geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
The notion of quantum field theory exists without reference to any predefined notion of configuration space of quantum fields, action functional, phase space etc.:
A quantum field theory in FQFT-axiomatization is simply a consistent assignment of spaces of quantum states, whereas in AQFT-axiomatization it is a consistent assignment of algebras of quantum observables, and that’s it.
However, most (or maybe all?) quantum field theories of interest in actual physics (as opposed to as devices of pure mathematics) are not random models of these axioms, but do arise under a process called quantization from a (local/extended) Lagrangian, hence from an action functional, defined on a configuration space of quantum fields, or else arise as holographic duals of quantum field theories that arise by quantization. Moreover, the extra information provided by the Lagrangian is commonly used (and is maybe strictly necessary) to interpret the mathematical structure of the axiomatic QFT in actual physics (though notably in AQFT there are results that re-extract at least parts of this data from the axiomatic QFT, for instance the Doplicher-Roberts reconstruction theorem which extract the global gauge group from the local net of quantum observables).
There are in turn two formalizations of the notion of quantization: algebraic deformation quantization and geometric quantization. In the latter one speaks of prequantization when referring to a precursor step to the actual quantization step, in which the symplectic form on phase space is lifted to differential cohomology, hence to a prequantum bundle. But in the context of higher geometry and higher geometric quantization this prequantization step is already part of the data of the Lagrangian itself: an extended Lagrangian already encodes not just the action functional but also the prequantum bundle and all the prequantum (n-k)-bundles in each dimension $k$. The action functional itself is the prequantum 0-bundle in this context.
Therefore, in the refined picture of higher geometry/extended quantum field theory it makes good sense to refer in a unified way to prequantum field theory for all of the data related to Lagrangians that is not yet the final quantum field theory.
In particular, an extended prequantum field theory of dimension $n$ is a rule that assigns
to every (suitably oriented) closed manifold of dimension $k$ a prequantum (n-k)-bundle
to every (suitably oriented) compact manifold with boundary $\Sigma_k$ a section of the prequantum $(n-k+1)$-bundle assigned to the boundary $\partial \Sigma_k$ and pulled back to the space of fields over $\Sigma_k$
such that this data is related suitably under transgression.
The actual extended quantum field theory would be obtained from such a data by passing from the assignment of a given prequantum $(n-k)$-bundle to that of the (n-k)-vector space of polarized sections of a suitable associated fiber bundle.
This is summarized in the following table:
extended prequantum field theory
$0 \leq k \leq n$ | (off-shell) prequantum (n-k)-bundle | traditional terminology |
---|---|---|
$0$ | differential universal characteristic map | level |
$1$ | prequantum (n-1)-bundle | WZW bundle (n-2)-gerbe |
$k$ | prequantum (n-k)-bundle | |
$n-1$ | prequantum 1-bundle | (off-shell) prequantum bundle |
$n$ | prequantum 0-bundle | action functional |
under construction
We discuss local (“extended”) topological prequantum field theory.
The following originates in the lecture notes (Schreiber Pittsburgh13) and draws on material that is discussed more fully in (Fiorenza-Valentino) and (hCSlpQFT).
After a technical preliminary to set the stage in
the first section gives the the definitions and general properties of
To digest this the reader may first or in parallel want to look at the simplest examples of these general considerations, which we discuss below in the first subsections of
After that we turn to the general case of examples of
Here the pattern of the discussion of examples is the following:
Prequantum field theory deals with “spaces of physical fields”. These spaces of fields are, in general, richer than just plain sets in two ways
Spaces of fields carry geometric structure, notably they may be smooth spaces, meaning that there is a way to determine which collections of fields form a smoothly parameterized collection. This is for instance the structure invoked (often implicitly) when performing variational calculus on spaces of fields in order to find their classical equations of motion.
Spaces of fields have gauge transformations between their points and possibly higher gauge transformations between these, meaning that they are in fact groupoids and possibly higher groupoids. In the physics literature this is best known in the infinitesimal approximation to these gauge transformations, in which case the spaces of fields are described by BRST complexes: the dg-algebras of functions on a Lie algebroid or L-∞ algebroid of fields.
Taken together this means that spaces of fields are geometric higher groupoids, such as orbifolds and more generally Lie groupoids, differentiable stacks, Lie 2-groupoids, … smooth ∞-groupoids.
A collection of all such geometric higher groupoids for a chosen flavor of geometry – for instance topology or differential geometry or supergeometry (for the description of fermion fields) or synthetic differential geometry or synthetic differential supergeometry, etc. – is called an ∞-topos.
Not quite every ∞-topos $\mathbf{H}$ serves as a decent context for collections (moduli stacks) of physical fields though. In the following we need at least that $\mathbf{H}$ has a reasonable notion of discrete objects so that we can identify the geometrically discrete spaces in there. We here need this to mean the following
An ∞-topos $\mathbf{H}$ is called locally ∞-connected and globally ∞-connetced if the locally constant ∞-stack-functor $LConst \colon$ ∞Grpd $\to \mathbf{H}$ is a reflective embedding.
The corresponding reflector we write
and also call the shape modality of $\mathbf{H}$. By the discussion at adjoint triple it follows that $LConst$ is also a coreflective embedding; the corresponding coreflector we write
and call the flat modality.
Every cohesive (∞,1)-topos is in particular globally and locally $\infty$-connected, by definition. Standard canonical examples to keep in mind are
$\mathbf{H} =$ ∞Grpd for ∞-Dijkgraaf-Witten theories;
$\mathbf{H} =$ Smooth∞Grpd for ∞-Chern-Simons theories;
$\mathbf{H} =$ SuperSmooth∞Grpd for ∞-Chern-Simons theories with fermions and supersymmetry;
$\mathbf{H} =$ SynthDiff∞Grpd for AKSZ sigma-models.
After sketching out the general
we formulate first
which concerns the case where the worldvolume/spacetime on which the physical fields propagate has no boundaries with boundary conditions imposed (no “branes” or “domain walls” or “defects”). The point of this section is to see how the “space of fields” – or rather: the moduli stack of fields – on a point induces the corresponding spaces/moduli stacks of fields on an arbitrary closed manifold, and, correspondingly, how the prequantum n-bundle on the space over fields over the point induces the action functional in codimension 0.
However, what makes local prequantum field theory rich is that it naturally incorporates extra structure on boundaries of worldvolume/spacetime. In fact, under suitable conditions there is another local prequantum field theory just over the boundary, which is related to the corresponding bulk field theory possibly by a kind of holographic principle. This general mechanism we discuss in
But plain boundaries are just the first example of a general phenomenon known as “defects” or “phase dualities” or “singularities” in field theories. Notably the boundary field theory itself may have boundaries, in which case this means that the original theory had corners where different boundary pieces meet. This we discuss in
Generally there are fields theories with general such singularties:
singularity | field theory with singularities |
---|---|
boundary condition/brane | boundary field theory |
domain wall/bi-brane | QFT with defects |
A prequantum field theory is, at its heart, an assignment that sends a piece of worldvolume/spacetime $\Sigma$ – technically a cobordism with boundary and corners – to the
space of field configurations over incoming and outgoing pieces of worldvolume/spacetime;
the space of field configurations over the bulk worldvolume/spacetime – the trajectories of fields;
an action functional that assigns to all these field configurations phases in a compatible manner.
These field configurations and spaces of trajectories between them are represented by spans/correspondences of (moduli-)spaces of fields (moduli stacks, really), hence diagrams of the form
Here $\mathbf{Fields}_{in}$ is to be thought of as the space of incoming fields, $\mathbf{Fields}_{out}$ that of outgoing fields, and $\mathbf{Fields}$ the space of all fields on some cobordism connecting the incoming and the outgoing pieces of worldvolume/spacetime. The left map sends such a trajectory to its starting configuration, and the right one sends it to its end configuration.
Given two such spans/correspondences, that share a common field configuration as in
can be composed, by forming consecutive trajectories from all pairs of trajectories that match in the middle. The space of these composed trajectories is the fiber product $\mathbf{Fields}_1 \underset{{\mathbf{Fields}_{out_1}} \atop {=\mathbf{Fields}_{in_2}}}{\times} \mathbf{Fields}_2$ which sits in a new span/correspondence
exhibiting the composite of the previous two. This way, spaces of fields with spans/correspondences between them form a category, which we denote $Span_1(\mathbf{H})$ if $\mathbf{H}$ denotes the ambient context (a topos) in which the spaces of fields live.
If two cobordisms run in parallel, then the field configurations on their union are pairs of the original field configurations, which are elements in the cartesian product of spaces of fields. Hence the operations
make this category of fields and correspondence into a monoidal category.
