In a Category with Fibered Products

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In summary, a fibered product in category theory is a way to compare two objects in a category by considering their relationships to a third object. It is represented as a pullback diagram and is important in defining concepts such as limits and colimits. Fibered products have applications in various areas of mathematics, but their use may be limited by their existence in certain categories and their complex construction. Additionally, understanding their behavior in larger categories may be challenging.
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Euge
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Let ##\mathscr{C}## be a category in which fibered products exist. If ##f : X \to Y## is a morphism in ##\mathscr{C}##, prove that ##f## is a monomorphism if and only if the diagonal morphism ##\Delta_f : X \to X\times_Y X## is an isomorphism.
 
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I have actually done some category theory, so I might have a go at the problem.

"Only if" part:

We assume ##f## is monomorphic.

(i) Let ##p_1 : X \times_Y X \rightarrow X## and ##p_2 : X \times_Y X \rightarrow X## be the canonical projection morphisms. See figure A. We have ##f \circ p_1 = f \circ p_2##. As ##f## is monomorphic we have ##p_1 = p_2 = p##. Thus there is a unique ##p## such that ##p \circ \Delta_f = \text{id}_X##.

Figure A.jpg

Figure A

(ii) See figure B. By definition of a fibered product, ##\Delta_f \circ p## is uniquely determined. By (i) we have ##p \circ \Delta_f \circ p = p##, meaning that by the uniqueness of ##\Delta_f \circ p##, we have ##\Delta_f \circ p = \text{id}_{X \times_Y X}##.

Figure B.jpg

Figure B

"If" part:

We assume that ##\Delta_f## is an isomorphism.

We wish to prove that if ##f \circ \alpha = f \circ \beta## (i.e. figure C) then ##\alpha = \beta##, when ##\Delta_f## is an isomorphism.

Figure C.jpg

Figure C

First note: that as ##\Delta_f## is an isomorphism it has a unique inverse morphism, and as such ##p_1 = p_2 = p##.

See figure D. By definition of a fibered product, there is a unique morphism ##\xi : W \rightarrow X \times_Y X## such that ##\alpha = p_1 \circ \xi## and ##\beta = p_2 \circ \xi##. As ##p_1 = p_2 = p##, we have ##\alpha = p \circ \xi = \beta##.

Figure D.jpg

Figure D
 
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Thanks @julian for participating. Your solution is correct! So the postgraduate problems are not all beyond you. :smile:
 
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Euge said:
Thanks @julian for participating. Your solution is correct! So the postgraduate problems are not all beyond you. :smile:
I guess I have read some graduate maths in my time. But reading graduate maths and doing problems are two different things. This gives me some confidence in at least trying the problems. I may not always solve them but I would probably learn some more maths in attempting doing so, which I guess is one of the main reasons for the problems!

This has given me incentive to go back and better understand category theory!
 
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Off topic. But any ideas for beginner books on cathegory theory and reasonable requirnents?
 
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You can start with Steve Awodey's Category Theory and/or Emily Riehl's Category Theory in Context.

It would help to have some background knowledge in abstract algebra, at least at the undergraduate level. A topology background is also useful, but in my view not required.
 
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What is the definition of a fibered product?

A fibered product is a mathematical construction that combines two objects in a category to create a new object. It is also known as a pullback or a limit and is denoted by the symbol ⊗.

What are the properties of a fibered product?

The fibered product has the following properties:

  • It is unique up to isomorphism.
  • It is commutative, meaning the order of the factors does not matter.
  • It is associative, meaning it can be iteratively applied to multiple objects.
  • It is distributive, meaning it interacts well with other operations in the category.

What is the significance of fibered products in mathematics?

Fibered products are important in many areas of mathematics, including algebraic geometry, topology, and category theory. They allow us to study relationships between objects in a category and can be used to define important concepts such as limits, colimits, and universal properties.

What are some examples of fibered products?

Some examples of fibered products include the Cartesian product of sets, the product of groups, the direct product of rings, and the product of topological spaces. In each case, the fibered product combines two objects in the category to create a new object with certain properties.

How are fibered products related to other mathematical concepts?

Fibered products are closely related to other concepts such as equalizers, pullbacks, and kernels. In fact, a fibered product can be thought of as a generalization of these concepts. They also have connections to other areas of mathematics, such as sheaf theory and homological algebra.

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