Equivalence of C & D: A Question

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Discussion Overview

The discussion revolves around the relationship between two full subcategories, C and D, and whether they can be considered equal or equivalent. It also touches on the concept of slice categories and their relation to fiber bundles, exploring theoretical aspects of category theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asserts that if C is a full subcategory of D and D is a full subcategory of C, then C and D must be equal, as full subcategories are defined by their objects.
  • Another participant requests clarification on slice categories, referencing a quote about A-indexed families of disjoint sets and the commutativity of a diagram involving functions between sets.
  • A different participant suggests that fiber bundles serve as a useful model for understanding slice categories, explaining that objects in the slice category correspond to projections from the original category to a base object.
  • This participant elaborates that each point of the base object corresponds to a fiber, and morphisms in the slice category act fiber-wise, maintaining consistency within fibers.

Areas of Agreement / Disagreement

There is a clear agreement among some participants regarding the equality of C and D based on their definitions as full subcategories. However, the discussion about slice categories and fiber bundles introduces different perspectives and does not reach a consensus on the interpretation of the quote or the implications of these concepts.

Contextual Notes

The discussion includes assumptions about the definitions of full subcategories and slice categories, which may not be universally agreed upon. The exploration of fiber bundles as a model for slice categories introduces additional complexity that remains unresolved.

phoenixthoth
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One question for now:
1. If C is a full subcategory of D and D is a full subcategory of C, what can be said, if anything, about C and D being either equal or equivalent in some way?
 
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They'd have to be equal.

Every object of C is an object of D, and vice cersa. Full subcategories are determined by their objects.
 
2. Could you explain slice categories?

I have a quote that I don't get:
It is useful to think of an object of Set/A as an A-indexed family of disjoint sets (the inverse images of the elements of A). The commutivity of the above diagram means that the function h is consistent with the decomposition of B and C into disjoint sets.

In the "above diagram," f:B-->A, g:C-->A, and h:B-->C s.t. gh=f.

Thanks in advance...
 
Have you worked much with fiber bundles? I think they're a good model to understand what's going on.

You can imagine the objects of the slice category are the projections from objects of the original category to the base object.

For a given projection, each "point" X of the base object corresponds to a "fiber", those "points" of the source object that project onto X. (Thus, we have a B-indexed family of fibers)

The morphisms of the slice category, then, are the morphisms that act "fiber-wise". That is, if P is in a fiber of X, then f(P) is also in a fiber of X.

Of course, in general you won't have points and fibers to manipulate, but I think this is the spirit behind it.
 
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