Then a choice of field configurations for a (not yet localized) field theory in dimension $n \in \mathbb{N}$ is a monoidal functor from a category of cobordisms of dimension $n$ to such a category of spans/correspondences
namely a consistent assignment that to each closed manifold $\Sigma_{n-1}$ of dimension $(n-1)$ assigns a space of field configurations $\mathbf{Fields}(\Sigma_{n-1})$ and that to each cobordism
assigns a span/correspondence of spaces of field configurations and trajectories
Apart from the field configurations themselves, prequantum field theory assigns to each trajectory a “phase” – an element in the circle group $U(1)$ – by a map called the (exponentiated) action functional. In order to nicely relate that to the expression of spaces of trajectories as spans/correspondences as above, it is useful to think of the circle group here as being the automorphisms of something. This is universally accomplished by taking it to be the automorphisms of the unique point in the delooping groupoid $\mathbf{B}U(1) = \{\ast \stackrel{c \in U(1)}{\to} \ast\}$. (A lightning review of groupoid-homotopy theory is below in Groupoids and basic homotopy 1-type theory.) In other words, we think of the group of phases $U(1)$ as the space of homotopies from the point to itself in the Eilenberg-MacLane space $\mathbf{B}U(1)$, expressed by the diagram (a homotopy fiber product diagram)
Using this, if we assume for simplicity that the in- and outgoing field configurations are sent constantly to the point in $\mathbf{B}U(1)$, then an (exponentiated) action functional on the space of trajectories $\exp(i S) \colon \mathbf{Fields} \to U(1)$ is equivalently a homotopy as shown on the left of the following diagram
Hence action functionals are naturally incorporated into spans/correspondences of moduli spaces of fields simply by regarding these to be formed not in the ambient topos $\mathbf{H}$ itself, but in its slice topos $\mathbf{H}_{/\mathbf{B}U(1)}$, where each object is equipped with a map to $\mathbf{B}U(1)$ and each morphism with a homotopy in $\mathbf{B}U(1)$ between the corresponding maps.
We write $\mathrm{Span}_1(\mathbf{H}, \mathbf{B}U(1))$ for the category of spans/correspondences as before, but now equipped with maps to, and transformations over, $\mathbf{B}U(1)$ as in the above diagram.
Then an action functional for a choice of field configurations that itself is given as a monoidal functor $\mathbf{Fields} \colon Bord_n^\otimes \to Span_1(\mathbf{H})$ as above is a monoidal functor
such that the spans of spaces of fields are those specified before, hence such that it fits as a lift into the diagram
where the right vertical functor forgets the phase assignments and just remembers the correspondences of field trajectories.
So far this is a non-local (or: not-necessarily local) prequantum field theory, since it assigns data only to entire $n$-dimensional cobordisms and $(n-1)$-dimensional closed manifolds, but is not guaranteed to be obtained by integrating up local data over little pieces of these manifolds. The latter possibility is however the characteristic property of local quantum field theory, which in turn is the flavor of quantum field theory that seems to matter in nature, and fundamentally.
In order to formalize this localization, we allow the cobordisms to contain higher-codimension pieces that are manifolds with corners. These then form not just a category of cobordisms, but an (∞,n)-category of cobordisms, which we will still denote $Bord_n^\otimes$. If we now have a cobordism with codimension-2 corners, then the field configurations over it now form a span-of-spans
Generally, for $n$-dimensional cobordism that are “localized” all the way to corners in codimension $n$, their field configurations and trajectories-of-trajectories etc. form $n$-dimensional cubes of spans-of-spans this way. We write $Span_n(\mathbf{H})$ for the resulting (∞,n)-category of spans.
In order to still have an action functional on trajectories is codimension-0 associated with this in the above fashion, we need to deloop $U(1)$ $n$-times to the n-groupoid $\mathbf{B}^n U(1)$ (the circle (n+1)-group). Accordingly a local prequantum field theory in dimension $n$ is given by a monoidal (∞,n)-functor
The point of local topological (prequantum) field theory is that by the cobordism theorem the above story reverses: the assignment of fields and their action functional in higher dimension is necessarily given by higher traces of the data assigned in lower dimension. Hence the whole assignment $S$ above is fixed by its value on the point, hence by a choice of one single map
the fully localized action functional. Or rather, this is the case for pure bulk field theory, with no branes or domain walls. If these are present, then each type of them in dimension $k$ is specified by a k-morphism in $Span_n(\mathbf{H}, \mathbf{B}^n U(1))$.
All this we now describe more formally.
We now first consider the formalization of prequantum field theory in the absence of any data such as boundary conditions, domain walls, branes, defects, etc. This describes either field theories in which no such phenomena are taken to be present, or else it describes that part of those field theories where such phenomena are present in principle, but restricted to the “bulk” of worldvolume/spacetime where they are not. Therefore it makes sense to speak of bulk field theory in this case.
For $n \in \mathbb{N}$, write
for the symmetric monoidal (∞,n)-category of cobordisms with $n$-dimensional framing. For $S \to O(n)$ a homomorphism of ∞-groups (may be modeled by a homomorphism of topological groups) to the general linear group (or homotopy-equivalently its maximal compact subgroup, the orthogonal group), we write
for the corresponding symmetric monoidal $(\infty,n)$-category of cobordisms equipped with S-structure on their $n$-stabilized tangent bundle.
In this notation we have an identification
because a framing of the $n$-stabilized tangent bundle is a trivialization of that bundle and hence equivalently a G-structure for $G$ the trivial group. In (LurieTFT) this is denoted by “$Bord_n^{fr}$”.
The cobordism theorem asserts, essentially, that $Bord_n$ is the symmetric monoidal (∞,n)-category with full duals which is free on a single generator, the point. In itself this is a deep statement about the homotopy type of categories of cobordisms. But for the following discussion the reader may just take this as the definition of $Bord_n$. This then makes $Bord_n$ a very simple object, as long as we are just mapping out of it, which we do.
What this means then is that a monoidal (∞,n)-functor
sends the point to some fully dualizable object $Z(\ast) \in \mathcal{C}$ and sends
the sphere $S^2$ to the 2-dimensional higher trace of the identity,
and so on.
For $\mathbf{H}$ an ∞-topos, and $n \in \mathbb{N}$, write
for the (∞,n)-category of spans in $\mathbf{H}$. From the cartesian monoidal category structure of $\mathbf{H}$ this inherits the structure of a symmetric monoidal (∞,n)-category which we write
Every object in $Span_n(\mathbf{H})$ is a self-fully dualizable object. The evaluation map/coevaluation map $k$-spans in dimension $k$ involve in top degree the spans
(…)
For $B \in Grp(\mathbf{H})$ an abelian ∞-group object in $\mathbf{H}$, spans in the slice (∞,1)-topos $\mathbf{H}_{/B}$ inherits a monoidal structure given on objects by
We write
for the resulting symmetric monoidal (∞,n)-category.
In the case that $\mathbf{H} =$ ∞Grpd this is a special case of (LurieTFT, around prop. 3.2.8), with the abelian ∞-group $B$ regarded as a special case of a symmetric monoidal (∞,1)-category.
Since the slice (∞,1)-category $\mathbf{H}_{/\flat \mathbf{B}^n U(1)}$ is itself an (∞,1)-topos – the slice (∞,1)-topos – we also have $Span_n(\mathbf{H}_{/\flat \mathbf{B}^n U(1)})$, according to def. 3. As an (∞,n)-category this is equivalent to $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ from def. 4, but the monoidal structure is different. The cartesian product in the slice is given by homotopy fiber product in $\mathbf{H}$ over $\flat \mathbf{B}^n U(1)$, not by the addition in the ∞-group structure on $\flat \mathbf{B}^n U(1)$, as in def. 4.
The central definition in the present context now is the following.
A local prequantum bulk field in dimension $n \in \mathbb{N}$ (in a given ambient cohesive (∞,1)-topos $\mathbf{H}$) is a monoidal (∞,n)-functor
from the (∞,n)-category of cobordisms (with S-structure), def. 2, to the (∞,n)-category of n-fold correspondences in $\mathbf{H}$.
A local action functional on such a local prequantum bulk field is a monoidal lift $S$ of this in
For $\mathbf{H} =$ ∞Grpd this is the perspective in (FHLT, section 3).
Since a monoidal $(\infty,n)$-functor $\mathbf{Fields} \colon Bord_n \to Span_n(\mathbf{H})$ is determined by its value on the point, we will often notationally identify it with this value and write
As a corollary of prop. 1 we have:
Given $\mathbf{Fields} \colon Bord_n^\otimes \to Span_n(\mathbf{H})^\otimes$, it assigns to a k-morphism represented by a closed manifold $\Sigma_k$ the internal hom (mapping stack) from $\Pi(\Sigma_k)$ (the shape modality of $\Sigma_k$, def. 1) into the moduli stack of fields
By the defining property of the mapping stack construction, this means that if $\mathcal{C}$ is an (∞,1)-site of definition of the (∞,1)-topos $\mathbf{H}$, then $[\Pi(\Sigma_k), \mathbf{Fields}]$ is the ∞-stack which to $U \in \mathcal{C}$ assigns the (∞,1)-categorical hom space
hence the ∞-groupoid of fields on $\Pi(\Sigma_k) \times U$.
