Of Topos and Torsors

Let X be a topological space. One goal of algebraic topology is to study the topology of X by means of algebraic invariants, such as the singular cohomology groups Hⁿ(X;G) of X with coefficients in an abelian group G. These cohomology groups have proven to be an extremely useful tool, due largely to the fact that they enjoy excellent formal properties, and the fact that they tend to be very computable. However, the usual definition of Hⁿ(X;G) in terms of singular G-valued cochains on X is perhaps somewhat unenlightening. This raises the following question: can we understand the cohomology group Hⁿ(X;G) in more conceptual terms?

As a first step toward answering this question, we observe that Hⁿ(X;G) is a representable functor of X. That is, there exists an Eilenberg-MacLane space K(G,n) and a universal cohomology class η ∈ Hⁿ(K(G,n);G) such that, for any topological space X, pullback of η determines a bijection

[X, K(G, n)] → Hⁿ(X; G)

Here [X,K(G,n)] denotes the set of homotopy classes of maps from X to K(G,n). The space K(G,n) can be characterized up to homotopy equivalence by the above property, or by the the formula π_kK(G,n)≃ ∗ if k̸ ≠ n

or

G if k = n.

In the case n = 1, we can be more concrete. An Eilenberg MacLane space K(G,1) is called a classifying space for G, and is typically denoted by BG. The universal cover of BG is a contractible space EG, which carries a free action of the group G by covering transformations. We have a quotient map π : EG → BG. Each fiber of π is a discrete topological space, on which the group G acts simply transitively. We can summarize the situation by saying that EG is a G-torsor over the classifying space BG. For every continuous map X → BG, the fiber product X^~ : EG × BG X has the structure of a G-torsor on X: that is, it is a space endowed with a free action of G and a homeomorphism X_~/G ≃ X. This construction determines a map from [X,BG] to the set of isomorphism classes of G-torsors on X. If X is a well-behaved space (such as a CW complex), then this map is a bijection. We therefore have (at least) three different ways of thinking about a cohomology class η ∈ H¹(X; G):

(1) As a G-valued singular cocycle on X, which is well-defined up to coboundaries.

(2) As a continuous map X → BG, which is well-defined up to homotopy.

(3) As a G-torsor on X, which is well-defined up to isomorphism.

The singular cohomology of a space X is constructed using continuous maps from simplices ∆^k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect singular cohomology to tell us very much about X. The second definition uses maps from X into the classifying space BG, which (ultimately) relies on the existence of continuous real-valued functions on X. If X does not admit many real-valued functions, then the set of homotopy classes [X,BG] is also not a very useful invariant. For such spaces, the third approach is the most powerful: there is a good theory of G-torsors on an arbitrary topological space X.

There is another reason for thinking about H¹(X;G) in the language of G-torsors: it continues to make sense in situations where the traditional ideas of topology break down. If X is a G-torsor on a topological space X, then the projection map X → X is a local homeomorphism; we may therefore identify X with a sheaf of sets F on X. The action of G on X determines an action of G on F. The sheaf F (with its G-action) and the space X (with its G-action) determine each other, up to canonical isomorphism. Consequently, we can formulate the definition of a G-torsor in terms of the category ShvSet(X) of sheaves of sets on X without ever mentioning the topological space X itself. The same definition makes sense in any category which bears a sufficiently strong resemblance to the category of sheaves on a topological space: for example, in any Grothendieck topos. This observation allows us to construct a theory of torsors in a variety of nonstandard contexts, such as the étale topology of algebraic varieties.

Describing the cohomology of X in terms of the sheaf theory of X has still another advantage, which comes into play even when the space X is assumed to be a CW complex. For a general space X, isomorphism classes of G-torsors on X are classified not by the singular cohomology H¹_sing(X;G), but by the sheaf cohomology H¹_sheaf(X; G) of X with coefficients in the constant sheaf G associated to G. This sheaf cohomology is defined more generally for any sheaf of groups G on X. Moreover, we have a conceptual interpretation of H¹_sheaf(X; G) in general: it classifies G-torsors on X (that is, sheaves F on X which carry an action of G and locally admit a G-equivariant isomorphism F ≃ G) up to isomorphism. The general formalism of sheaf cohomology is extremely useful, even if we are interested only in the case where X is a nice topological space: it includes, for example, the theory of cohomology with coefficients in a local system on X.

Let us now attempt to obtain a similar interpretation for cohomology classes η ∈ H² (X ; G). What should play the role of a G-torsor in this case? To answer this question, we return to the situation where X is a CW complex, so that η can be identified with a continuous map X → K(G,2). We can think of K(G,2) as the classifying space of a group: not the discrete group G, but instead the classifying space BG (which, if built in a sufficiently careful way, comes equipped with the structure of a topological abelian group). Namely, we can identify K(G, 2) with the quotient E/BG, where E is a contractible space with a free action of BG. Any cohomology class η ∈ H²(X;G) determines a map X → K(G,2), and we can form the pullback X^~ = E × _BG X. We now think of X as a torsor over X: not for the discrete group G, but instead for its classifying space BG.

