I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.
Alexander Grothendieck, Récoltes et semailles, 1985–1987, pp. 552-3-1 (“The Rising Sea”)
I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory.
The tools of category theory and higher category theory serve to organize other structures. There is a plethora of applications that have proven to be much more transparent when employing the nPOV. Higher category theory has helped foster entire new fields of study that would have been difficult to conceive otherwise. This page lists and discusses examples.
The following is a (incomplete) list of examples of topics for which higher category theory has proven to be useful.
The field of differential geometry has long managed to avoid the change to an $n$-point of view that had been found to be unavoidable, natural and fruitful in algebraic geometry long ago. But more recently – not the least due to the recognition of differential higher geometric structures in the physics of gauge theory and supergravity (such as that of orbifolds and orientifolds, of smooth gerbes and smooth principal ∞-bundles) – sheaf and topos theoretic concepts, such as synthetic differential geometry, diffeological spaces and differentiable stacks are gaining wider recognition and appreciation.
For instance the ordinary category Diff of smooth manifolds fails to have all pullbacks, it only has pullbacks along transversal maps. This observation is usually the starting point for realizing that differential geometry is in need of a bit of category theory in the form of higher geometry.
In all notions of generalized smooth spaces all pullbacks do exist. But they may still not be the “right” pullbacks. For instance cohomology of pullback objects may not have the expected properties. This is solved by passing to smooth derived stacks, such as derived smooth manifolds.
Recent developments in higher category theory, such as the concept of higher Structured Spaces based on Higher Topos Theory, put all these notions of generalized geometries into a unified picture of higher geometry that realizes old ideas about how category theory provides a language for space and quantity in great detail and powerful generality and sheds new light on old classical problems such the description of the derived moduli stack of derived elliptic curves and the construction of the tmf spectrum from it. This construction has benefited tremendously from the adoption of the nPOV. Using this point of view, the general strategy becomes naturally evident.
Much of topological vector space theory, e.g., the theory of distributions, nuclear spaces, etc. has its origins in partial differential equation theory and is intensely conceptual (categorical) in spirit. It is routine these days to accept distributional solutions, but it wasn’t always so, and it was the efficacy of the abstract TVS theory which changed people’s minds.
Way back Cartan studied differential equations in terms of exterior differential systems. From the $n$POV, these may be understood naturally as sub Lie ∞-algebroids of a tangent Lie algebroid.
Bill Lawvere noticed in the 1960s that the notion of differential equation makes sense in any smooth topos (as described here). In his highly influential article Categorical dynamics he promoted the point of view that all things differential geometric can be formulated in abstract category theory internal to a suitable topos. This is the origin of synthetic differential geometry. It may be understood as providing the fundamental characterization of the notion of the infinitesimal.
Closely related to both these perspectives, a modern point of view on differential equations that is proving to be very fruitful regards them as part of the theory of D-modules.
A multitude of notions of cohomology and its variants are unified from the $n$POV when viewed as ∞-categorical hom-spaces in (∞,1)-topoi. See cohomology.
Specifically, the subject of Hochschild cohomology, when generalized to higher order Hochschild cohomology effectively merges into the canonical concept of (∞,1)-powering of an (∞,1)-topos over ∞Grpd. See Hochschild cohomology for details.
The study of homotopy theory originated in the study of categories such as those of topological spaces and other objects such as chain complexes whose morphisms were known to admit a notion of homotopy. Historically, in a sequence of steps formalisms were proposed that would organize the rich interesting structure found in such situations. As a first approximation the notion of homotopy category and derived category was introduced in order to deal with structures “up to homotopy”. But it was clear that the homotopy category captured only a very small part of the interesting information. Quillen introduced the notion of model category as a formalization of the full structure, and this formalization turned out to yield a powerful theory that today provides a powerful toolset for dealing with homotopy theoretic situations.
But also the notion of model category was seen to not be the full answer. For instance a model category in a sense retains too much non-intrinsic information. Equivalence classes of model categories under Quillen equivalence are a more intrinsic characterization of a given homotopy theory. But this means that one needs some higher categorical notion for the collection of all model categories. This problem came to be known as the search for the homotopy theory of homotopy theories.
