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Idea

The cochain cohomology of the framed knot graph complex (sometimes called the “Wilson graoh complex”) in the bidegree corresponding to strictly trivalent graphs (i.e. Jacobi diagrams) coincides with the space of framed weight systems on round chord diagrams

H ,0deg=0i.e.trivalent(KnotGraphs)(𝒲 c) weightsystemsonroundchorddiagrams(𝒲 t) weightsystemsonJacobidiagrams H^{\bullet, \overset{ \mathclap{ {deg=0} \atop {i.e.\; trivalent} } }{ \overbrace{ 0 }} } \big( KnotGraphs \big) \;\simeq\; \underset{ \color{blue} { {\weight\;systems\;on} \atop {round\;chord\;diagrams} } }{ (\mathcal{W}^c)^\bullet } \;\simeq\; \underset{ \color{blue} { {\weight\;systems\;on} \atop {Jacobi\;diagrams} } }{ (\mathcal{W}^t)^\bullet }

References

  • Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 (arXiv:q-alg/9603010)

  • Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (arXiv:math/9910139)

The Green-Schwarz-type action functional for the supermembrane in 11d, i.e. for the fundamental M2-brane sigma model, turns out to elegantly arise as the supersymmetric trivialization of the C-field superspace 4-cocycle restricted from 11d-superspace to the M2 brane’s super-worldvolume. This is known as the “super-embedding construction”. An analogous construction of the M5-brane’s GS sigma-model action had remained an open problem. In this talk I review a recent result (arxiv.org/abs/1908.00042) which shows that the M5-brane action, too, does arise from a super-embedding construction – but after first passing from plain 11d superspace to super-exceptional superspace.

This is joint work with H. Sati and D. Fiorenza. Slides will be available at ncatlab.org/schreiber/show/Super-exceptional+embedding+construction+of+the+M5-brane

{ΩConf {1,,n}( 3)trajectoriesofcollisionlesspointsobservableΣ H}H (ΩConf {1,,n}( 3),)realcohomologyofloopspaceofconfigurationspaceofnorderedpoints \left\{ \underset{ \color{blue} {trajectories\;of} \atop {\color{blue}collisionless\;points} }{ \underbrace{ \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^3) } } \overset{ \color{blue} observable }{\longrightarrow} \Sigma^\bullet H \mathbb{R} \right\} \;\simeq\; \underset{ { {\color{blue}real\,cohomology\,of} \atop {\color{blue}loop\,space\,of} } \atop { {\color{blue}configuration\,space\,of} \atop {\color{blue}n\,ordered\,points} } }{ \underbrace{ H^\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^3) , \mathbb{R} \big) } }
H (nΩConf {1,,n}( 3))cohomologyofloopspaceofconfigurationspaceofnorderedpoints𝒲 pb horizontalweightsystems \underset{ { {\color{blue}cohomology\,of} \atop {\color{blue}loop\,space\,of} } \atop { {\color{blue}configuration\,space\,of} \atop {\color{blue}n\,ordered\,points} } }{ \underbrace{ H^\bullet \big( \underset{ n \in \mathbb{N} }{\sqcup} \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^3) \big) } } \;\simeq\; \underset{ {\color{blue} horizontal} \atop {\color{blue}weight\,systems} }{ \underbrace{ \mathcal{W}^\bullet_{pb} } }

τ\tau

(14p 1[K3×X 4]) 2+χ 8[K3×X 4] =(14p 1[X 4]12) 2+24χ 4[X 4] =(14p 1[X 4]) 2+12(12p 1[X 4]12)+24χ 4[X 4] =(14p 1[X 4]) 2+12(12p 1[X 4]12)+24c 2[X 4] =(14p 1[X 4]) 2+12(12p 1[X 4]12)12p 1[X 4] \begin{aligned} & - \big( \tfrac{1}{4} p_1[K3 \times X^4] \big)^2 + \chi_8[K3 \times X_4] \\ & = - \big( \tfrac{1}{4} p_1[X^4] - 12 \big)^2 + 24 \cdot \chi_4[X^4] \\ & = - \big( \tfrac{1}{4} p_1[X^4] \big)^2 + 12 \cdot \big( \tfrac{1}{2} p_1[X^4] - 12 \big) + 24 \cdot \chi_4[X^4] \\ & = - \big( \tfrac{1}{4} p_1[X^4] \big)^2 + 12 \cdot \big( \tfrac{1}{2} p_1[X^4] - 12 \big) + 24 \cdot c_2[X^4] \\ & = - \big( \tfrac{1}{4} p_1[X^4] \big)^2 + 12 \cdot \big( \tfrac{1}{2} p_1[X^4] - 12 \big) - 12 \cdot p_1[X^4] \end{aligned}

\infty

\ldots

\subset


M-Theory and Mathematics 2020

Research Institute of NYUAD

New York University, Abu Dhabi

Confirmed speakers:

Katrin BeckerDavid BermanPeter Bouwknegt
Martin CederwallChong-Sun ChuJacques Distler
Michael DuffJosé Figueroa-O'FarrillDomenico Fiorenza
Sergei GukovFei HanChris Hull
Amer IqbalBranislav JurčoNeil Lambert
Varghese MathaiChristian SaemannHenning Samtleben
Hisham SatiUrs SchreiberAshoke Sen
Eric SharpeDmitri SorokinMeng-Chwan Tan

The explanation by the laws of topology of surprising stability effects in dynamical physical systems goes back, at least, to Lord Kelvin, who observed that vortices in what was then thought to be the space-filling ether, would take the shape of knots, which, much like smoke rings in air, would move and vibrate but not break their linked and knotted structure.