If $\mathbf{Fields}$ is a moduli ∞-stack of gauge fields for some smooth ∞-group $G$, hence of the form $\mathbf{B}G_{conn}$, then this an $\infty$-groupoid of a kind of smoothly (or else geometrically) $U$-parameterized collections of flat ∞-connections on $\Sigma_k$.
(…)
(…)
(…)
Traditional classical mechanics (Hamiltonian mechanics, Lagrangian mechanics, Hamilton-Jacobi theory) is naturally understood as a special case of – and in fact as deriving from – local prequantum field theory over $\mathbf{B}U(1)_{conn}$. This is discussed in some detail at
We discuss here aspects of higher Dijkgraaf-Witten theory-type prequantum field theories, which are those prequantum field theories whose moduli stack $\mathbf{Fields}$ is a discrete ∞-groupoid (and usually also required to be finite, especially if its quantization is considered). This is a special case of the higher Chern-Simons theories discussed below in Higher Chern-Simons local prequantum field theory, and hence strictly speaking need not be discussed separately. We use it here as a means to review some of the relevant homotopy theory by way of pertinent examples.
The original Dijkgraaf-Witten theory is that in dimension 3 (reviewed in 3d Local prequantum field theory below), which was introduced in (Dijkgraaf-Witten 90) as a toy version of standard 3d Chern-Simons theory for simply connected gauge group. A comprehensive account with first indications of its role as a local (extended, multi-tiered) field theory then appeared in (Freed-Quinn 93), and ever since this has served as a testing ground for understanding the general principles of local field theory, e.g. (Freed 94), independently of the subtleties of giving meaning to concepts such as the path integral when the space of fields is not finite. In section 3 of (FHLT 10), the general prequantum formalization as in def. 5 is sketched for Dijkgraaf-Witten type theories, and in section 8 there the quantization of these theories to genuine local quantum field theories is sketched.
Dijkgraaf-Witten theory in dimension 1 is what results when one regards a group character of a finite group $G$ as a local action functional in the sense of def. 5. We give now an expository discussion of this simple but instructive example of a local prequantum field theory and in the course of it introduce some of the relevant basics of the homotopy theory of groupoids (homotopy 1-types).
Essence of gauge theory: Groupoids and basic homotopy 1-type theory
Action functionals on spaces of trajectories: Correspondences of groupoids over the space of phases
The punchline of this section is little theorem 1 at the very end, which states that the 1d local prequantum field theory whose local action functional is the delooping of a group character assigns to the circle the action functional which is again that group character. The proof of this statement is an unwinding of the basic mechanisms of local prequantum field theories.
First some brief remarks, before we dive into the formalism.
A group character on a finite group $G$ is just a group homomorphism $G \to U(1)$ to the circle group (taken here as a discrete group). In order to regard this as an action functional, we are to take $G$ as the gauge group of a physical field theory. The simplest such case is a field theory such that on the point there is just a single possible field configuration, to be denoted $\phi_0$. The reader familiar with basics of traditional gauge theory may think of the fields as being gauge field connections (“vector potentials”), hence represented by differential 1-forms. But on the point there is only the vanishing 1-form, hence just a single field configuration $\phi_0$.
Even though there is just a single such field, that $G$ is the gauge group means that for each element $g \in G$ there is a gauge transformation that takes $\phi_0$ to itself, a state of affairs which we suggestively denote by the symbols
Again, the reader familiar with traditional gauge theory may think of gauge transformations as in Yang-Mills theory. Over the point these form, indeed, just the gauge group itself, taking the trivial field configuration to itself.
That the gauge group is indeed a group means that gauge transformations can be applied consecutively, which we express in symbols as
Regarded this way, we say the gauge group acting on the single field $\phi_0$ forms a groupoid, whose single object is $\phi_0$ and whose set of morphisms is $G$.
Of course in richer field theories there may be more than one field configuration, clearly, with gauge transformations between them. If $\phi_0$ and $\phi_1$ are two field configurations and $g$ is a gauge transformation taking one to the other, we may usefully denote this by
Similarly then for yet another gauge configuration to another field configuration
then composing them gives the picture
We now discuss this notion of groupoids more formally.
The following is a quick review of basics of groupoids and their homotopy theory (homotopy 1-type-theory), geared towards the constructions and fact needed for 1-dimensional Dijkgraaf-Witten theory.
A (small) groupoid $\mathcal{G}_\bullet$ is
a pair of sets $\mathcal{G}_0 \in Set$ (the set of objects) and $\mathcal{G}_1 \in Set$ (the set of morphisms)
equipped with functions
where the fiber product on the left is that over $\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1$,
such that
$i$ takes values in endomorphisms;
$\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{G}_0)$ the identities; in particular
$s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$;
every morphism has an inverse under this composition.
This data is visualized as follows. The set of morphisms is
and the set of pairs of composable morphisms is
The functions $p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1$ are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.
For $X$ a set, it becomes a groupoid by taking $X$ to be the set of objects and adding only precisely the identity morphism from each object to itself
For $G$ a group, its delooping groupoid $(\mathbf{B}G)_\bullet$ has
$(\mathbf{B}G)_0 = \ast$;
$(\mathbf{B}G)_1 = G$.
For $G$ and $K$ two groups, group homomorphisms $f \colon G \to K$ are in natural bijection with groupoid homomorphisms
In particular a group character $c \colon G \to U(1)$ is equivalently a groupoid homomorphism
Here, for the time being, all groups are discrete groups. Since the circle group $U(1)$ also has a standard structure of a Lie group, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on
to mean explicitly the discrete group underlying the circle group. (Here “$\flat$” denotes the “flat modality”.)
For $X$ a set, $G$ a discrete group and $\rho \colon X \times G \to X$ an action of $G$ on $X$ (a permutation representation), the action groupoid or homotopy quotient of $X$ by $G$ is the groupoid
with composition induced by the product in $G$. Hence this is the groupoid whose objects are the elements of $X$, and where morphisms are of the form
for $x_1, x_2 \in X$, $g \in G$.
As an important special case we have:
For $G$ a discrete group and $\rho$ the trivial action of $G$ on the point $\ast$ (the singleton set), the coresponding action groupoid according to def. 4 is the delooping groupoid of $G$ according to def. 3:
Another canonical action is the action of $G$ on itself by right multiplication. The corresponding action groupoid we write
The constant map $G \to \ast$ induces a canonical morphism
This is known as the $G$-universal principal bundle. See below in 11 for more on this.
The interval $I$ is the groupoid with
For $\Sigma$ a topological space, its fundamental groupoid $\Pi_1(\Sigma)$ is
For $\mathcal{G}_\bullet$ any groupoid, there is the path space groupoid $\mathcal{G}^I_\bullet$ with
$\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\}$;
$\mathcal{G}^I_1 =$ commuting squares in $\mathcal{G}_\bullet$ = $\left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,.$
This comes with two canonical homomorphisms
which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.
For $f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet$ two morphisms between groupoids, a homotopy $f \Rightarrow g$ (a natural transformation) is a homomorphism of the form $\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet$ (with codomain the path space object of $\mathcal{K}_\bullet$ as in example 8) such that it fits into the diagram as depicted here on the right:
Here and in the following, the convention is that we write
$\mathcal{G}_\bullet$ (with the subscript decoration) when we regard groupoids with just homomorphisms (functors) between them,
$\mathcal{G}$ (without the subscript decoration) when we regard groupoids with homomorphisms (functors) between them and homotopies (natural transformations) between these
The unbulleted version of groupoids are also called homotopy 1-types (or often just their homotopy-equivalence classes are called this way.) Below we generalize this to arbitrary homotopy types (def. 23).
For $X,Y$ two groupoids, the mapping groupoid $[X,Y]$ or $Y^X$ is
A (homotopy-) equivalence of groupoids is a morphism $\mathcal{G} \to \mathcal{K}$ which has a left and right inverse up to homotopy.
The map
which picks any point and sends $n \in \mathbb{Z}$ to the loop based at that point which winds around $n$ times, is an equivalence of groupoids.
Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example 3 – a skeleton.
The statement of prop. 3 becomes false as when we pass to groupoids that are equipped with geometric structure. This is the reason why for discrete geometry all Chern-Simons-type field theories (namely Dijkgraaf-Witten theory-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the AKSZ sigma-models. But even so, Dijkgraaf-Witten theory is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. 3 is not canonical.
Given two morphisms of groupoids $X \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y$ their homotopy fiber product
hence the ordinary iterated fiber product over the path space groupoid, as indicated.
An ordinary fiber product $X_\bullet \underset{B_\bullet}{\times}Y_\bullet$ of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:
For $X$ a groupoid, $G$ a group and $X \to \mathbf{B}G$ a map into its delooping, the pullback $P \to X$ of the $G$-universal principal bundle of example 5 is equivalently the homotopy fiber product of $X$ with the point over $\matrhbf{B}G$:
Namely both squares in the following diagram are pullback squares
(This is the first example of the more general phenomenon of universal principal infinity-bundles.)