Recently, this problem was fully solved and homotopy theory fully understood as the special case of higher category theory that deals with (∞,1)-categories:
the notion of model category, in particular when refined to that of a simplicial model category serves as a presentation of the notion of (∞,1)-category;
the “homotopy theory of homotopy theories” is accordingly the (∞,1)-category of (∞,1)-categories $(\infty,1)Cat$;
better yet: there is an (∞,2)-category of all $(\infty,1)$-categories;
in $(\infty,1)Cat$ two $(\infty,1)$-categories presented by model categories are equivalent precisely if the presenting model categories may be connected by a zig-zag sequence of Quillen equivalences;
all “homotopy”-constructions in model category theory, such as homotopy limits, mapping cones etc. are tools for constructing the corresponding higher categorical intrinsic notions, such as limit in an (∞,1)-category.
all variant notions find their intrinsic higher categorical interpretation this way: for instance stable homotopy theory is the study of stable (∞,1)-categories;
the homotopy category of a model category is simply the decategorification of the corresponding $(\infty,1)$-category to just a 1-category;
and for instance the notion of homotopy category of a stable $(\infty,1)$-category reproduces the notion of triangulated category, thus incorporating also a large toolset from homological algebra into the picture.
… The study of rational homotopy theory is naturally understood as the study of the localizations of (∞,1)-toposes at morphism that induce equivalences in cohomology with certain line-object coefficients. See rational homotopy theory in an (∞,1)-topos. …
In full generality, (algebraic) K-theory is a universal assignment of spectra to stable (∞,1)-categories.
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… see Tannaka duality …
See at
cohesion on (∞,1)-toposes solves the Simons-Sullivan question characterization on the characterization of generalized (Eilenberg-Steenrod-type) differential cohomology. See at differential cohomology hexagon for details.
In deformation theory it was early on recognized that for a good theory the notion of Kähler differentials has to be generalized to the notion of cotangent complex. With the advent of the study of derived moduli spaces, such as the derived moduli space of derived elliptic curves, this needed to be further generalized to notions of cotangent complexes not just of rings, but of E-∞-rings.
It turns out that all these concepts are special cases of a construction obtained from a simple higher categorical notion, that of left adjoint sections of a tangent (∞,1)-category.
While it is common to view logic as the study of absolute truth, in fact logic can have many different interpretations, or semantics. A particular semantics for logic can be useful both to inform the study of logic, and to prove facts logically about the semantics. One very fruitful semantics of this sort is categorical semantics for logic and type theory, according to which every category (and especially every topos) has an internal language and internal logic. Interpreting “ordinary” mathematical statements in the internal language of exotic categories can make it much easier to study those categories, while on the other hand it can provide new insight into otherwise mysterious logical notions.
In particular, the internal logic of a category (such as a topos) is, in general, constructive, i.e. the principle of excluded middle (and also stronger statements, such as the axiom of choice) are generally false. Thus, in order for a theorem to be interpretable internally in such categories, its proof must be constructive. So while the original “constructivists” believed that classical mathematics was “wrong,” nowadays there are good reasons to care about constructive mathematics even if one believes that excluded middle and the axiom of choice are “true,” since regardless of their “global” truth they will not be true in the internal logic of many interesting categories. Conversely, category-theoretic models have provided new insight into the independence of various axioms in constructive mathematics, such as differing forms of the axiom of choice.
As another example, the identity types in Martin-Löf’s original constructive dependent type theory construct, from any type $A$ and terms $a, b \in A$, a new type $Id_A(a, b)$. According to the propositions as types interpretation, the elements of $Id_A(a,b)$ are proofs that $a$ and $b$ are propositionally equal; thus $Id_A(a,b)$ is a replacement for the truth value of the proposition $(a=b)$. There are type-theoretic functions $1 \to Id(a, a)$, $Id(b, c) \times Id(a, b) \to Id(a, c)$ and $Id(a, b) \to Id(b, a)$ expressing the reflexivity, transitivity and symmetry of this propositional equality, but in general an identity type (even the “reflexive” identity type $Id(a,a)$) can have many distinct elements. This has long been a source of discomfort to type theorists. However, from a higher-categorical point of view, it is natural to view the terms of identity types as isomorphisms in a groupoid—or, more precisely, an ∞-groupoid, since identity types have their own identity types, and all the laws of associativity, exchange, etc. only hold up to terms of these higher identity types. This suggests that the nonuniqueness of identity proofs should be embraced rather than denigrated, producing a theory at least related to the “internal logic” of (∞,1)-category theory and homotopy theory; see identity type for more details.