(See e.g. Lomonaco 96 http://www.ams.org/books/psapm/051/) Kraph 03 onlinelibrary.wiley.com/doi/abs/10.1034/j.1600-0498.2002.440102.x

While Kelvin and his contemporaries were famously wrong about the existence of a space-filling ether, the idea of knotted topological structures appearing in physical substances and stabilizing their shapes turned out to be spot-on and has seen dramatic confirmation in a multitude of of physical systems, ranging over all scales, from fundamental particle physics, over solid state physics to cosmology.

In fundamental particle physics it remains an open problem to explain the existence of protons and neutrons, hence of atomic nuclei and hence of ordinary matter(!), as stable bound states of fundamental quarks. In the guise of the Mass Gap Problem, this is one of the Millenium Problems whose solution awaits a million dollar award by the Clay Maths Institute. While still open, the best contender for an answer are knotted topological structures in the pion field known as Skyrmions (e.g. Atiyah-Manton 89). Recently Skyrmions have been shown to correctly explain the properties of atomic nuclei all the way from Helium up to Carbon (Lau-Manton 14), a feat that no other model achieves. Moreover, the Skyrmion model of nucleons has been found to be implied by realizations of quantum chromodynaics in string theory (Sakai-Sugimoto 04, 05, Sugimoto 16).

In solid state physics, there have been three phases of matter known since antiquity (solid, fluid, gas) and one more discovered in the 20th century (plasmas) but the 21st century saw the discovery of an entirely new class of phases of matter visible at low tempterature in quantum crystals and known as topological phases of matter. Here the stable knotted structures appear in patterns of the energy bands that electrons in these materials may occupy. Topological phases in particular control the behaviour of graphene, the new ultralight ultrastrong carbon crystal that was discovered in 2004.

Finally, on the very largest scales of the universe, topological defects in te space-filling Higgs field which was famously discovered a few years ago, are thought to play an important role in the explanation of properties of the cosmic microwave background. This includes cosmic monopoles, vertices and cosmic strings.

\ldots

  1. Indented block with citation at end

I would have written you a shorter letter, but I didn’t have enough time.

Other indented bloc

Hit the road, jack, and don’t you come back no more.

  1. Yet another
A(ni=1Sym((H ni(X))[i]))i=22nSym((H 2ni(X))[i1]) A \;\coloneqq\; \left( \underoverset{n}{i = 1}{\bigoplus} Sym \Big( \big( H_{n-i}(X) \big) [i] \Big) \right) \otimes \underoverset{i = 2}{2n}{\bigotimes} Sym \Big( \big( H_{2n - i}(X) \big)[i - 1] \Big)
H n(S 3,)={ | n{0,3} 0 | otherwise H_n\big( S^3, \mathbb{R}\big) \;=\; \left\{ \array{ \mathbb{R} &\vert& n \in \{0,3\} \\ 0 &\vert& \text{otherwise} } \right.
a 3 a_{{}_{3}}
b 43,b 40 b_{{}_{4-3}} , b_{{}_{4-0}}
v 84,v 81 v_{{}_{8-4}} , v_{{}_{8-1}}
s 6=b 43b 43a 3a 3 s_{{}_{6}} \;=\; - b_{{}_{4-3}} \vee b_{{}_{4-3}} \otimes a_{{}_{3}} \wedge a_{{}_{3}}
CE(𝔩Maps(S 3,S 4))(db 4 =0 db 1 =0 dv 4 =2b 4b 1 dv 7 =2b 4b 4) CE \Big( \mathfrak{l} Maps \big( S^3, S^4 \big) \Big) \left( \begin{aligned} d\, b_4 & = 0 \\ d\, b_1 & = 0 \\ d\, v_{{}_{4}} & = 2 \cdot b_{{}_{4}} \wedge b_{{}_{1}} \\ d\, v_{{}_{7}} & = 2 \cdot b_{{}_{4}} \wedge b_{{}_{4}} \end{aligned} \right)
(1)Graphs n(Σ)A graph complex of n-point Feynman diagrams for Chern-Simons theory onΣ qiassign Feynman amplitudes of free CS/AKSZ theory AΩ PA (Conf n(Σ))A de Rham algebra of semi-algebraic differential forms on the FM-compactification of the configuration space of n points inΣ. \underset{ \color{blue} \array{ \phantom{A} \\ \text{graph complex} \\ \text{of n-point Feynman diagrams} \\ \text{for Chern-Simons theory} \\ \text{on} \; \Sigma } }{ Graphs_n(\Sigma) } \underoverset{ \simeq_{\mathrlap{qi}} } { \color{blue} \array{ \text{assign Feynman amplitudes} \\ \text{of free CS/AKSZ theory} \\ \phantom{A} } } { \longrightarrow } \underset{ \color{blue} \array{ \phantom{A} \\ \text{de Rham algebra} \\ \text{of semi-algebraic differential forms} \\ \text{on the FM-compactification} \\ \text{of the configuration space of n points} \\ \text{in}\; \Sigma } }{ \Omega^\bullet_{PA} \big( Conf_n\big( \Sigma \big) \big) } \,.

