# Questions tagged [forgetful-functors]

For questions involving the forgetful functor, the functor that drops some of the properties of the input structure before mapping to the output.

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### Forgetful functor from Pos to Set does not have a right adjoint

I'm trying to show that the forgetful functor from $Pos$ to $Set$ does not have a right adjoint, by showing that it does not preserve coequalizers. The hint in the lecture notes I am studying, ...
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### How does the forgetful functor from $\mathbf{C}/C$ to $\mathbf{C}$ forgets the object $C$?

First, sorry for duplication. I've noticed How does the functor $F: \textbf{C}/C \to \textbf{C}$ "forget about the base object" $C$?, but answers there didn't solve my confusion, and my ...
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### 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 ...
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### 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 ...
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### 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 $G = \langle V, E, \operatorname{src}, \operatorname{trg}\rangle$ ...
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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 ...
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### $\lambda$-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 $\lambda$ ...
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### 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 ...
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### 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 ...
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### 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 $A\in \cal K$ ?
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### 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 ...
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### 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 ...