## Posts filed under ‘algebraic topology’

### Monads in mathematics 4 : the bar and cobar constructions

The use of monads and comonads in homological algebra is as old as the theory: Godement’s standard construction refers to the use of monads in cohomology theory, and is said to be the first study of a general method for constructing acyclic resolutions. Later the theory was explored in greater depth by Eilenberg, Moore, Barr, Beck. The term bar resolution is now most commonly used to describe the process. Monads derived from operads were also studied by Lawvere, Mitchell, Bénabou, under the name of algebraic theories, with aim towards topoi and logic.

Bar resolutions (see J. Baez website and the LNM Seminar on Triples and Categorical Homology Theory) are a way to canonically (i.e. functorially) describe an arbitrary algebra over a monad using only free algebras. It is a particular case of definition with generators and relations which is often the only way to describe infinite mathematical objects with finite expressions (computer algebra systems usually deal with finitely generated objects, and use generators and relations to describe their elements), but bar resolutions somehow are the universal way of doing this.

Most cohomology theories fit in the framework of bar constructions, though in various apparently unrelated ways. However, a visible common denominator of most constructions is the use of simplicial methods. This makes them of some use in homotopical algebra: bar resolutions are used to define canonical cofibrant resolutions of objects, which explains their uses in definition of derived functors.
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### Ten constructions of the cohomology of varieties

When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitions lead to equivalent results). The case of differentiable manifolds or algebraic varieties is particularly impressive, since there were a lot of equivalent cohomology theories defined during last century in order to simplify proofs or allow generalisations. Most cohomology theories, if not all, are defined as the cohomology of a complex : i.e. a sequence of vector spaces or modules $(V_n)$ with a boundary map $d$. The kernel of $d$ is called the set of cycles, while the images by $d$ are called boundaries : the cohomology $H^n$ is then the quotient of cycles in $V_n$ by the subspace of boundaries.

Classical topologists, for example, will use preferably (see MacLane, Homology or the book of Allen Hatcher) :

• simplicial (co)homology : it is defined for a triangulated space, i.e. the manifold is cut by curves, surfaces, etc. which make it isomorphic to a sort of polyhedron (a complex); simplicial homology describe non-triviality (holes) in the combinatorial structure of this polyhedron; here the boundary map is really the boundary map.
• singular (co)homology : a more abstract version of simplicial homology; now we consider the set of all possible simplexes (curves, polygons, polyhedra, and their generalisations…) drawn on the manifold; this definition was given by Eilenberg; I don’t know who first defined simplicial homology, but MacLane mentions that Poincaré and Noether gaves important contributions to this theory.

Then come sheaves, which were explicitly defined by Leray (tales for young mathematicians help remember that Leray made considerable efforts as a prisoner in concentration camps to focus his work on especially “useless” subjects to avoid helping Nazis). (more…)

### Extensions of sheaves and the local-global spectral sequence

GIven a topological space $X$, algebraic topologists would sometimes be interested in sheaves over this space. In most cases, these sheaves are sheaves of functions with some special property, or sheaves of modules over these rings. We could casually say thet the notion of sheaf is some mix of topology and algebra (more generally, categories), which allows to remember where are localized objects. For example, the set of continuous functions over a compact space can sometimes allow to recover an isomorphic space, but its mere normed algebra structure does not immediately say things like: “i am greater here than there”, “i am positive here and negative there”…

So a sheaf is not only determined by a set of fonctions (the commonly used term is section) defined over the whole space, but also by sections over open sets, and by the restriction maps between these various sets of sections. A sheaf has also a gluing theory : if we choose sections over open sets with equal restrictions on the intersections, they must glue to a section over the union of the open sets (if not, our object does not deserve to be a sheaf, so it is only a presheaf).

Virtually anything which can be restricted to open sets and defined locally defines a sheaf: continuous functions form a sheaf, but functions with $||f||_{\infty} = 1$ do not usually form a sheaf (this property is not local). A sheaf of abelian groups is a sheaf of things which can be (locally) added and substracted, a sheaf of rings has sections which can be multiplied, and we can also define sheaves of modules over a sheaf of rings.

A morphism of sheaves $f: F \to G$ is a map which is determined by the image of localized sections. In the case of abelian groups, the kernel sheaf is the sheaf of sections of F which are locally in the kernel of f, while the quotient sheaf (cokernel) is something looking like G where sections are identified if locally it should be so. A morphism is thus injective or surjective if locally it is so. Sheaves of abelian groups (or sheaves of modules over some sheaf of rings) form an abelian category, and it makes sense to speak of the extension group $\mathrm{Ext}^1(F,G)$ for two such sheaves F and G. Suppose we are working with sheaves of modules (especially coherent sheaves) over a sheaf of rings $\mathcal O_X$. How should we compute this group ?