Σ\Sigma

A̲:C𝒞(A)C\underline{A} \colon C \in \mathcal{C}(A) \mapsto C

{ρ C|CA}\{\rho_C \vert C \subset A \; \}

II'

ρ:A\rho \colon A \to \mathbb{C}

ρ C:CA\rho_C \colon C \subset A \to \mathbb{C}

sfrac12\sfrac{1}{2}

\cdots
{Γ μ,Γ ν}=±2g μν \{\Gamma_{\mu}, \Gamma_\nu\} = \pm 2 g_{\mu \nu}
(Γ 0) 2=±1 (\Gamma_0)^2 = \pm 1
\wedge \;\; \Leftrightarrow
(MM)M α M(MM) 1μ MM μ1 μ MM μM \array{ & (M \otimes M) \otimes M & \stackrel{\alpha}{\longrightarrow} & M \otimes (M \otimes M) & \stackrel{1 \otimes \mu}{\longrightarrow} & M \otimes M \\ & {}_{\mu \otimes 1}\searrow && && \swarrow_{\mu} & \\ && M \otimes M & \stackrel{\mu}{\longrightarrow} M && }
HGH \subset GN GHN_G HW GHW_G H|W GH|\left\vert W_G H\right\vert
1 21 \subset \mathbb{Z}_2 2\mathbb{Z}_2 2\mathbb{Z}_222
2 2\mathbb{Z}_2 \subset \mathbb{Z}_2 2\mathbb{Z}_21111
Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A0?
=
D1?
U(1)
\simeq
Spin(2)
A1×\timesA1
=
D2?
SU(2)×\timesSU(2)
\simeq
Spin(4)
A3
=
D3
cyclic group of order 4
4\mathbb{Z}_4
cyclic group of order 4
2D 2 42 D_2 \simeq \mathbb{Z}_4
SU(4)
\simeq
Spin(6)
D4dihedron on
bigon
Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
D 6D_6
binary dihedral group of order 12
2D 62 D_6
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
D 8D_8
binary dihedral group of order 16
2D 162 D_{16}
SO(12), Spin(12)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group, spin group
SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

|ϕ|=const\left\vert \nabla \phi \right\vert = const

n=n=01234
DI(n)=DI(n)=1Z/2SO(3)G2G3
= Aut(R)= Aut(C)?= Aut(H)= Aut(O)?

\varnothing

\to

Coend calculus

Guest post by Fosco Loregian and Bryce Clarke.

Pastro, Craig, and Ross Street. Doubles for monoidal categories. Theory and applications of categories 21.4 (2008): 61-75.

Coend calculus rules the behaviour of certain universal objects associated to functors of two variables T:C op×CDT : C^{op}\times C \to D; intuitively, end(T)end(T) stands to TT as the limit limF\lim\, F of F:ABF : A \to B stands to FF; the major difference is that end(T)end(T) takes into account the fact that TT eats at the same time two “terms” of the same “type” CC, once covariantly in the second component, and once contravariantly in the first: on arrows f:ccf : c\to c' the functor TT acts in fact as follows:

T(c,c) T(f,c) T(c,f) T(c,c) T(c,c) \begin{array}{ccccc} && T(c',c) && \\ &\overset{T(f,c)}\swarrow && \overset{T(c',f)}\searrow &\\ T(c,c) &&&& T(c',c') \end{array}

Now, a distinguising feature of the objects that depend contra-covariantly on the same variable is that they can be integrated: given a sufficiently regular function f(x)f(x), its dependence from xx can be thought as “covariant” (and defined, say, on a topological vector space VR nV\cong \mathbf{R}^n), whereas the “dxd x” in the symbol

f(x)dx \int f(x)d x

is contravariant (it belongs to a certain dual space of covectors on VV); altogether, the integral can be thought as exhibiting a contra-covariant dependence from xx.

