior to a theory of sets. Fully formalized, ETCS makes no mention of sets, only a [[signature]] and a list of [[axioms]]; the only background is [[predicate logic|first-order logic]] (which gives the rules for deducing theorems). All of these remarks are perfectly obvious, but the misunderstanding above is common enough that working out the details of full formalization and recording them might be a useful exercise, and that is what we carry out here.

### Choices in presentation

While we're at it, however, we should also point out features of categorical set theory which distinguish it from traditional forms of set theory such as ZFC. Conceptually and philosophically, there is a big difference in that ZFC emphasizes membership between sets as the basic relation (founded on the idea that elements can themselves have elements), whereas in ETCS elements of sets are intuited as essentially featureless (sets being "bags of dots" so to speak, where elements are distinguishable but have no internal structure), and the emphasis is rather on mappings and patterns of mappings where mathematically relevant structure resides.

These conceptual distinctions are discussed more fully elsewhere in the nLab; see for example [[structural set theory]] and pages and references linked therein. Our focus in this page will be on more formalist concerns, but even after disengaging from possibly contentious philosophical issues, experience has shown that logicians not familiar with the categorical background might easily misunderstand what we are doing and why we make the choices we do, and worse, deride those choices from lack of understanding. We thus feel some advance explanation might be of some benefit.

#### Choosing sequent calculus

Even if we restrict focus to formal aspects, there are notable differences between axioms of ETCS and axioms of ZFC in terms of the strength of the underlying logic needed to express them, or in terms of what [internal logical](http://ncatlab.org/nlab/show/internal+logic#kinds_of_internal_logics_in_different_categories_20) assumptions are needed on a background environment in which to interpret them. The relevant distinctions between different types of internal theories (e.g., finite limit theories, coherent theories, etc.) is often conveniently expressed via a [[sequent calculus]]-style presentation of first-order logic that is common currency among categorical logicians, and that is the mode we will adopt here.

When we write a sequent $\varphi \vdash \psi$, it is understood that predicates $\varphi$, $\psi$ have the same type (we could write $\varphi \vdash_X \psi$ to indicate that $\varphi$ and $\psi$ are both of type $X$). Roughly speaking, this means they have the same free variables, although if we see a sequent of the form

$$\varphi(x, y) \vdash \psi(x)$$

where a free variable $y$ does not appear explicitly, then we can match up the types by adding it in freely, replacing $\psi(x)$ by $\psi(x) \wedge [y = y]$. This replacement process corresponds to a "substitution" operation in a [[hyperdoctrine]], a way of doing categorical logic. In fact, a sequent calculus gives rise to a functor

$$Pred: Types^{op} \to Preord$$

taking each type $X$ to the set of predicates of that type, preordered by the entailment relation $\vdash_X$, and where $Pred(f)$ is a substitution map of preorders for each map $f: X \to Y$ between types. Composing with the posetal reflection $Preord \to Pos$ gives a hyperdoctrine

$$Types^{op} \to \Pos$$

which plays the role of a Lindenbaum algebra. Thus the close connection between sequent calculus presentations and the sorts of hyperdoctrine presentations that have proved effective in categorical logic.

In a categorical semantics of a sequent calculus, we have a category of types
$C$ and a hyperdoctrine $Sub: C^{op} \to Pos$ that assigns to each object $c$ its subobject lattice, and a map between hyperdoctrines $Pred \to Sub$. Thus each formula represents a subobject (of an object represented by the formula's type), and a sequent represents an inclusion between subobjects. Logical operations represent categorical operations on subobject lattices in a natural way in the semantics of sequent calculus, e.g., $\exists$ represents taking the image of a subobject along a map, and $\wedge$ represents intersections of subobjects.

#### Choosing the multiplicity of predicates

In addition, there are different choices of signature in which one might present a theory. Our own choices of signature are driven by the desire to have the model theorist's notion of homomorphism match the category theorist's notion of morphism appropriate for a given situation. For example: it is possible to define a monoid as a structure consisting of a associative binary operation $m$ for which a two-sided identity element _exists_ (so the signature of the theory of monoids consists of just a binary function symbol $m$); one can then prove the identity is unique. However, homomorphisms of models of this theory, under this signature, need not preserve the identity; if we want identities to be preserved, then a constant for the identity needs to be explicitly added to the signature. In the same way, one _could_ define a category by recourse to a single ternary relation $c(f, g, h)$ whose intended interpretation is that $f$ and $g$ are composable morphisms such that $f \circ g = h$, and add axioms stipulating existence of identities, but then homomorphisms fail to be functors.