For $X$ a groupoid and $\ast \to X$ a point in it, we call
the loop space groupoid of $X$.
For $G$ a group and $\mathbf{B}G$ its delooping groupoid from example 3, we have
Hence $G$ is the loop space object of its own delooping, as it should be.
We are to compute the ordinary limiting cone $\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast$ in
In the middle we have the groupoid $(\mathbf{B}G)^I_\bullet$ whose objects are elements of $G$ and whose morphisms starting at some element are labeled by pairs of elements $h_1, h_2 \in G$ and end at $h_1 \cdot g \cdot h_2$. Using remark 9 the limiting cone is seen to precisely pick those morphisms in $(\mathbf{B}G_\bullet)^I_\bullet$ such that these two elements are constant on the neutral element $h_1 = h_2 = e = id_{\ast}$, hence it produces just the elements of $G$ regarded as a groupoid with only identity morphisms, as in example 2.
The free loop space object is
Notice that $\Pi_1(S^0) \simeq \ast \coprod \ast$. Therefore the path space object $[\Pi(S^0), X_\bullet]^I_\bullet$ has
objects are pairs of morphisms in $X_\bullet$;
morphisms are commuting squares of such.
Now the fiber product in def. 10 picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore $X_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X$ is the groupoid whose
objects are diagrams in $X_\bullet$ of the form
morphism are cylinder-diagrams over these.
One finds along the lines of example 10 that this is equivalent to maps from $\Pi_1(S^1)$ into $X_\bullet$ and homotopies between these.
Even though all these models of the circle $\Pi_1(S^1)$ are equivalent, below the special appearance of the circle in the proof of prop. 4 as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an evaluation map with a coevaluation map.
For $G$ a discrete group, the free loop space object of its delooping $\mathbf{B}G$ is $G//_{ad} G$, the action groupoid, def. 4, of the adjoint action of $G$ on itself:
For an abelian group such as $\flat U(1)$ we have
Let $c \colon G \to \flat U(1)$ be a group homomorphism, hence a group character. By example 3 this has a delooping to a groupoid homomorphism
Unde the free loop space object construction this becomes
hence
So by postcomposing with the projection on the first factor we recover from the general homotopy theory of groupoids the statement that a group character is a class function on conjugacy classes:
With some basic homotopy theory of groupoids in hand, we can now talk about trajectories in finite gauge theories, namely about spans/correspondences of groupoids and their composition. These correspondences of groupoids encode trajectories/histories of field configurations.
Namely consider a groupoid to be called $\mathbf{Fields} \in$ Grpd, to be thought of as the moduli space of fields in some field theory, or equivalently and specifically as the target space of a sigma-model field theory. This just means that for $\Sigma$ any manifold thought of as spacetime or worldvolume, the space of fields $\mathbf{Fields}(\Sigma)$ of the field theory on $\Sigma$ is the mapping stack (internal hom) from $\Sigma$ into $\mathbf{Fields}$, which means here for DW theory that it is the mapping groupoid, def. 9, out of the fundamental groupoid, def. 7, of $\Sigma$:
We think of the objects of the groupoid $[\Pi_1(\Sigma), \mathbf{Fields}]$ as being the fields themselves, and of the morphisms as being the gauge transformations between them.
The example to be of interest in a moment is that where $\mathbf{Fields} = \mathbf{B}G$ is a delooping groupoid as in def. 3, in which case the fields are equivalently flat principal connections. In fact in the discrete and 1-dimensional case currently considered this is essentially the only example, due to prop. 3, but for the general idea and for the more general cases considered further below, it is useful to have the notation allude to more general moduli spaces $\mathbf{Fields}$.
The simple but crucial observation that shows why spans/correspondences of groupoids show up in prequantum field theory is the following.
If $\Sigma$ is a cobordism, hence a manifold with boundary with incoming boundary component $\Sigma_{in} \hookrightarrow \Sigma$ and outgoing boundary components $\Sigma_{out} \hookrightarrow \Sigma$, then the resulting cospan of manifolds
is sent under the operation of mapping into the moduli space of fields
to a span of groupoids
Here the left and right homomorphisms are those which take a field configuration on $\Sigma$ and restrict it to the incoming and to the outgoing field configuration, respectively. (And this being a homomorphism of groupoids means that everything respects the gauge symmetry on the fields.) Hence if $[\Pi_1(\Sigma_{in,out}),\mathbf{Fields}]$ is thought of as the spaces of incoming and outgoing field configurations, respectively, then $[\Pi_1(\Sigma), \mathbf{Fields}]$ is to be interpreted as the space of trajectories (sometimes: histories) of field cofigurations over spacetimes/worldvolumes of shape $\Sigma$.
This should make it plausible that specifying the field content of a 1-dimensional discrete gauge field theory is a functorial assignsment
from a category of cobordisms of dimension one into a category of such spans of groupoids. It sends points to spaces of field configurations on the point and 1-dimensional manifolds such as the circle as spaces of trajectories of field configurations on them.
Moreover, for a local field theory it should be true that the field configurations on the circle, says, are determined from gluing the field configurations on any decomposition of the circle, notably a decomposition into two semi-circles. But since we are dealing with a topological field theory, its field configurations on a contractible interval such as the semicircle will be equivalent to the field configurations on the point itself.
The way that the fields on higher spheres in a topological field theory are induced from the fields on the point is by an analog of traces for spaces of fields, and higher traces of such correspondences (the “span trace”). This is because by the cobordism theorem, the field configurations on, notably, the n-sphere are given by the $n$-fold span trace of the field configurations on the point, the trace of the traces of the … of the 1-trace. This is because for instance the 1-sphere, hence the circle is, regarded as a 1-dimensional cobordism itself pretty much manifestly a trace on the point in the string diagram formulation of traces.
Here $\ast^+$ is the point with its potitive orientation, and $\ast^-$ is its dual object in the category of cobordisms, the point with the reverse orientation. Since, by this picture, the construction that produces the circle from the point is one that involves only the coevaluation map and evaluation map on the point regarded as a dualizable object, a topological field theory $Z \colon Bord_n \to Span_n(\mathbf{H})$, since it respects all this structure, takes the circle to precisely the same kind of diagram, but now in $Span_n(\mathbf{H})^\otimes$, where it becomes instead the span trace on the space $\mathbf{Fields}(\ast)$ over the point. This we discuss now.
Before talking about correspondences of groupoids, we need to organize the groupoids themselves a bit more.
A (2,1)-category $\mathcal{C}$ is
a collection $\mathcal{C}_0$ – the “collection of objects”;
for each tuple $(X,Y) \in \mathcal{C}_0 \times \mathcal{C}_0$ a groupoid $\mathcal{C}(X,Y)$ – the hom-groupoid from $X$ to $Y$;
for each triple $(X,Y,Z) \in \mathcal{C}_0 \times \mathcal{C}_0 \times \mathcal{C}_0$ a groupoid homomorphism (functor)
called composition or horizontal composition for emphasis;
for each quadruple $(W,X,Y,Z,)$ a homotopy – the associator –
(…) and similarly a unitality homotopy (…)
such that for each quintuple $(V,W,X,Y,Z)$ the associators satisfy the pentagon identity.
The objects of the hom-groupoid $\mathcal{C}(X,Y)$ we call the 1-morphisms from $X$ to $Y$, indicated by $X \stackrel{f}{\to} Y$, and the morphisms in $\mathcal{C}(X,Y)$ we call the 2-morphisms of $\mathcal{C}$, indicated by
If all associators $\alpha$ can and are chosen to be the identity then this is called a strict (2,1)-category.
Write Grpd for the strict (2,1)-category, def. 11, whose
1-morphisms are functors $f \colon \mathcal{G} \to \mathcal{K}$;
2-morphisms are homotopies between these.
Write $Span_1(Grpd)$ for the (2,1)-category whose
1-morphisms are spans/correspondences of functors, hence
2-morphisms are diagrams in Grpd of the form
composition is given by forming the homotopy fiber product, def. 10, of the two adjacent homomorphisms of two spans, hence for two spans
and
their composite is the span which is the outer part of the diagram
There is the structure of a symmetric monoidal (2,1)-category on $Span_1(Grpd)$ by degreewise Cartesian product in Grpd.
An object $X$ of a symmetric monoidal (2,1)-category $\mathcal{C}^\otimes$ is fully dualizable if there exists
another object $X^\ast$, to be called the dual object;
a 1-morphism $ev_X \colon X^\ast \otimes X \to \mathbb{I}$, to be called the evaluation map;
a 1-morphism $coev_X \colon \mathbb{I} \to X \otimes X^\ast$, to be called the coevaluation map;
and
and
(the saddle?)
and
(the co-saddle)
such that these exhibit an adjunction and are themselves adjoint (…).