Ends and coends are associated to functors T:C op×CDT : C^{op}\times C \to D in a similar fashion that resembles integration: they are certain objects cT(c,c)\int_c T(c,c) (the end) and cT(c,c)\int^c T(c,c) (the coend), canonically associated to TT, treating cc as a mute variable (meaning that cT(c,c)\int_c T(c,c) and cT(c,c)\int_{c'} T(c',c') are the same object), and satisfying a “commutative rule of integrals” analogous to

f(x,y)dxdy=(f(x,y)dx)dy=(f(x,y)dy)dx \int f(x,y) d x d y = \int \Big(\int f(x,y)d x\Big) d y = \int \Big(\int f(x,y)d y\Big) d x

The end of TT is endowed with projections on the “symmetrized components” cT(c,c)T(c,c)\int_c T(c,c) \to T(c',c'), one for each object cc'; dually, the coend cT(c,c)\int^c T(c,c) is endowed with injections T(c,c) cT(c,c)T(c',c') \to \int^c T(c,c).

All in all, this happens also for colimits, so the two constructions are -at least intuitively- tightly related: this intuition can of course be made more precise. Every universal object built in category theory can be thought either as a subobject of a product (a limit), or as a quotient of a coproduct (a colimit), and co/ends make no exception:

  • An end arises as an “object of invariants” cT\int_c T for the action of TT given by the functions on arrows T xy:hom C op×C(x,y)hom D(Tx,Ty)T_{x y} : \hom_{C^{op}\times C}(x,y) \to \hom_D(T x, T y), and it is defined as the subobject of cCT(c,c)\prod_{c\in C} T(c,c) of those elements invariant under this action.
  • Dually, a coend arises as a “quotient space” of cCT(c,c)\coprod_{c\in C}T(c,c) by a suitable equivalence relation generated by the same functions T xy:hom C op×C(x,y)hom D(Tx,Ty)T_{x y} : \hom_{C^{op}\times C}(x,y) \to \hom_D(T x, T y), i.e. as a space of orbits for the action.

More on this will be explained later on in the discussion.

What is more important now, and quite astounding, is that such contra-covariant actions arise at every corner of category theory: using co/ends it is possible to re-state the Yoneda lemma and the theory of Kan extensions, and to find plenty of applications to algebra, topology, geometry… and functional programming. :-)

Dinaturality

As already said, a functor T:C op×CDT : C^{op}\times C \to D acts on morphisms as

T(c,c) T(f,c) T(c,f) T(c,c) T(c,c) \begin{array}{ccccc} && T(c',c) && \\ &\overset{T(f,c)}\swarrow && \overset{T(c',f)}\searrow &\\ T(c,c) &&&& T(c',c') \end{array}

given two such functors, say P,Q:C op×CDP,Q : C^{op}\times C \to D, we can consider the two diagrams

P(c,c) P(f,c) P(c,f) P(c,c) P(c,c) \begin{array}{ccccc} && P(c',c) && \\ &\overset{P(f,c)}\swarrow && \overset{P(c',f)}\searrow &\\ P(c,c) &&&& P(c',c') \end{array}

and

Q(c,c) Q(c,c) Q(c,f) Q(f,c) Q(c,c) \begin{array}{ccccc} Q(c,c)&&&& Q(c',c')\\ &\underset{Q(c,f)}\searrow&& \underset{Q(f,c')}\swarrow& \\ &&Q(c,c')&& \end{array}

Given two functors F,G:ABF,G : A \to B a natural transformation can be seen as a family of maps that “fill the gap” between F(f)F(f) and G(f)G(f) in a commutative square; in a similar fashion, a dinatural transformation between the above P,QP,Q can be seen as a way to close a certain diagram that testifies a transformation from the arrow action of PP to the arrow action of QQ:

P(c,c) P(f,c) P(c,f) P(c,c) P(c,c) α c α c Q(c,c) Q(c,c) Q(c,f) Q(f,c) Q(c,c) \begin{array}{ccccc} && P(c',c) && \\ &\overset{P(f,c)}\swarrow && \overset{P(c',f)}\searrow &\\ P(c,c) &&&& P(c',c')\\ \!\!\!\!{\color{red} \alpha_c\downarrow} &&&&\,\,\,\,\,\,\,{\color{red}\downarrow\alpha_{c'}} \\ Q(c,c)&&&& Q(c',c')\\ &\underset{Q(c,f)}\searrow&& \underset{Q(f,c')}\swarrow& \\ &&Q(c,c')&& \end{array}

Just as co/limits are defined via suitable transformation to/from a constant, so are co/ends:

  • If Q:C op×CDQ : C^{op}\times C \to D, a wedge for QQ with base xx consists of a dinatural transformation from the constant functor on xDx\in D, i.e. of a family of morphisms α c:xQ(c,c)\alpha_c : x\to Q(c,c) such that for each f:ccf :c\to c' the above hexagon reduces to a commutative square:
    x Q(c,c) Q(c,c) Q(c,c) \begin{array}{ccc} x &\to& Q(c,c)\\ \downarrow && \downarrow\\ Q(c',c') &\to & Q(c,c') \end{array}
  • If P:C op×CDP : C^{op}\times C \to D, a cowedge for PP with tip yy consists of a dinatural transformation to the constant functor on yDy\in D, i.e. of a family of morphisms α c:P(c,c)y\alpha_c : P(c,c) \to y such that for each f:ccf :c\to c' the above hexagon reduces to a commutative square:
    P(c,c) P(c,c) P(c,c) y \begin{array}{ccc} P(c',c) &\to& P(c,c)\\ \downarrow && \downarrow\\ P(c',c') &\to & y \end{array}

    There exists a category Wd(Q)Wd(Q) of wedges, defined with an obvious choice of morphisms between bases; similarly, there is a category of cowedges Cwd(P)Cwd(P).