Likewise, if we are presenting a theory of finitely complete categories, then it is not enough to say they are categories for which finite limits exist: if we want the relevant homomorphisms of models of the theory to be functors that preserve finite limits, then appropriate predicates for finite limits (e.g., for pullbacks and terminal objects) need to be adjoined to the signature. And so on. Note that such requirements add (slightly) to the length of the presentation, but we regard that as a superficial consideration -- it is more important to have the signature and axioms reflect the correct notion of homomorphism for each layer of structure.

#### Whether on-the-nose or up to isomorphism?

A final consideration is whether we want a categorical set theory to have _chosen_ products, _chosen_ power objects, etc. or we want such structures to be uniquely given only up to isomorphism. The prevailing tendency among category theorists is to go with the latter; if we want to reflect that practice, then the signature ought to be relational; for example, we might have a ternary predicate $prod(a, b, c)$ which says $c$ is _a_ product of objects $a$ and $b$, rather than have a binary function $a \times b$ which specifies _the_ product.

The present author (Todd Trimble) feels the difference isn't such a big deal; while choosing particular products, pullbacks, power objects, etc. as part of the structure might not seem "natural", such strict or rigid structure is basically harmless. This observation can be formalized in terms of suitable equivalences (in much the same way that while monoidal categories and monoidal functors occur more often in nature, there are strictification 2-functors which allow for them to be formally and equivalently replaced by strict monoidal categories and strict monoidal functors). There are in fact benefits to working with strict structures because they can be expressed in very weak fragments of logic. For example, the theory of strict toposes is a finite limit theory (it is finitary-algebraic over the category of directed graphs), meaning the notion of strict topos object can be internalized within any finitely complete category.

Thus, we will present fragments of the theory ETCS along two parallel tracks, one being with structure considered only up to isomorphism (expressed in a coherent theory), and the other involving specific choices of structure (expressed in a finite limit theory). That will bring us up to the theory of toposes with a natural numbers object. This in itself is a reasonable notion of category of sets; to use words of Lawvere, we can view it as a categorical theory of "variable sets".

Two further axioms are adjoined to the theory of toposes with NNO to get ETCS (bringing one closer to a category of Zermelo sets; to use words of Lawvere, ETCS is a categorical theory of "constant sets"): the axiom of choice and the axiom of well-pointedness. These axioms are non-algebraic axioms that take us outside of finite limit logic: the axiom of choice is manifestly written in coherent logic, and the well-pointedness axiom is not even coherent (although ETCS can, via "Morleyization", be rewritten in coherent logic if excluded middle is assumed in the meta-logic).

### Layering ETCS

ETCS as presented here is built up in stages:

* The theory of categories, as a one-sorted theory. (The usual presentation involves two sorts, objects and  morphisms. Our excuse is that many introductions to logic do not discuss any except one-sorted theories -- thus these are more familiar to logical novices, the very ones subject to the misconceptions recounted above.) In a morphisms-only approach, we have "source" and "target" operations and a ternary predicate $c(f, g, h)$, meaning "$f$ and $g$ are composable and their composite is $h$".

* The theory of categories with finite limits. This builds on the previous theory and adds some new predicates, namely an unary predicate $1$ where $1(x)$ means "$x$ is terminal", and a quaternary predicate $p$ where $p(f, g, h, k)$ means "$f$, $g$, $h$, $k$ are the arrows of a pullback square".

* The theory of toposes with a [[natural numbers object]] (NNO). Here we add in still more predicates to capture the notion of power object up to isomorphism.

* The theory of toposes with NNO that are well-pointed (terminal objects are generators) and satisfy AC (every epi has a section).

The sections below reflect this multi-stage approach.

## Preliminaries

We assume familiarity with the standard notion of a [[algebraic theory|(finitary) first-order theory]] with equality; this includes the standard rules of deduction in the Gentzen [[sequent calculus]] for first-order logic[^note2].

