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Questions tagged [forgetful-functors]

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2
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1answer
73 views

Completeness of over category (slice)

Let $\mathcal J$ be a (small) category (denote $I:= \mathcal J_0$) and $\mathcal C$ a category that has all (small) limits (all limits of shape $\mathcal J$ for all $\mathcal J$). Prop 3.4 states then ...
1
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2answers
41 views

Why does $\mathbb{Z}$ represent the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$

This is from Emily Riehl's Category theory in context The forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ is represented by the group $\mathbb{Z}$ thanks to the natural isomorphism $\alpha:\mathbf{...
0
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1answer
56 views

Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
6
votes
1answer
121 views

Is there a notion of a transversal of subobjects?

If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
4
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0answers
29 views

Is the category of small graphs a concrete category?

Let $\mathbf{Grf}$ be the category whose objects are small graphs and whose arrows are graph homomorphisms. A small graph is tuple $G = \langle V, E, \operatorname{src}, \operatorname{trg}\rangle$ ...
3
votes
1answer
84 views

Composition of monadic functors isn't monadic

Disclaimer: this question already has a solution here: Composition of monadic functors may not be monadic . However, I would like to understand how to solve this using another characterisation of ...
2
votes
1answer
93 views

$\lambda$-accessible categories, unclear proof

In the context of $\lambda$-accessible categories consider the proof of the proposition $1.22$ here. How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ ...
0
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2answers
57 views

Underlying set of the free monoid, does it contain the empty string?

In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element. Is the empty string an element of the underlying set of the free monoid? In ...
4
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1answer
83 views

How to see that endotransformations of fiber functor have a coalgebra structure?

This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless ...
0
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0answers
35 views

Subfunctor from a category to Set

I'm looking for the definition of a subfunctor $R$ of a functor $U:\cal K \to \mathbb {Set}$. Are there any other conditions beyond that $R(A)\subseteq U(A)$ for all objects $A\in \cal K$ ?
3
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2answers
91 views

Forgetful functor from complete metric space to metric space

I am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to ...
2
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1answer
120 views

A formal definition of forgetful functor

I have read in many category theory textbooks the term "forgetful functor". But no has ever given me a precise definition of this term. I want a rigorous definition, not merely an answer that ...