The end cQ(c,c)\int_c Q(c,c) of QQ is now defined as a terminal object in the category of its wedges; dually, the coend cP(c,c)\int^c P(c,c) of PP is the initial object of the category of its cowedges. Of course, we say “the” end because such an initial object is unique up to unique isomorphism when it exists.

So far, so good. In fact, we didn’t stray much far from plain old category theory, as it is possible to show the following:

Lemma (co/ends are colimits): Given Q:C op×CDQ : C^{op}\times C \to D there exist a category twC\text{tw}\, C and a functor Q τ:twCDQ^\tau : \text{tw}\, C \to D such that

cQ(c,c)limQ τ \int_c Q(c,c) \cong \lim \,Q^\tau

For those who know: the end of QQ is the weighted colimit of Q:C op×CDQ : C^{op}\times C \to D with the hom C\hom_C functor C op×CSetC^{op}\times C \to Set, and thus the category twC\text{tw}\, C is nothing more, nothing less than the category of elements of hom C\hom_C; this allows for a very explicit presentation of twC\text{tw}\, C:

  • objects: the arrows of CC, f:ccf : c \to c';
  • morphisms: the commutative squares
    c d f g c d \begin{array}{ccc} c &\leftarrow& d \\ {}^f\downarrow && \downarrow^g\\ c' &\to& d' \end{array}

Corollary (hom commutes with all ends): As a consequence of the fact that cQ\int_c Q is a limit, there is an isomorphism

hom C(y, cP(c,c)) chom C(y,P(c,c))(ccnt) \hom_C\Big(y, \int_c P(c,c)\Big) \cong \int_c \hom_C(y, P(c,c)) \qquad\qquad{(ccnt)}

natural in yy. Dually,

hom C( cP(c,c),y) chom C(P(c,c),y).(ccnt) \hom_C\Big( \int^c P(c,c), y \Big )\cong \int_c \hom_C(P(c,c), y). \qquad\qquad{(ccnt)}

natural in yy.

(of course, a coend is just an end in the opposite category!)

But why are co/ends denoted as integrals? The notation dates back to Yoneda,

Yoneda, Nobuo. “On Ext and exact sequences.” J. Fac. Sci. Univ. Tokyo Sect. I 8.507-576 (1960): 1960.

(in particular, see §4 but beware that the notation is reversed; a coend is denoted a\int_a and an end a *\int_a^\ast!) and it is essentially motivated by the fact that an end behaves like an integral:

Theorem (Fubini rule): Let P:C op×C×D op×DEP : C^{op}\times C \times D^{op}\times D \to E be a functor; then

c( dP(c,c,d,d)) (c,d)P(c,d,c,d) d( cP(c,c,d,d))(Fub) \int^c\left(\int^d P(c,c,d,d)\right) \cong \int^{(c,d)}P(c,d,c,d) \cong \int^d\left(\int^c P(c,c,d,d)\right) \qquad\qquad{(Fub)}

in the sense that if one of the three objects exists, so do the other two, and they are all canonically isomorphic (the category C op×C×D op×DC^{op}\times C \times D^\text{op}\times D is of course equal to (C×D) op×(C×D)(C\times D)^\text{op}\times( C \times D)). Similarly, there is such a rule for ends.

Thus, in category theory integration with respect to a variable is a process that can happen in whatever order we desire: given a permutation σ\sigma of the set {1,,n}\{1,\dots,n\}, whenever the integral

c σ1 c σ2 c σnP(c σ1,c σ1,c σ2,c σ2,,c σn,c σn) \int^{c_{\sigma 1}}\int^{c_{\sigma 2}}\cdots \int^{c_{\sigma n}} P(c_{\sigma 1}, c_{\sigma 1}, c_{\sigma 2}, c_{\sigma 2}, \dots, c_{\sigma n}, c_{\sigma n})

exists, then so does

(c 1,,c n)P(c 1,c 1,c 2,c 2,,c n,c n) \int^{(c_1,\dots, c_n)} P(c_1, c_1, c_2, c_2, \dots, c_n, c_n)

Proof. The slickest proof I know for this goes as follows: assume all coends exist; then, sending a functor P:C op×CDP : C^{op}\times C \to D to its coend is a functor C:[C op×C,D]D\int^C : [C^{op}\times C, D] \to D, and it is easy to see that it is a left adjoint (for those who know, C\int^C is a particular kind of weighted colimit, and every such weighted colimit admits a right adjoint expressed in terms of the weight: but now it’s easy to prove that these right adjoints commute, thus yielding the Fubini rule by uniqueness of adjoints).