Given a symmetric monoidal (2,1)-category $\mathcal{C}$, and a fully dualizable object $X \in \mathcal{C}$ and a 1-morphism $f \colon X \to X$, the trace of $f$ is the composition
Every groupoid $X \in Grpd \hookrightarrow Span_1(Grpd)$ is a dualizable object in $Span_1(Grpd)$, and in fact is self-dual.
The evaluation map $ev_X$, hence the possible image of a symmetric monoidal functor $Bord_1 \to Span_1(Grpd)$ of a cobordism of the form
is given by the span
and the coevaluation map $coev_X$ by the reverse span.
For $X \in Grpd \hookrightarrow Span_1(Grpd)$ any object, the trace (“span trace”) of the identity on it, hence the image of
is its free loop space object, prop. 4:
The second order covaluation map on the span trace of the identity is
By prop. 5 the trace of the identity is given by the composite span
By prop. 4 we have
Along these lines one checks the required zig-zag identities.
We have now assembled all the ingredients need in order to formally regard a group character $c \colon G \to U(1)$ on a discrete group as a local action functional of a prequantum field theory, hence as a fully dualizable object
in a (2,1)-category of correspondences of groupoids as in def. 13, but equipped with maps and homotopies between maps to the coefficient over $\mathbf{B}\flat U(1)$. This is described in def. 17 below. Before stating this, we recall for the 1-dimensional case the general story of def. 5.
Given a discrete groupoid $X$, functions
are in natural bijection with homotopies of the form
where the function corresponding to this homotopy is that given by the unique factorization through the homotopy fiber product $\flat U(1) \simeq \ast \underset{\mathbf{B}\flat U(1)}{\times} \ast$ (example 12) as shown on the right of
This means that if we have an action functional on a space of trajectories, and if these trajectories are given by spans/correspondences of groupoids as discussed above, then the action functional is naturally expressed as the homotopy filling a completion of the span to a square diagram over $\mathbf{B}\flat U(1)$. Therefore we cosider the following.
Write $Span_1(Grpd, \flat\mathbf{B}U(1))$ for the (2,1)-category whose
objects are groupoids $X$ equipped with a morpism
morphisms are spans $X_1 \leftarrow Y \rightarrow X_2$ equipped with a homotopy $\phi$ in
2-morphisms are morphism of spans compatible with the maps to $\mathbf{B}\flat U(1)$ in the evident way.
The operation of composition is as in $Span_1(Grpd)$, def. 13 on the upper part of these diagrams, naturally extended to the whole diagrams by composition of the homotopies filling the squares that appear.
$Span_1(Grpd, \mathbf{B}\flat U(1))$ carries the structure of a symmetric monoidal (2,1)-category where the tensor product is given by
There is an evident forgetful (2,1)-functor
which forgets the maps to $\mathbf{B}\flat U(1)$ and the homotopies between them. This is a monoidal (2,1)-functor.
As generalization of prop. 5 we now have the following:
Every object
is a dualizable object, with dual object
and with evaluation map given by
In conclusion we may now compute what the 1-dimensional prequantum field theory defined by a group character $c \colon G \to U(1)$ regarded as a local action functional assigns to the circle.
The prequantum field theory defined by a group character
assigns to the circle the trace of the identity on this object, which under the identifications of example 12, example 15, and example 17 is the group character itself:
Here the action functional on the right sends a field configuration $g \in G = [\Pi(S^1), \mathbf{B}G]_0$ to its value $c(g) \in U(1) = (\flat \mathbf{B}U(1))_1$ under the group character.
It follows that in a discussion of quantization the path integral for the partition function of 1d DW theory is given by the Schur integral over the group character $c$.
In conclusion, 1-dimensional Dijkgraaf-Witten theory as a prequantum field theory comes down to be essentially a geometric interpretation of what group characters are and do. One may regard this as a simple example of geometric representation theory. Simple as this example is, it contains in it the seeds of many of the interesting aspects of richer prequantum field theories.
The group character $c : G \to U(1)$ which defines 1-dimensional prequantum Dijkgraaf-Witten theory in 1d Dijkgraaf-Witten theory is equivalently a cocycle in degree-1 group cohomology
More familiar are maybe cocycles in higher degree. In view of the above it is plausible that one may interpret a cocycle in degree-$n$ group cohomology, for all $n \in \mathbb{N}$ as a higher order action functional $\mathbf{B}G \to \flat\mathbf{B}^n U(1)$ and induce an $n$-dimensional local prequantum Dijkgraaf-Witten-type theory from it.
Here we discuss the case of $n = 2$ where a group 2-cocycle is regarded as the local action functional of a 2-dimensional Digjkgraaf-Witten field theory. We use this as occasion to introduce a bit of the theory of 2-groups and their homotopy theory (homotopy 2-type-theory). Below in 3d DW theory we then turn to the fully general case of ∞-groupoid-theory.
(…)
In view of the above it is plausible that one may interpret a cocycle in degree-$n$ group cohomology, for all $n \in \mathbb{N}$ as a higher order action functional $\mathbf{B}G \to \flat\mathbf{B}^n U(1)$ and induce an $n$-dimensional local prequantum Dijkgraaf-Witten-type theory from it.
Here we review how to formalize this and then consider the example of DW theory in arbitrary dimension $n$.
We briefly recall here some basic definitions and facts of ∞-groupoids and their homotopy theory, geared towards their use in 3-dimensional Dijkgraaf-Witten theory and generally in ∞-Dijkgraaf-Witten theory.
An ∞-groupoid is first of all supposed to be a structure that consists of k-morphisms for all $k \in \mathbb{N}$, which for $k \geq 1$ go between $(k-1)$-morphisms.
In the context of Kan complexes, the tool for organizing such collections of k-morphisms is the notion of a simplicial set, which models $k$-morphisms as being of the shape of $k$-simplices – a vertex for $k = 0$, an edge for $k = 1$, a triangle for $k = 2$, a tetrahedron for $k = 3$, and so on.
This means that a simplicial set $K_\bullet$ is a sequence of sets $\{K_n\}_{n \in \mathbb{N}}$ (sets of $k$-simplex shaped $k$-morphisms for all $k$) equipped with functions $d_i \colon K_{k+1} \to K_{k}$ that send a $(k+1)$-simplex to its $i$-th face, and functions $s_i \colon K_k \to K_{k+1}$ that over a $k$-simplex “erects a flat $(k+1)$-simplex” in all possible ways (hence which inserts “identities” called “degeneracies” in this context).
If we write $\Delta$ for the category whose objects are abstract cellular simplices and whose morphisms are all cellular maps between these, then such a simplicial set is equivalently a functor of the form
Hence we think of this as assigning
a set $[0] \mapsto K_0$ of objects;
a set $[1] \mapsto K_1$ of morphism;
a set $[2] \mapsto K_2$ of 2-morphism;
a set $[3] \mapsto K_3$ of 3-morphism;
and generally
as well as specifying
functions $([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n)$ that send $n+1$-morphisms to their boundary $n$-morphisms;
functions $([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1})$ that send $n$-morphisms to identity $(n+1)$-morphisms on them.
The fact that $K$ is supposed to be a functor enforces that these assignments of sets and functions satisfy conditions that make consistent our interpretation of them as sets of $k$-morphisms and source and target maps between these. These are called the simplicial identities.
But apart from this source-target matching, a generic simplicial set does not yet encode a notion of composition of these morphisms.
For instance for $\Lambda^1[2]$ the simplicial set consisting of two attached 1-cells
and for $(f,g) : \Lambda^1[2] \to K$ an image of this situation in $K$, hence a pair $x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2$ of two composable 1-morphisms in $K$, we want to demand that there exists a third 1-morphisms in $K$ that may be thought of as the composition $x_0 \stackrel{h}{\to} x_2$ of $f$ and $g$. But since we are working in higher category theory (and not to be evil), we want to identify this composite only up to a 2-morphism equivalence
From the picture it is clear that this is equivalent to demanding that for $\Lambda^1[2] \hookrightarrow \Delta[2]$ the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets
A simplicial set where for all such $(f,g)$ a corresponding such $h$ exists may be thought of as a collection of higher morphisms that is equipped with a notion of composition of adjacent 1-morphisms.
For the purpose of describing groupoidal composition, we now want that this composition operation has all inverses. For that purpose, notice that for
the simplicial set consisting of two 1-morphisms that touch at their end, hence for
two such 1-morphisms in $K$, then if $g$ had an inverse $g^{-1}$ we could use the above composition operation to compose that with $h$ and thereby find a morphism $f$ connecting the sources of $h$ and $g$. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form
Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in $K$.
In order for this to qualify as an $\infty$-groupoid, this composition operation needs to satisfy an associativity law up to coherent 2-morphisms, which means that we can find the relevant tetrahedrons in $K$. These in turn need to be connected by pentagonators and ever so on. It is a nontrivial but true and powerful fact, that all these coherence conditions are captured by generalizing the above conditions to all dimensions as in the definition of Kan complexes.
In order to conceive of the $k$-simplices for higher $k$ as “globular k-morphism” going from a source to a target one needs a bit of combinatorics. This provided by the orientals (due to Ross Street).