This is now known as homotopy type theory, see there for more.
See also higher category theory and physics.
By the end of the 19th century a fairly complete, powerful and elegant mathematical formulation of classical mechanics: in terms of symplectic geometry. By the middle of the 20th century, the passage to the corresponding quantum theory was pretty well modeled by the geometric quantization of symplectic geometries.
But there were some loose ends. Notably the fully general theory involved Poisson manifolds, not just symplectic manifolds. And the mechanics of relativistic classical field theory was realized to be more naturally described by multisymplectic geometry.
Both these generalizations have a natural common higher categorical formulation: that of Lie ∞-algebroids: a Poisson geometry is naturally encoded in its corresponding Poisson Lie algebroid. Its higher categorical versions – the n-symplectic manifolds – encode the corresponding multisymplectic geometry.
Moreover, the quantization step of geometric quantization was understood to be effectively the Lie integration of these Lie ∞-algebroids to the corresponding Lie ∞-groupoids (currently this is well understood for low $n$).
The basic structure of quantum mechanics and quantum information theory is encoded in the theory of dagger-compact categories.
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Maxwell realized that the electromagnetic field is controlled by a degree 2-cocycle in de Rham cohomology: the electromagnetic field strength. Later Dirac noticed that this is one part of a degree 2-cocycle in differential cohomology that characterize a connection on a line bundle.
Later the Yang-Mills field was understood to similarly be a connection on a bundle, this time on a $G$-principal bundle for $G$ some possibly nonabelian group.
While thinking about the mathematical structures possibly underlying standard model of particle physics and gravity, theoretical physicists considered more general hypothetical gauge fields, such as the Kalb-Ramond field, the RR-field or the supergravity C-field. Today all these gauge fields are understood to be modeled, mathematically, by generalized differential cohomology.
Theories of supergravity have been known to require higher gauge fields in the above sense – hence the term supergravity C-field. A powerful formalism for handling these theories is the D'Auria-Fre formulation of supergravity. As described there, this is secretly (but evidently) nothing but a description of supergravity as a theory of connections on nonabelian $G$-principal ∞-bundles for $G$ some super Lie ∞-group. For instance Cremmer-Scherk 11-dimensional supergravity theory is governed by the super Lie 3-group $G$ whose L-∞-algebra is the supergravity Lie 3-algebra.
The BV-BRST formalism is secretly a way to talk about the fact that configuration spaces of gauge theories are not naive spaces such as manifolds, but are general spaces in the sense of higher geometry:
the configuration space is really an object $Conf \in Sh_{(\infty,1)}((dgAlg^-)^{op})$ in the ∞-stack (∞,1)-topos on the (∞,1)-site $(dgAlg^-)^{op}$ of certain ∞-algebras modeled as dg-algebras. The BV-BRST-complex of a physical system is the global derived function algebra
(many more aspects go here, eventually)…
There are essentially two axiomatizations of what quantum field theory is, both of which are inherently $\infty$-categorical:
in the FQFT picture – the Schrödinger picture – a quantum field theory is described as an (∞,n)-functor on an (∞,n)-category of cobordisms. The cobordism hypothesis – now a theorem that characterizes central properties of these (∞,n)-categories, has been a major driving force in the development of higher category theory.
in the AQFT/factorization algebra picture – the Heisenberg picture – a quantum field theory is described as an $\infty$-copresheaf of observables on its parameter space.
3-dimensional TFT such as Chern-Simons theory and Dijkgraaf-Witten theory and the global aspects of 2-dimensional conformal field theory are inherently governed by the theory of modular tensor categories.
The local aspects of 2-dimensional conformal field theory are governed by vertex operator algebras. A vertex operator algebra is really the algebra over an operad, for the operad of holomorphic pointed spheres (as described there).
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