The building blocks of co/end calculus

Here we explore how co/ends allow to rediscover category theory from scratch.

Natural transformations

Theorem (Natural transformations as an end) Let F,G:CDF,G : C \to D be two functors; then, there is an isomorphism

Nat(F,G) chom D(Fc,Gc)(nat) Nat(F,G) \cong \int_c \hom_D(F c,G c) \qquad\qquad{(nat)}

Proof. There is a natural choice for a wedge ω:Nat(F,G)hom(Fc,Gc)\omega : Nat(F,G) \to \hom(F c,G c), that sends a natural transformation to its cc-component; it remains to show that this is indeed a terminal wedge. Given another wedge α:Ahom(Fc,Gc)\alpha : A \to \hom(F c, G c), the wedge condition translates into the equation

A α ac hom(Fc,Gc) α ac Gf hom(Fc,Gc) Ff hom(Fc,Gc) \begin{array}{ccc} A &\overset{\alpha_{a c}}\to& \hom(F c,G c) \\ {}_{\alpha_{a c'}}\downarrow && \downarrow_{G f \circ -} \\ \hom(F c', G c') &\underset{- \circ F f}\to & \hom(F c, G c') \end{array}

valid for every aAa\in A; but this is only a convoluted way to say that for every aAa\in A the family

{α a,c:FcGc} \{\alpha_{a,c} : F c \to G c\}

is a natural transformation.

Two important remarks:

  1. In an additive setting, the wedge condition for α\alpha can be easily translated into the fact that natural transformations appear form the kernel of a certain map; the intuition that naturality is a cocycle condition is more or less what led Yoneda to study ends and coends in homological algebra.

  2. Even in a non-additive setting, one can easily see that a natural transformation α:FG\alpha : F \Rightarrow G is a map that equalizes the action of F,GF,G on arrows; this means that the following diagram

    Nat(F,G) cChom(Fc,Gc)Ff *Gf * cchom(Fc,Gc) Nat(F,G) \to \prod_{c\in C} \hom(F c, G c) \underset{G f_\ast}{\overset{F f^\ast}\rightrightarrows} \prod_{c\to c'} \hom(F c, G c')

    is an equalizer; there is nothing special here, as for every functor T:C op×CDT : C^{op}\times C \to D there is a similar equalizer diagram

    cT(c,c) cCT(c,c)Tf *Tf * ccT(c,c) \int_c T(c,c) \to \prod_{c\in C} T(c,c) \underset{T f_\ast}{\overset{T f^\ast}\rightrightarrows} \prod_{c\to c'} T(c, c')

Here‘s a discussion on what is the coend of the hom functor; I claim that the following object represents the coend of hom(F,G)\hom(F-,G-), perhaps you know where the same object appears under a different name, and where it is used for some purpose? I find this particularly intriguing in the case of a monoid MM regarded as single-object category:

  • the end of hom M\hom_M is the center of the monoid, i.e. the subset {mMmx=xmxM}\{m\in M\mid mx=xm \forall x\in M\};
  • the coend of hom M\hom_M corresponds to something like the π 0\pi_0 of the monoid.

I didn’t expect these construction to be dual, and yet they are!

Claim (but also: exercise for the reader). Let CC be a small category. The coend

chom C(c,c) \int^c \hom_C(c,c)

is the set of connected components of the “endo-comma” category whose objects are endomorphisms of CC, and whose morphisms (u:xx)(v:yy)(u : x \to x) \to (v : y \to y) are the f:xyf : x \to y such that fu=vff u = v f. More generally, if F,G:CDF,G: C \to D are functors there is an isomorphism

chom(Fc,Gc)π 0((F/G) end) \int^c \hom(F c, G c) \cong \pi_0((F/G)_{end})

where the endomorphism comma is defined similarly.

The Yoneda lemma and Kan extensions

On the first day He created the Yoneda lemma, and He saw that it was good:

Theorem (The ninja Yoneda lemma) Let F:C opSetF : C^{op} \to Set; then for every object aCa\in C,

Fa cFc×hom C(a,c) cSet(hom C(c,a),Fc) F a\cong \int^c F c \times \hom_C(a,c) \cong \int_c \Set(\hom_C(c,a), F c)

Proof. The set cSet(hom C(c,a),Fc)\int_c \Set(\hom_C(c,a), F c) is the set of natural transformations from hom(,c)\hom(-,c) to FF, and thus the non-ninja Yoneda lemma yields an isomorphism between this set and FaF a. Dually, call FyF\otimes y the functor

a xFx×hom C(a,x) a\mapsto \int^x F x\times \hom_C(a,x)

Then, we have

Nat(Fy,G) aSet(Fx×hom C(a,x),Ga) a xSet(Fx×hom C(a,x),Ga) a xSet(Fx,[hom C(a,x),Ga]) xSet(Fx, a[hom C(a,x),Ga]) xSet(Fx,Nat(y(x),G)) Nat(F,G) \begin{array}{rl} Nat(F\otimes y, G) &\cong \int_a Set(F x\times \hom_C(a,x), G a)\\ &\displaystyle\cong\int_a\int_x Set(F x\times \hom_C(a,x), G a)\\ &\displaystyle\cong\int_a\int_x Set(F x, [\hom_C(a,x), G a])\\ &\displaystyle\cong\int_x Set(F x, \int_a[\hom_C(a,x), G a])\\ &\displaystyle\cong\int_x Set(F x, Nat(y(x), G))\\ &\displaystyle\cong Nat(F, G) \end{array}

Each of these steps can be easily justified in light of what we already proved:

  • the fact that the hom functor preserves ends;
  • the Fubini rule for ends;
  • the fact that the set of natural transformations between two functors is an end.