The $k$-oriental $O(k)$ is precisely the prescription for how exactly to think of a $k$-simplex as being a k-morphism in an omega-category. The first few look like this:
In fact, the omega-nerve $N(K)$ of an omega-category $K$ is the simplicial set whose collection of $k$-cells $N(K)_k := Hom(O(k),K)$ is precisely the collection of images of the $k$th oriental $O(k)$ in $K$.
This is fully formally the prescription of how to think of a Kan complex as an $\infty$-groupoid: the Kan complex $C$ is the omega-nerve of an omega-category in which all morphism are invertible:
the $k$-cells in $C_k$ are precisely the collection of $k$-morphisms ihn the omega-category of shape the $k$th oriental $O(k)$;
the horn-filler conditions satisfied by these cells is precisely a reflection of the fact that
there exists a notion of composition of adjacent k-morphisms in the omega-category;
under this composition all $k$-morphisms have an inverse.
This is easy to see in low dimensions:
a 1-cell $\phi \in C_1$ in the simplicial set $C$ has a single source 0-cell $x := d_1 \phi$ and a single target 0-cell $y := d_0 \phi$ and hence may be pictured as a morphism
a 2-cell $\phi \in C_2$ in the simplicial set $C$ has two incoming 1-cells $d_2 \phi, d_0 \phi \in C_1$ and one outgoing 1-cell $d_1 \phi \in C_1$, and if we think of the two incoming 1-cells as representing the composite of the corresponding 1-morphisms, we may picture te 2-cell $\phi$ here as a globular 2-morphism
More in detail, one may think of the incoming two adjacent $1$-cells here as not being the composite of these two morphism, but just as a composable pair, and should think of the existence of the 2-morphism $\phi$ here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism $d_1 \phi$.
So if an $\infty$-groupoid is thought of as a globular ω-category in which all k-morphisms are invertible, then the corresponding Kan complex is the nerve or rather the ω-nerve of this ω-category.
Notably if $C$ is to be regarded as (the nerve of) an ordinary groupoid, every composable pair of morphisms has a unique composite, and hence there should be a unique 2-cell
that is the unique identity 2-morphism
More generally, in a 2-groupoid there may be non-identity 2-morphisms, and hence for any 1-morphism $k _ x_0 \to x_2$ 2-isomorphic to $h$, there may be many 2-morphisms $g \circ f \Rightarrow k$, hence many 2-cells
All we can say for sure is that at least one such 2-cell exists, and that the 2-cells themselves may be composed in some way. This is precisely what the horn-filler conditions in a Kan complex encode.
We have already seen in low dimension how the existence of composites in an $\omega$-category is reflected in the fact that in a Kan-complex certain 2-simplices exist, and how the non-uniqueness of these 2-simplices reflects the existence of nontrivial 2-morphisms.
To see in a similar fashion that the Kan condition ensures the existence of inverses consider an outer horn in $C$, a diagram of 1-cells of the form
In general given such a diagram in a category, there is no guarantee that the corresponding triangle as above will exist in its nerve. But if the category is a groupoid, then it is guaranteed that the missing 1-face can be chose to be the inverse of $f$ composed with the morphism $h$, and there is at least one 2-morphism
A similar analysis for higher dimensional cells shows that the fact that a Kan complex has all horn fillers encodes precisely the fact that it is the omega-nerve of an omega-category in which all k-morphisms for all $k$ are composable if adjacent and have a weak inverse.
We review how 1-groupoids are incarnated as Kan complexes via their nerve.
A (small) groupoid $\mathcal{G}_\bullet$ is
a pair of sets $\mathcal{G}_0 \in Set$ (the set of objects) and $\mathcal{G}_1 \in Set$ (the set of morphisms)
equipped with functions
where the fiber product on the left is that over $\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1$,
such that
$i$ takes values in endomorphisms;
$\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{G}_0)$ the identities; in particular
$s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$;
every morphism has an inverse under this composition.
For $\mathcal{G}_\bullet$ a groupoid, def. 18, we write
for the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$; schematically:
For each $n \geq 1$, the two maps $d_0$ and $d_n$ that forget the first and the last morphism in such a sequence and the $n-1$ maps $d_k$ that form the composition of the $k$th morphism in the sequence with the next one, constitute $(n+1)$ functions denoted
Moreover, the assignments $s_i$ that insert an identity morphism in position $i$ constitute functions denoted
These collections of maps in def. 18 satisfy the simplicial identities, hence make the nerve $\mathcal{G}_\bullet$ into a simplicial set. Moreover, this simplicial set is a Kan complex, where each horn has a unique filler (extension to a simplex).
(A 2-coskeletal Kan complex.)
The nerve operation constitutes a full and faithful functor
Write
for the category of Kan complexes, which is the full subcategory of that of simplicial sets on the Kan complexes.
This means that for $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, an element $f_\bullet \colon X_\bullet \to Y_\bullet$ in the hom-set $Hom_{KanCplx}(X_\bullet,Y_\bullet)$ is
such that
For $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, their mapping space
is the simplicial set given by
The construction in def. 20 defines an internal hom of Kan complexes.
As such it is also common to write $Y^X$ for $Maps(X,Y)$, as well as $[X,Y]$. Notice that the latter notation is sometimes used instead for just the set of connected components of $Maps(X,Y)$.
It follows that the category $KanCplx$ is naturally enriched over itself.
We now have the following immediate generalizations of the corresponding constructions seen above for 1-groupoids.
Write
for the Kan complex which is 1-groupoid with two objects and one nontrivial morphism and its inverse between them. This comes with two inclusions
of its endpoints.
Then for $X_\bullet \in KanCplx$ any other Kan complex, the mapping space $[I,X]_\bullet$ from def. 20 is the path space object of $X_\bullet$.
A 1-cell in the mapping Kan complex $[X_\bullet, Y_\bullet]_\bullet$ is a homotopy between two morphisms of Kan complexes:
For $f_\bullet, g_\bullet \colon X_\bullet \to Y_\bullet$ two morphisms between two Kan complexes, hence $f_\bullet,g_\bullet \in Hom_{KanCplx}(X,Y)$, a (right-)homotopy $\eta \colon f \Rightarrow g$ is a morphism $\eta_\bullet \colon X_\bullet \to [I_\bullet,X_\bullet]_\bullet$ into the path space object of def. 18 such that we have a commuting diagram
Hence a homotopy between two maps $X_\bullet \to Y_\bullet$ of Kan complexes is precisely a 1-cell in the mapping space $[X_\bullet, Y_\bullet]_\bullet$ of def. 20.
We say that a map $X_\bullet \to Y_\bullet$ of Kan complexes is a homotopy equivalence if it has a left and right inverse up to homotopy, hence an ordinary inverse in $\pi_0[X_\bullet, Y_\bullet]$.
For Kan complexes which are 1-groupoids hence which are nerves of groupoids, homotopy equivalence of Kan complexes is equivalently homotopy equivalence of these groupoids according to def. 9.
We may write ∞Grpd for $KanCplx$ regarded as a $KanCplx$-enriched category, hence as fibrant sSet-enriched category.
We write $X$ (without the subscript) for a Kan complex $X_\bullet$ regarded as an object of $\infty Grpd$. As such, $X$ (or its equivalence class) is alse called a homotopy type.
The category ∞Grpd itself “is” the canonical homotopy theory. (For more on this see also at homotopy hypothesis.)
The following is the immediate generalization of def. 10.
Given two morphisms of Kan complexes $X \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y$ their homotopy fiber product
hence the ordinary iterated fiber product over the path space Kan complex, as indicated.
An important class of examples of ∞-groupoids are those which are presented under the Dold-Kan correspondence by chain complexes of abelian groups.
Write $Ch_{\bullet \geq 0}$ for the category of chain complexes (of abelian groups in non-negative degree).
As usual, for $A \in$ Ab an abelian group, we write $A[n]$ for the chain complex with $A$ in degree $n$ and 0 in all other degrees (the suspension of a chain complex).
Write sAb for the category of simplicial abelian groups, hence simplicial objects in abelian groups. Finally write
for the normalized chain complex-functor, which sends a simplicial abelian group $A_\bullet$ to the chain complex whose $n$-chains are the non-degenerate elements of $A_n$ and whose differential is the alternating sum of the face maps of $A_\bullet$:
Of relevance now are the following two standard facts.
Dold-Kan: The functor $N \colon Ch_{\bullet \geq 0} \to sAb$ is an equivalence of categories. The inverse functor
sends a chain complex to the simplicial abelian group whose $n$-simplices are the images of the normalized chain complex $N(\mathbb{Z}(\Delta^n))$ of chains $\mathbb{Z}(\Delta^n)$ the $n$-simplex.
Moore: The forgetful functor $sAb \to sSet$ which sends a simplicial abelian group to its underlying simplicial set factors through Kan complexes
Taken together, this provides us with the following very useful construction.
