Examples of microcosm12/13/2022 ![]() ![]() So, let’s see how this fact is ‘explained’ in terms of lax maps. The monoid object (the ‘microcosm’) can only thrive in a category (a ‘macrocosm’) that resembles itself. We can define a monoid object inside any monoidal category - but a monoidal category is like a categorified monoid! So, for a monoid object to make sense in some category, that category needs to be like a categorified monoid object. Here’s the classic example of the microcosm principle. Let me try to explain this, because it’s really not as terrifying as it sounds. When Jim and I gave an n n-categorical formulation of the microcosm principle in terms of n n-categorical algebras of an operad O O, the key idea was to use lax maps from the terminal n n-categorical O O-algebra to a fixed n n-categorical O O-algebra. ![]() I’m glad more people are trying to find setups where the microcosm principle becomes a theorem - and thanks for pointing this out, David. And now you get the evident notion of rig object in a category with finite products to which you alluded (as well as all the less interesting examples which you enumerated). With δ \delta sending ( a, b, c ) ∈ A × ( B × C ) (a,b,c)\in A\times (B\times C) to ( a, b, a, c ) ∈ ( A × B ) × ( A × C ) (a,b,a,c)\in(A\times B)\times(A\times C). On the other hand, in the slides and associated paper, Hasuo, Jacobs and Sokolova give the principle a 2-categorical formulation as a lax natural transformation X : 1 ⇒ ℂ X: \mathbf is a category with finite products, then it has one of these “lax rig-category” structures with ⊗ \otimes Later, in section 4.3, they give a formal treatment of the principle using operads. the human soul) corresponds to some feature of the macrocosm. We name this principle the microcosm principle, after the theory, common in pre-modern correlative cosmologies, that every feature of the microcosm (e.g. 11), as with monoid objects in a monoidal category. Recall their claim in Higher-Dimensional Algebra III that “certain algebraic structures can be defined in any category equipped with a categorified version of the same structure” (p. The audience was selected to create a microcosm of American society.Ģ) anything that is regarded as a world in miniature.Įxample: We regarded the struggle in prison as a microcosm of the struggle as a whole.įarmville, the town that Dorothy left behind in the 1940s, had become in the 1950s a microcosm of Americas struggle over integration in its public schoolģ) human beings, humanity, society, or the like, viewed as an epitome or miniature of the world or universe.Furthering my study of coalgebra, I came across slides for a couple of talks ( here and here) which put John and Jim’s microcosm principle into a coalgebraic context. The dynamic acts as a microcosm for Edinburgh's own theological development (expanded version: A small, representative system having analogies to a larger system in constitution, configuration, or development)Įxamples: "He sees the auto industry as a microcosm of the U.S. ![]() (Unfortunately I wasn't able to find any sample sentences for this definition of macrocosm)ġ) a little world a world in miniature (opposed to macrocosm) Ecological microcosms can be as small as a few milliliters (Taub 1969a, 1969b) or as large as an armory (Chapter. ![]() You can see its effects in the macrocosm of HIV infections.īut the flower choker holds a unique place in the macrocosm of the early aughts revivalģ) a representation of a smaller unit or entity by a larger one, presumably of a similar structure. The physics that works for falling bodies and pirouetting ice skaters down here in the microcosm of the Earth makes galaxies up there in the macrocosm of the universe.Ģ) the total or entire complex structure of something A system reflecting on a large scale one of its component systems or parts. Contrasted with microcosm.)Įxamples: In other words, the macrocosm of the cosmos is reflected in the microcosm of individual experience (Expanded version: the whole of a complex structure, especially the world or the universe, contrasted with a small or representative part of it. the great world or universe the universe considered as a whole (opposed to microcosm). What's the difference between "macrocosm" and "microcosm"? I've done some sleuthing on some online dictionaries to make sense of these words, but I fail to see the difference.ġ) The entire world the universe. ![]()
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