This natural deduction-style kind of proof is half-jokingly called “coend-fu” ((端楔術 : literally “the art [of handling] terminal wedges”) in my note, soon-ish a book, on coends.

Incidentally, the isomorphism Fa cFc×hom C(a,c)F a\cong \int^c F c \times \hom_C(a,c) is precisely the sense in which “every presheaf is a colimit of representable functors”; the colimiting diagram has domain the category of elements of FF, and the natural functor Σ:El(F)C[C op,Set]\Sigma : El(F) \to C \to [C^{op},Set] has colimit FF.

On the second day, He created Kan extensions, and category theory was complete.

Theorem (Kan extensions are co/ends) Let AGBFCA \xleftarrow{G} B\xrightarrow{F} C be a span of functors; if the coend

bhom A(Gb,a)×Fb \int^b \hom_A(G b, a) \times F b

exists, then it is the value at aa of the left Kan extension of FF along GG; dually, is the end

bFb hom A(a,Gb) \int_b F b ^ {\hom_A(a, G b)}

exists, then it is the value at aa of the right Kan extension of FF along GG.

Proof. The proof is another lengthy, but completely straightforward, kata using the coend-fu we already know:

Nat(Lan GF,H) ahom(Lan GF(a),Ha) ahom( chom(Gc,a)×Fc,Ha) Fub achom(hom(Gc,a)×Fc,Ha) achom(Fc,[hom(Gc,a),Ha]) ccnt chom(Fc, a[hom(Gc,a),Ha]) nat chom(Fc,Nat(hom(Gc,),H)) nat chom(FC,HGC)=Nat(F,HG) \begin{array}{rl} Nat(\text{Lan}_G F, H) &\textstyle \cong \int_a \hom(\text{Lan}_G F( a ), H a ) \\ & \textstyle \cong \int_a \hom\Big( \int^c \hom(G c, a )\times F c, H a \Big) \\ Fub & \textstyle \cong \int_{ a c} \hom\Big( \hom(G c, a )\times F c, H a \Big) \\ & \textstyle \cong \int_{ a c} \hom\Big( F c,[\hom(G c, a ), H a ] \Big) \\ ccnt & \textstyle \cong \int_c \hom\Big( F c,\int_a [\hom(G c, a ), H a ] \Big) \\ nat & \textstyle \cong \int_c \hom\Big( F c,Nat(\hom(G c,-), H) \Big) \\ nat & \textstyle \cong \int_c \hom\Big( F C,H G C \Big) = Nat(F, H G) \\ \end{array}

Bimodules

This Pastro and Street paper makes heavy use of the theory of bimodules. Let’s dig deeper into their structure. First of all, let us define a bicategory as follows:

  • objects are (unitary) rings R,S,R,S,\dots
  • 1-cells M:RSM : R \to S are modules SM R{}_S M_R, left over RR and right over SS.
  • 2-cells φ: RM S RN S\varphi : {}_R M_S \to {}_R N_S are RR-SS-linear homomorphisms of modules.

The composition of 1-cells is the tensor product of modules: RM SN T{}_R M \otimes_S N_T is a RR-TT-bimodule for every RR-SSbimodule MM and every SS-TT-bimodule NN (so in particular this is not a strict 2-category); 2-cells are composed horizontally and vertically, using the obvious function composition, and bifunctoriality of the \otimes operation.

Good! There is a 2-categorical characterization of modules. What is good for? Well: rings are monoids in the category of abelian groups. We could have done the very same thing taking (plain) monoids, i.e. monoids in the category of sets, and defining a category of “bimodules” as sets with a left action of some monoid, and a right action of some other monoid.

What is interesting now is that we can generalize the same construction, with no additional cognitive loading, and take X,Y,ZX,Y,Z\dots to be multi-object monoids: we define a bicategory ModMod as follows:

  • objects are categories C,D,E,C,D, E,\dots
  • 1-cells P:CDP : C \mathrel{⇸} D are functors P:D op×CSetP : D^{op}\times C \to Set
  • 2-cells are natural transformations between functors.

A functor P:D op×CSetP : D^{op}\times C \to Set is a multiobject module on which the multiobject monoids C,DC,D act.

So categories really are monoids, and are also eager to act on sets.