We write
for the composite of the two functors of prop. 11.
We refer to this as the “Dold-Kan map”, or say “by Dold-Kan”, etc. It provides us with a rich supply of Kan complexes, hence of ∞-groups. In fact, it embeds homological algebra into the homotopy theory of ∞-groupoids in that it is a homotopical functor:
The Dold-Kan map of def. 27 sends quasi-isomorphisms of chain complexes to homotopy equivalences of Kan complexes, def. 22.
Notably we have the following example
For $n \in \mathbb{N}$ write
for the $n$-fold suspension of the discrete circle group, regarded by Dold-Kan as the $n$-fold delooping Kan complex of $\flat U(1)$.
Since the first simplex that contains a non-degenerate $n$-cell is $\Delta^n$, it follows that $\mathbf{B}^n\flat U(1)$ is trivial in degrees $\lt n$
Then since there is a unique non-degenerate $n$-cell in $\Delta^n$ we next have
Next there are $(n+2)$ non-degenrate $n$-simplices in $\Delta^{n+1}$, but known $n+1$ of their images in $\flat U(1)$ determines the last one (as their oriented product), hence next
Similarly for arbitary $k$ the set $(\mathbf{B}^n \flat U(1))_k$ is some Cartesian power of $\flat U(1)$. But cince here we are mostly interested in $\mathbf{B}^n \flat U(1)$ as an n-groupoid, hence only for mapping $n$-groupoids into it, it is mostly enough to just know its cells up to degree $(n+1)$.
Below in Higher Chern-Simons theory the smooth version of the circle group plays a role. In order to amplify that here we are just dealing with the discrete group underlying $U(1)$, we write
for it.
By the Dold-Kan correspondence each $\mathbf{B}^n \flat U(1)$ is naturally an ∞-group and its delooping is indeed $\mathbf{B}^{n+1}\flat U(1)$
For $G$ a discrete group and $A$ a discrete abelian group, there is a natural isomorphism
between the degree-$n$ group cohomology of $G$ with coefficients in $A$ and the connected components of maps of ∞-groupoids from the delooping of $G$ to the $n$-fold delooping of $A$.
This means that local action functionals for higher Dijkgraaf-Witten type theories, hence maps of $\infty$-groupoids of the form $\exp(i S_{DW}^n) \colon \mathbf{B} \flat G \to \mathbf{B}^n \flat U(1)$ are equivalently cocycles $[c] \in H^n_{Grp}(\flat G,\flat U(1))$ in degree-$n$ group cohomology:
In particular the original 3d Dijkgraaf-Witten theory appears this way as the theory of a group 3-cocycle.
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Let $G \in Grp(\mathbf{H})$ be a simply connected compact simple Lie group. and write $\mathbf{B}G \in Smooth\infty Grpd$ for its delooping stack. By the discussion at Lie group cohomology there is a bijection
Write
for a representative of the second Chern class under this bijection. We may regard this map as a local Lagrangian, hence as an object
in the (∞,n)-category of correspondences in the slice (∞,1)-topos $\mathbf{H}_{/B^4 \mathbb{Z}}$ equipped with the phased tensor product.
Since $Corr_3(\mathbf{H}_{/B^4 \mathbb{Z}})^{\otimes_{phased}}$ is an (∞,n)-category with duals, this defines a framed-topological local prequantum field theory
This sends a closed manifold $\Sigma$ of dimension 2 (a surface) to the class of a line bundle
on the moduli stack of flat connections (“local systems”) of $\Sigma$, which is the phase space of $G$-Chern-Simons theory.
By the discussion at cobordism hypothesis (this corollary in view of this proposition) an extension of this to an oriented-topological local preqauntum field theory is equivalent to
choices of $SO(3)$-∞-actions on $\mathbf{B}G$;
choices of equivariant extensions
There are in general not too many $SO(3)$-∞-action on $\mathbf{B}G$, so consider the trivial one. Then
and an equivariant extension is given by a map
hence an element in $H^4(B SO(3), \mathbb{Z})$. This is essentially given by the first Pontryagin class $p_1$. Hence it follows that the extension of $\mathbf{c}_2$-Chern-Simons local prequantum fields theory (on the level of levels) to oriented cobordisms is given by
The moduli space of fields that this assigns to an oriented manifold $\Sigma$ is the space of maps
Hence in codimension-0 this is still just $\mathbf{Loc}_G(\Sigma)$.
In order to trivialize the $p_1$-contribution appearing here, consider the homotopy fiber sequence
exhibiting an $SO(3)$-∞-action on $B^3 \mathbb{Z}$. Then there is an oriented-topological local prequantum field theory with equivariant local Lagrangian
given by the total composite in the diagram
The moduli spaces of fields of this oriented theory on $\Sigma$ now are
with 3-stabilized $p_1$-structures as in Atiyah 2-framing.
A further variant of this is given by the model for the supergravity C-field
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By def. 5, a local prequantum bulk field theory in dimension $(n+1)$ is equivalently a morphism of the form
for any object $\mathbf{Fields} \in \mathbf{H}$.
First we now observe in
that there is a fairly canonical such morphism $S^{n+1}_{tYM}$, namely the “atlas relative to manifolds” of $\flat \mathbf{B}^{n+1}U(1)$ given by the sheaf of closed differential (n+1)-forms. Analyzing what this morphism is like, when regarded as a local prequantum field theory by def. 5, shows that it is the “universal” higher topological Yang-Mills prequantum field theory. What this means becomes clear when we analyze the possible boundary field theories of this theory.
Universal boundary condition for $S_{tYM}$: Differential cohomology and Cheeger-Simons field theory
General boundary conditions: Higher Chern-Weil theory and $\infty$-Chern-Simons theory
where we discuss how the boundary theories for $S^{n+1}_{tYM}$ are precisely the prequantum field theories of higher Chern-Simons theory-type, the ∞-Chern-Simons theories. These include ordinary 3d Chern-Simons theory, higher dimensional Chern-Simons theory on ordinary gauge fields but also higher Chern-Simons theory on higher gauge fields such as the String 2-group 7-dimensional Chern-Simons theory, the AKSZ sigma-models, and also closed string field theory.
This shows how $\infty$-Chern-Simons theories arise canonically as precisely the local boundary field theories of the canonical local field theory which exists in any differential cohesive (∞,1)-topos.
Continuing in this vein we can then work out what all the further higher codimension boundary field theories and defect field theories of this universal higher topological Yang-Mills theory and hence of ∞-Chern-Simons theories are. We find
which are given by generalized “Bohr-Sommerfeld isotropic subspaces” of the moduli stacks of $\infty$-Chern-Simons fields.
Then…
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There is a special and especially simple map to the coefficient object $\flat \mathbf{B}^{n+1} U(1)$ for flat local action functionals/prequantum n-bundles, namely the map
which in components is the inclusion of closed differential forms into de Rham hypercohomology cocoycles.
We here introduce and describe this map and then regard it as a local action functional of a local prequantum field theory according to def. 5. Below in Higher Chern-Simons prequantum field theory we find that this field theory is such that close to its boundaries it looks like (higher) topological Yang-Mills theory for every possible higher gauge group and every possible invariant polynomial on it, as one considers every possible boundary condition. Therefore we here refer to this as the “universal topological Yang-Mills theory”.
The notion of a sheaf of chain complexes or equivalently of a chain complex of sheaves over a fixed topological space has a long tradition in homological algebra. Many sheaves however are naturally considered not on one fixed space, but on “all of them”. For instance differential forms in any degree may be “pulled back” along any smooth function between smooth manifolds. Accordingly if we regard the whole category SmoothMfd of smooth manifolds as a replacement for and generalization of the category of open subsets of any given one, then differential forms constitute a sheaf on that site, hence a functor
In particular for $n = 0$ this is just the sheaf of smooth functions
One way to think of this is that this sheaf is the real line $\mathbb{R}$ first of all as a set – which is the value of $\underline{\mathbb{R}}$ on the point $\ast$ – and secondly equipped with its canonical smooth structure which is encoded by the system of all sets $C^\infty(X,\mathbb{R})$ of smooth functions from any smooth manifold $X$, and the information $C^\infty(\phi, \mathbb{R}) \colon C^\infty(Y, \mathbb{R}) \to C^\infty(X,\mathbb{R})$ of how these functions pull back (precompose) along any smooth function of test manifolds $\phi \colon X \to Y$.
Regarding the smooth manifold $\mathbb{R}$ this way means regarding it as what is sometimes called a diffeological space, and what here more generally call a smooth space.
Therefore we will often just write $\mathbb{R}$ when we really mean the sheaf $\underline{\mathbb{R}}$ represented by it.
The Dold-Kan map of def. 27 directly extends to (pre-)sheaves which regard a (pre-)sheaf of chain complexes as a presheaf of Kan complexes
By the previous reasoning, a presheaf of Kan complexes on the category of smooth manifolds, we may think of as being a plain Kan complex, hence ∞-groupoid (the value of the presheaf on the point), together with a rule for what the smooth functions into it are, for each smooth testmanifold. Hence we think of it as a smooth ∞-groupoid.