First-order [[theory|theories]] are typically presented in terms of a [[signature]] consisting of function and predicate symbols, together with axioms each of which is a closed formula in the first-order language generated by the signature. That mode of presentation will be followed here. Standard logical conventions are followed; for example, it is common to state axioms as formulas $P(x_1, \ldots, x_n)$ in which some of the variables $x_i$ may appear freely, with the implicit understanding that such variables are bound by unwritten [[universal quantifiers]] at the heads of formulas, to make the formulas closed.

To make the exposition more readable, we add some explanations (in parentheses) which do not belong to the formal theory, but which may aid comprehension. We also employ the standard device of abbreviating formulas, by declaring at various junctures definitions and notational conventions.

In formalizing universal properties, one very helpful purely logical abbreviation is the "exists-unique" quantifier $\exists !$. Namely, if $Q(x)$ is a formula in which a variable $x$ appears freely, then $\exists ! x: Q(x)$ is shorthand for the conjunction of

$$(\exists_x Q(x))$$ \wedge

$$\,$$

$$Q(x) = Q(y) \vdash x = y$$

## The theory of categories

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

We define the [[theory]] $Th(Cat)$ of [[categories]].

The [[signature]] of $Th(Cat)$ consists of

* Unary function symbols $s$, $t$;

* A ternary predicate $c$.

(The [[theory]] of [[categories]] is commonly presented as a two-sorted theory, with an object sort and a morphism sort. Here we instead use [[single-sorted definition of a category|the single-sorted theory]], whose terms are to be interpreted as _morphisms_ of a category. We generally use letters $f, g, h, \ldots$ for variable terms. If $e$ is any term, then the intended interpretation of $s(e)$ is: the identity morphism of the domain of $e$. Similarly, $t(e)$ means the identity morphism of the codomain of $e$. The intended interpretation of $c(f, g, h)$ is that $h = f \circ g$.)

The axioms of $Th(Cat)$ are as follows:

1. $\vdash (s s(f) = s(f) = t s (f)) \wedge (t t(f) = t(f) = s t(f))$.

1. $c(f, g, h) \vdash (s(f) = t(g) \wedge t(f) = t(h) \wedge s(g) = s(h))$

1. $s(f) = t(g) \vdash \exists !_h c(f, g, h)$

1. ([[identity|Identity]] laws) $\vdash c(f, s(f), f) \wedge c(t(f), f, f)$

1. ([[associativity|Associativity]]) $(c(f, g, j) \wedge c(g, h, k)) \vdash (c(j, h, m) \Leftrightarrow c(f, k, m))$

(Notice that an **object-term** may be defined as a term $e$ for which either of the following equivalent equations holds: $e = s(e)$, $e = t(e)$.)

=--

## The theory of finitely complete categories

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

We define the [[theory]] $Th(Lex)$ of [[finitely complete categories]].

(The axioms say there is a [[terminal object]] $1$, and a [[pullback]] for each pair $f, g$ with common target.)

The axioms of $Th(Lex)$ are the axioms of $Th(Cat)$ plus the following:

1. $\exists_1 (1 = s(1)) \wedge (\forall_f (f = s(f)) \Rightarrow
(\exists ! g: (s(g) = f \wedge t(g) = 1))$

1. (Pullbacks exist) Define $p(f, g, h, k)$ to be the conjunction of the formulas
$$(\exists_j c(f, h, j) \wedge c(g, k, j))$$
$$\,$$
$$(\exists_{j'} (c(f, h', j') \wedge c(g, k', j')) \Rightarrow (\exists ! i: (c(h, i, h') \wedge c(k, i, k')))$$
Then
$$t(f) = t(g) \vdash \exists_h \exists_k p(f, g, h, k)$$

=--

## The theory of elementary toposes

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

We define the [[theory]] $Th(Topos)$ of [[elementary toposes]].

The signature of $Th(Topos)$ is obtained by adding to the signature of $Th(Lex)$ unary function symbols $P, \lambda, \rho$.

(The intended interpretation is that $P(f: X \to Y)$ is the [[power object]] of $Y$, and $\langle \lambda(Y), \rho(Y) \rangle: E(Y) \hookrightarrow P(Y) \times Y$ is the elementhood relation, which we rewrite as a jointly [[monomorphism|monic]] [[span]], and consider as a [[universal property|universal]] relation from $P(Y)$ to $Y$.)