To complete the analogy with our analysis in the case n = 1, we would like to interpret the fibration X → X as defining some kind of sheaf F on the space X. This sheaf F should have the property that for each x ∈ X, the stalk F_x can be identified with the fiber X^~_x ≃ BG. Since the space BG is not discrete (or homotopy equivalent to a discrete space), the situation cannot be adequately described in the usual language of set-valued sheaves. However, the classifying space BG is almost discrete: since the homotopy groups π_iBG vanish for i > 1, we can recover BG (up to homotopy equivalence) from its fundamental groupoid. This suggests that we might try to think about F as a “groupoid-valued sheaf” on X, or a stack (in groupoids) on X.

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Political Ideology Chart

It displays anarchism (lower end) and authoritarianism (higher end) as the extremes of another (vertical) axis as a social measure while left-right is the horizontal axis which is an economic measure.

Anarchism is about self-governance, having as little hierarchy as possible. As you go to the left, the means of production are distrubuted more equally; and as you go to the right, individuals and corporations own more of the means of production and accumulate capital.

On the upper left you have an authoritarian state, distributing the means of production to the people as equally as possible; on the lower left you have the collectives, getting together voluntarily utilizing their local means of production and sharing the products; on the lower right you have anarchocapitalists, with no state, tax or public service, everything operated by private companies in a completely free and global market; and finally on the top right you both have powerful state and corporations (pretty much all the countries).

But after all, these terms change meanings through history and different cultures. Under unrestrained capitalism the accumulation of wealth both creates monopolies and more importantly political influence. So that influences state interference and civil liberties also. It also aspires for infinite growth which leads to the depletion of natural resources which is another diminishing fact for the quality of living for the people. At that point it favors conservatism rather than progressive scientific thinking. Under collective anarchism, since it’s localized, it is quite difficult to create global catastrophes, and this is why in today’s world, the terms anarchism and capitalism seems as opposite.

Holism. Note Quote.

It is a basic tenet of systems theory/holism as well as of theosophy that the whole is greater than the sum of its parts. If, then, our individual minds are subsystems of larger manifestations of mind, how is it that our own minds are self-conscious while the universal mind (on the physical plane) is not? How can a part possess a quality that the whole does not? A logical solution is to regard the material universe as but the outer garment of universal mind. According to theosophy the laws of nature are the wills and energies of higher beings or spiritual intelligences which in their aggregate make up universal mind. It is mind and intelligence which give rise to the order and harmony of the physical universe, and not the patterns of chance, or the decisions of self-organizing matter. Like Capra, the theosophical philosophy rejects the traditional theological idea of a supernatural, extracosmic divine Creator. It would also question Capra’s notion that such an extracosmic God is the self-organizing dynamics of the physical universe. Theosophy, on the other hand, firmly believes in the existence of innumerable superhuman, intracosmic intelligences (or gods), which have already passed through the human stage in past evolutionary cycles, and to which status we shall ourselves one day attain. There are two opposing views of consciousness: the Western scientific view which considers matter as primary and consciousness as a by-product of complex material patterns associated with a certain stage of biological evolution; and the mystical view which sees consciousness as the primary reality and ground of all being. Systems theory accepts the conventional materialist view that consciousness is a manifestation of living systems of a certain complexity, although the biological structures themselves are expressions of “underlying processes that represent the system’s self-organization, and hence its mind. In this sense material structures are no longer considered the primary reality” (Turning Point). This stance reaffirms the dualistic view of mind and matter. Capra clearly believes that matter is primary in the sense that the physical world comes first and life, mind, and consciousness emerge at a later stage. That he chooses to call the self-organizing dynamics of the universe by the name “mind” is beside the point. If consciousness is regarded as the underlying reality, it is impossible to regard it also as a property of matter which emerges at a certain stage of evolution. Systems theory accepts neither the traditional scientific view of evolution as a game of dice, nor the Western religious view of an ordered universe designed by a divine creator. Evolution is presented as basically open and indeterminate, without goal or purpose, yet with a recognizable pattern of development. Chance fluctuations take place, causing a system at a certain moment to become unstable. As it “approaches the critical point, it ‘decides’ itself which way to go, and this decision will determine its evolution”. Capra sees the systems view of the evolutionary process not as a product of blind chance but as an unfolding of order and complexity analogous to a learning process, including both independence from the environment and freedom of choice. However, he fails to explain how supposedly inert matter is able to “decide,” “choose,” and “learn.” This belief that evolution is purposeless and haphazard and yet shows a recognizable pattern is similar to biologist Lyall Watson’s belief that evolution is governed by chance but that chance has “a pattern and a reason of its own”.

While the materialistic and mystical views of mind seem incompatible and irreconcilable, mind/matter dualism may be resolved by seeing spirit and matter as fundamentally one, as different grades of consciousness-life-substance. Science already holds that physical matter and energy are interconvertible, that matter is concentrated energy; and theosophy adds that consciousness is the highest and subtlest form. From this view there is no absolutely dead and unconscious matter in the universe. Everything is a living, evolving, conscious entity, and every entity is composite, consisting of bundles of forces and substances pertaining to different planes, from the astral-physical through the psychomental to divine-spiritual. Obviously the degree of manifested life and consciousness varies widely from one entity to another; but at the heart of every entity is an indwelling spiritual atom or consciousness-center at a particular stage of its evolutionary unfoldment. More complex material forms do not create consciousness, but merely provide a more developed vehicle through which this spiritual monad can express its powers and faculties. Evolution is far from being purposeless and indeterminate: our human monads issued from the divine Source aeons ago as unself-conscious god-sparks and, by taking embodiment and garnering experience in all the kingdoms of nature, we will eventually raise ourselves to the status of self-conscious gods.