A basic example is a Lie groupoid $\mathcal{G}_\bullet \in Grpd(SmoothMfd)$, which represents a presheaf of Kan complexes on smooth manifolds by
Previously we have considered higher Dijkgraaf-Witten as taking place in the homotopy theory of plain ∞-groupoids (geometrically discrete ∞-groupoids). Now we would like to have “a homotopy theory of smooth ∞-groupoids”.
(A modern term for “a homotopy theory” is “an (∞,1)-category”, but for a heuristic idea of what is going on some readers may find it helpful to think of “a homotopy theory” instead.)
In order to define “a homotopy theory” of smooth ∞-groupoids is, to be denoted Smooth∞Grpd, we need to say what a homotopy and hence what a homotopy equivalence is supposed to be. Since smooth structure should be a local property witnessed on small smooth open balls, we declare that
is the homotopy theory obtained by “universally turning stalk-wise homotopy equivalences of Kan complexes, def. \ref{spring}, into actual homotopy equivalences”. The formal definition of this idea is called the simplicial localization of $pSh(SmoothMfd, KanCplx)$ at the stalkwise homotopy equivalences.
We call ∞Grpd also the differential cohesive ∞-topos of smooth ∞-groupoids. For brevity and since most everything we discuss in the following holds for arbitrary differential cohesive (∞,1)-toposes, we from now on denote it by
We now immediately turn to a simple example that illustrates some basic aspects of this construction.
As a first example for how to work in the homotopy theory of smooth ∞-groupoids we have the following.
The canonical chain map
where the left map includes the sheaf of (locally) constant $U(1)$-valued functions into that of all $U(1)$-valued smooth functions, is a stalk-wise quasi-isomorphism. Hence under the Dold-Kan correspondence, def. 27, both present the same smooth ∞-groupoid
By definition and using prop. 12 we need to check that over a small enough smooth open ball, the chain map becomes a quasi-isomorphism. But on an open ball this is the statement of the Poincaré lemma.
Accordingly we have a canonical inclusion:
For $n \in \mathbb{N}$, write
for the sheaf of closed smooth differential forms of degree $n$, regarded as a smooth space, regarded as a smooth ∞-groupoid. Write
for the image in morphisms of Smooth∞Grpd under the Dold-Kan correspondence, def. 27, of the chain map
The map $\Omega^n_{cl} \to \flat \mathbf{B}^n U(1)$ in def. 29 may be characterized more abstractly as follows:
for every smooth manifold $\Sigma$, theinduced morphism of mapping spaces
is a 1-epimorphism, hence a stalk-wise epimorphism on connected components.
By def. 5 we may now regard the map
of prop. 29 as a local action functional for an $(n+1)$-dimensional local prequantum field theory. We call this the “universal higher topological Yang-Mills theory” for reasons that become clear as we anaylize its boundary field theories now.
We consider now the boundary field theories for the “universal topological Yang-Mills theory” of def. 30.
Where the plain (∞,n)-category of cobordisms $Bord_n$ is freely generated from the point $\ast$ alone, so the $(\infty,n)$-category of cobordisms with possibly a marked boundary is freely generated from the point and one new morphism
which we think of as being the interval $D^1$ with one end “marked”. Now a local field theory with local action functional according to def. 5 encodes not only the value on the point, which we now take to be
but moreover one morphism in $Span_n(\mathbf{H}, \mathbf{B}^{n+1}\flat U(1))$ from the trivial field configuration with trivial action to this data, hence (as amplified in FV) a diagram in $\mathbf{H}$ of the form
Therefore defining such boundary data means defining a moduli stack $\mathbf{Fields}^{\partial}$ of boundary field configurations, together with a homotopy filling the above diagram which encodes the relative action functional on this boundary data.
In order to find all possible such boundary data for $\exp(i S_{tYM})$, we can make use of the homotopy fiber product construction of def. 24 to find the universal such boundary data, the one through which any other one factors.
The universal boudnary condition for $\exp(i S_{tYM})$, hence the homotopy fiber product $\ast \underset{\mathbf{B}^{n+1}\flat U(1)}{\times} \Omega^{n+1}_{cl}$, is given by the image under the Dold-Kan map, def. 27, of the Deligne complex
hence the universal boundary data is
The boundary field theory defined this way we may call Cheeger-Simons field theory.
By the universal property of the homotopy fiber product we then have the following statement.
Boundary field theories for $\exp(i S_{tYM})$ are equivalently moduli stacks $\mathbf{Fields} \in \mathbf{H}$ equipped with maps
hence equipped with a circle n-bundle with connection (the prequantum n-bundle of the boundary theory).
Moreover, the universal property of the Cheeger-Simons field theory identifies all these boundary theories as being of higher Chern-Simons type, in that they have a curvature associated to them which is an invariant differential form (invariant polynomial) on the moduli stack
We call these theories of ∞-Chern-Simons theory-type.
For $G$ a simply connected simple Lie group and
the moduli stack of $G$-principal connections, the local action functional of ordinary 3d Chern-Simons theory is of the form
This prequantum 3-bundle is the Chern-Simons circle 3-bundle.
Many more examples… e.g. 7d Chern-Simons theory, AKSZ sigma-model, etc….
While we may think of the (∞,n)-category of cobordisms $Bord_n$ as built from smooth manifolds, the cobordism theorem clearly states that these just serve as a presentation of a structure that is not intrinsically related to smooth or even topological structure. This is made manifest by prop. 2: the value of a local prequantum field theory on a k-morphism in $Bord_n$ depends only on the homotopy type of the $k$-dimensional manifold that presents this $k$-morphism. Of course this is precisely the property that the term “topological” in topological field theory is referring to.
But boundaries and defects of a topological field theory may add extra structure to the theory which is not “purely topological” in this way. Here we consider a canonical class of defects for universal higher topological Yang-Mills theory, def. 30, and $\infty$-Chern-Simons theory, def. 17, which implements the expected higher parallel transport/higher holonomy of the higher Chern-Simons type action functionals over actual smooth manifolds.
For $k \leq n$ and for $\Sigma_k$ and oriented closed manifold, there is a morphism of smooth moduli stacks
which is compatible with the standard fiber integration of differential forms and with transgression in ordinary cohomology in that it fits into a commuting diagram
More generally, if $\Sigma_k$ is a manifold with boundary then there is a diagram
where the bottom left map is the fiber integration from before, applied to the closed boundary, and where the homotopy filling the diagram is such that it reproduces this fiber integration in the case that the boundary is empty, in that
This follows by unwinding the traditional formulas for fiber integration in differential cohomology, reformulating them in homotopy theory and observing that they are natural in their arguments, hence extend to morphisms of higher stacks, as discussed here.
A homotopy/gauge equivalence between a circle n-bundle with connection $(P,\nabla)$ and a trivial circle $n$-bundle with connection given by a globally defined differential form $(0,\omega)$ is equivalently a section/trivialization of the underlying circle n-bundle. Therefore the above says that the fiber integration of an $n$-connection over a manifold with boundary is equivalently a section of the transgression of the underlying bundle to the boundary.
We may combine this with the $\infty$-Chern-Simons action functional:
Let $\exp(i S_{CS}) \colon \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}$ be an ∞-Chern-Simons theory local action functional as in prop. 17. Then for $\Sigma_k$ an oriented smooth $k$-dimensional manifold with boundary, the corresponding transgression defect is the pasting-composite
or rather its further pullback to the shape modality
This is a defect between the boundary transgression, def. 18, of the $\infty$-Chern-Simons theory and the tautological higher differential higher Chern-Simons theory.
We see below that both the Wess-Zumino-Witten theory as well as Wilson lines in Chern-Simons theory arise from transgression defects this way.
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defect given by transgression over circle
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defect given by transgression over sphere
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locality of local field theory is obsrvationally related to causal locality
traditional Lagrangian mechanics and Hamiltonian mechanics are naturally embedding into higher prequantum field theory by the notion of prequantized Lagrangian correspondences
quantization of prequantum field theory is naturally formulated in terms of motivic quantization
The idea of formulating local prequantum field theory by spans in a slice over a “space of phases” in higher geometry has been expressed in the unpublished note
A formulation of the idea for Dijkgraaf-Witten theory-type field theories is indicated in section 3 of
based on the considerations in section 3.2 of
Based on the general formulation of the more general field theory with defects described in section 4.3 there, in
the structure of such domain walls/defects/branes are analyzed in the prequantum theory, hence with coefficients in an (∞,n)-category of spans.
The study of local prequantum ∞-Chern-Simons theory with its codimension-1 ∞-Wess-Zumino-Witten theory and codimension 2-Wilson line-theory in this fashion, in an ambient cohesive (∞,1)-topos is discussed in
Much of the content of this entry here serve as, or arose as, lecture notes for
This forms one section of the more comprehensive lecture notes at