The axioms of $Th(Topos)$ are obtained by adjoining to $Th(Lex)$ the following further axioms. In what follows, $M(f, g)$ denote the following formula in $Th(Cat)$:

$$s(f) = s(g) \wedge (\forall_{h, i, j, k} (c(f, h, j) \wedge c(f, i, j) \wedge c(g, h, k) \wedge c(g, i, k)) \Rightarrow h = i)$$

($M(f, g)$ means that the pair $(f, g)$ is a jointly monic span, aka a relation. In other words, that $\langle f, g \rangle \colon s(f) \to t(f) \times t(g)$ is monic, and $(f, g)$ form a relation between $t(f)$ and $t(g)$.)

1. ($P(f: X \to Y) = P(Y)$ is an object that depends only on the codomain of $f$, and $\lambda(f), \rho$ are morphisms that span between $P(Y)$ and $Y$, comprising a local elementhood relation)
$$\vdash (P(f) = P(t(f)) = t(P(f)) = t(\lambda(f))) \wedge (t(\rho(f)) = t(f))$$

1. $\vdash M(\lambda(f), \rho(f))$

1. (Universal property of power objects)
$$M(f, g) \vdash \exists ! h: \exists_k p(h, \lambda(f), g, k) \wedge c(\rho(f), k, f)$$

=--

## The theory ETCS

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

Finally, we define the [[theory]] $Th(ETCS)$: [[ETCS]].

The signature of $Th(ETCS)$ is obtained by adjoining to the signature of $Th(Topos)$ the following

* Nullary function symbols (i.e., constants) $\mathbb{N}, 0, \sigma$.

The axioms of $Th(ETCS)$ are obtained by adjoining to the axioms of $Th(Topos)$ the following further axioms:

1. ([[well-pointed topos|Well-pointedness]]: equality of maps $x \to y$ is detected by points $1 \to x$)
$$(s(f) = s(g) \wedge t(f) = t(g)) \vdash \forall_h (s(h) = 1 \wedge t(h) = s(f) \Rightarrow f \circ h = g \circ h) \Rightarrow f = g$$

1. ([[axiom of choice|Axiom of Choice]]: every epi $f$ admits a section $i$)
$$(\forall_{g, h} (t(f) = s(g) = s(h) \;\wedge\; t(g) = t(h) \;\wedge\; g \circ f = h \circ f) \;\Rightarrow\; g = h) \vdash \exists_i (s(i) = t(f) \;\wedge\; t(i) = s(f) \;\wedge\; f \circ i = t(f))$$

1. ([[natural numbers object|Zero and successor]])
$$\vdash s(0) = 1 \wedge t(0) = \mathbb{N} = s(\sigma) = t(\sigma)$$

1. ([[natural numbers object|Recursion: existence]])
$$s(f) = 1 \wedge t(f) = s(g) = t(g) \vdash \exists_h (s(h) = \mathbb{N} \wedge t(h) = t(f) \wedge f = h \circ 0 \wedge h \circ \sigma = g \circ h)$$

1. (Recursion: uniqueness)
$$(s(f) = \mathbb{N} = s(g) \wedge t(f) = t(g) \wedge f \circ 0 = g \circ 0 \wedge f \circ \sigma = g \circ \sigma) \vdash f = g$$

This completes the formal specification of $Th(ETCS)$.

=--

## Remarks

### On $Th(Cat)$

The theory of categories is usually presented as a two-sorted theory, where the sorts $O$ and $M$ are for objects and morphisms.

Despite the length of the list of axioms, it should be noted that they generally have an extremely simple logical structure. Indeed, with some slight modification, the theory $Th(Topos)$ may be rewritten so that it becomes an [[essentially algebraic theory]], at least in the sense that the models of $Th(Topos)$ are categorially equivalent (as categories = models of $Th(Cat)$) to models of an essentially algebraic theory. That is to say, the notion of "topos" can be [[internalization|internalized]] in any category with finite limits; this is even true of toposes with natural numbers object.

Hence, the only "irremovable" existential clauses (that necessitate passing from essentially algebraic logic to a larger fragment of first-order logic) are well-pointedness and the axiom of choice. The axiom of choice belongs to [[geometric logic]].

This is in marked contrast to ZFC, whose axiom list is superficially shorter (ignoring the fact that axiom schemas are technically infinite lists of axioms!) but whose logical complexity is much greater.