A bimodule is also called a profunctor, a distributor, a correspondence, a span,… and possibly with many other names; each of these names comes from a certain intuition behind their nature that leads to the definition of the same bicategory:

  • they are called profunctors because they generalize functors: some profunctors are called representable, and they are the ones of the form hom(b,Fa)\hom(b, F a) for some functor F:ABF : A \to B between categories. A pro-functor thus works “on behalf of a functor”, as well as a relation generalizes a function.

  • …which is why some people prefer to call them relators: just as a func-tion is a special kind of rela-tion, a func-tor is a special kind of rela-tor.

  • they are called distributors: as the nLab says,

    > Jean Bénabou, who invented the term and originally used “profunctor,” now prefers “distributor,” which is supposed to carry the intuition that a distributor generalizes a functor in a similar way to how a distribution generalizes a function.

    There’s in fact a beautiful story about this: Lawvere defined a notion of distribution between toposes, such that the points of a topos p:Setp : Set \to \mathcal{E} behave like Dirac delta functions, and such that distributions between presheaf toposes are exactly profunctors.

  • they are called correspondences or spans because of the Grothendieck construction: every presheaf P:D op×CSetP : D^{op}\times C \to Set has a category of elements El(P)El(P) that in this case is a discrete fibration over D op×CD^{op}\times C

Well… until now we cheated a bit. In order to get a bicategory we need to define a composition law for 1-cells and show that it is bifunctorial, and I didn’t tell you how to do it: but it turns out it is really easy, if you know coend-fu! Indeed, the intuition of a bimodule as a “matrix indexed by its domain” and the rule to compose two relations guide us to find an expression to meaningfully compose Q:AB,P:BCQ : A \mathrel{⇸} B, P : B \mathrel{⇸} C. We can define

(PQ)(a,c)= bP(a,b)×Q(b,c) (P \diamond Q) (a,c) = \int^b P(a,b)\times Q(b,c)

boils down, on discrete domains A,BSetA,B\in\Set to a “matrix product of sets” like

(PQ) ac= bBP ab×Q bc (P \diamond Q)_{a c} = \sum_{b\in B} P_{a b}\times Q_{b c}

There is also a connection between the ways in which profunctors compose, and the way in which relations do. Indeed, look how the two concepts closely resemble each other:

(x,z)SR yY ((x,y)R) ((y,z)S) (PQ)(x,z) = y Q(x,y) × P(y,z) \begin{array}{cccccc} (x,z)\in S\circ R & \iff & \exists y\in Y & \big((x,y)\in R\big) & \wedge & \big((y,z)\in S\big) \\ (P\diamond Q)(x,z) & = & \int^y & Q(x,y) & \times & P(y,z) \\ \end{array}

Finally: yes, you can make this precise by saying that sets are discrete categories, or even more precisely categories enriched over truth values, and that relations are precisely the {0,1}\{0,1\}-enriched version of profunctors, as they are functions A×B{0,1}A\times B \to \{0,1\}!

Doubles for monoidal categories

One of the key results of the paper Doubles for Monoidal Categories is the canonical equivalence of categories:

Tamb(𝒞)[Doub(𝒞),Set] \mathbf{Tamb}(\mathcal{C}) \simeq [\mathbf{Doub}(\mathcal{C}), \mathbf{Set}]

This theorem has since been labelled by some as the “fundamental theorem of optics”, as it provides the link between the category of Tambara modules Tamb(𝒞)\mathbf{Tamb}(\mathcal{C}) and the double Doub(𝒞)\mathbf{Doub}(\mathcal{C}) of the monoidal category 𝒞\mathcal{C}, now also known as the category of optics. To unpack this theorem, we first begin with the definition of a Tambara module.

Tambara modules

The category of (bi)modules Mod\mathbf{Mod} forms a bicategory, however when we choose a particular category 𝒞\mathcal{C} we may instead consider the monoidal category Mod(𝒞)\mathbf{Mod}(\mathcal{C}) whose:

  • objects are endomodules P:𝒞 op×𝒞SetP \colon \mathcal{C}^{op} \times \mathcal{C} \rightarrow \mathbf{Set};
  • morphisms are natural transformations;
  • monoidal product is module composition:
    (PQ)(X,Y)= ZP(X,Y)×Q(Y,Z) (P \diamond Q) (X,Y) = \int^Z P(X,Y)\times Q(Y,Z)

    One may ask the question: what happens when 𝒞\mathcal{C} has the structure of a monoidal category?

Definition: Let 𝒞\mathcal{C} be a monoidal category. A (left) Tambara module on 𝒞\mathcal{C} consists of:

  • a profunctor P:𝒞 op×𝒞SetP \colon \mathcal{C}^{op} \times \mathcal{C} \rightarrow \mathbf{Set};
  • a family of functions τ A(X,Y):P(X,Y)P(AX,AY)\tau_{A}(X,Y) \colon P(X, Y) \rightarrow P(A \otimes X, A \otimes Y) called the Tambara structure maps, which are natural in X,YX, Y and dinatural in AA, satisfying the equations:

Backus’ 1977 lecture Backus' 1977 lecture? Backus' 1977 lecture

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