It must be added that this fully formal presentation masks much of the conceptual clarity afforded by the underlying categorical and structural insights that are actually at work, particularly the systematic use of universal mapping properties. To put it more sharply: the traditional syntactic machinery used to present first-order theories is not really a natural or well-adapted 'home' for presenting categorical theories such as ETCS. The natural mode of presentation would use diagrams of arrows from the outset, as formalized say by Freyd's [[Q-sequence]]s, and that is as fully formal as the more traditional mode adopted here, which after all goes back to Frege and hasn't changed in a century.

## References

Over the years various people at

* _Foundations of Mathematics_ [mailing list](http://www.math.psu.edu/simpson/fom/) (around 1998)
{#FOM}

have challenged [[category theory|category theorists]] to write down a fully formal presentation of ETCS.

note1: Note how cumbersome are the traditional "machine-level" logical scripts for expressing this theory, as opposed to the more visual and easily apprehended arrow-theoretic notations (which are just as rigorous). No doubt this was anticipated by those in the FOM debate who wished to portray categorical foundations in the worst possible light: traditional 1-dimensional logical notations tend to make the theory look as clumsy as possible!

note2: We use $\vdash$ to denote an entailment between formulas; axioms of our theory are written in the form $\Gamma \vdash P$ where $\Gamma$ is a finite, possibly empty list of formulas which provide the _context_ in which $P$ is asserted. Obviously one could eschew this symbol entirely, and instead give each axiom in the form of a single formula with appropriately embedded implication symbols $\Rightarrow$. But, as explained in this article, this would obscure the later point that large fragments of ETCS, in fact all axioms except well-pointedness, are manifestly expressible in _coherent logic_.

[[!redirects fully formal ETCS]]
[[!redirects fully formal definition of ETCS]]


fully normal spaces are equivalently paracompact

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## Statement

Let $X$ be a $T_1$ [[topological space]].

Assuming the [[axiom of choice]] then the following are equivalent:

1. $X$ is a [[fully normal topological space]];

2. $X$ is a [[paracompact Hausdorff topological space]].

([Stone 48](#Stone48))

Since [[metric spaces are fully normal]] it follows as a corollary that [[metric spaces are paracompact]]. Accordingly, this statement is now also known as _Stone's theorem_.

Note that without a [[separation axiom]] such as $T_1$, the result fails to hold.  For example, any [[compact topological space]] is paracompact, and any [[fully normal topological space]] is [[normal topological space|normal]], so any non-normal compact space is a paracompact space that's not fully normal.

The [[Hausdorff space|Hausdorff]] condition from statement 2 cannot be dropped, again because there are compact $T_1$ spaces that are not normal, such as an infinite set with the [[cofinite topology]].

## References

* {#Stone48} A. H. Stone, _Paracompactness and product spaces_, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. ([Euclid](http://projecteuclid.org/euclid.bams/1183512390))


fully normal topological space

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## Definition

A [[topological space]] is called _fully normal_ if every [[open cover]] $\{U_i \subset X\}_{i \in I}$ is a __normal cover__, i.e., has a [[refinement]] by an open cover $\{V_j \subset X\}_{j \in J}$ such that every _star_ (eq:Star) in the latter cover is contained in a patch of the former.
Furthermore, the resulting cover $\{V_j\}_{j\in J}$ also admits such a star
refinement, and this process can be continued indefinitely.

Here, for $x \in X$ a point, the **star** of $x$ is the [[union]] of the patches that contain $x$:

\[
  \label{Star}
  star(x,\mathcal{V})
  \;\coloneqq\;
  \left\{
     V_j \in \mathcal{V} \;\vert\; x \in V_J
  \right\}
\]

Normal covers are also known as [[numerable covers]],
since they are precisely the [[open covers]] that admit
a subordinate [[partition of unity]].

## In pointfree topology

Any [[completely regular locale]] has a largest [[uniformity]],
the [[fine uniformity]], which consists of all normal covers.

If a [[completely regular locale]] admits a [[complete uniformity]],
then the [[fine uniformity]] is complete.

A [[locale]] is [[paracompact]] if and only if it admits a [[complete uniformity]].
In this case, we can take the [[fine uniformity]].

## Examples

* [[metric spaces are fully normal]]

## Properties

* [[fully normal spaces are equivalently paracompact]]

## Related conceps

* [[numerable cover]]

* [[paracompact space]]

* [[paracompact locale]]

* [[separation axiom]]

[[!redirects fully normal topological spaces]]

[[!redirects fully normal space]]
[[!redirects fully normal spaces]]