# Questions tagged [functors]

This tag is for questions relating to functors, which is a mapping from one category into another that is compatible with the category structure. Functors exist in both covariant and contravariant types.

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### Why I should not study embedded categories?

In my lecture course of representation theory of $K$-algebras, we embedded the category of $A$-modules (where $A$ is an associative unital $K$-algebra) into the category of $K<X_i|i\in I>$-...
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### If F is a functor from $C_{1} \times C_{2}$ does this equlity hold: $F(f, h \circ g) =F(f,h) \circ F(f,g)$?

Lets say we have a functor $F:C_{1} \times C_{2} \rightarrow D$, where $C_{1},C_{2},D$ are catagories. Does this equation hold for funtors: $F(f, h \circ g) =F(f,h) \circ F(f,g)$, where $f,h,g$ are ...
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### Additivization of functors in an abelian monoidal category

Crossposted on MathOverflow here. I'm having trouble with the proof of Lemma 2.9 in "Cohomology of Monoids in Monoidal Categories" by Baues, Jibladze, and Tonks, and I was wondering if someone could ...
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I would like to understand what are here on the page $266$ the types of items in the adjoint euqation $$\mu^{Lim}_{\cal K}\vdash Lim\ \eta_{\cal K}^{Lim}.$$ They should be functors by the definition ...
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### $G \to G'$ is a functor from $\text{Grp}$ to $\text{Grp}$

I'm not sure if this question has asked before, but I've just start to study category theory and I'm still learning what is a functor, so I have some specific questions about that. Ok, $\text{Grp}$ ...
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### Why is el(-)=$\int(-)$ a functor from functors to a slice cateory?

I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ...
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### Intuitive notion of functoriality in topological data analysis

For school, we have to give a presentation about topological data analysis and I am in charge of motivating why topological data analysis is cool and useful. Most of what I say is based on "Topology ...
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### Colimit of Disjoint Metric Spaces vs Topological Coproduct

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of pairwise disjoint metric spaces. Here we use the convention that a metric space can assume infinite distance. Let Met be the category with metric spaces ...
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### Interpreting a Group Action as a Functor

A group $G$ can be thought of as a groupoid $G$ with a single object. How is defining an action of $G$ on an object of a category $C$ the same thing as defining a functor $G \rightarrow C$. I know ...
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### Product functor and diagonal functor

Let $C$ be a category and consider the product category $C \times C$. There is a diagonal functor associating to each object $X$ of $C$ the pair $(X,X)$ as an object of $C \times C$. On the other ...
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### Is an equalizer really a special case of a limit?

Let $I$ be a category with $2$ objects $A,B$ and four morphisms $1_A, 1_B, u,v$ where $u,v \in Hom_I(u,v)$. Define a functor $F: I \to \mathcal{C}$ by $$F(A) = X, F(B) = Y, F(u) = f, F(v) = g$$ My ...
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### Suppose every functor which preserves $\oplus$ arbitrarily is additive when restricted to fin. gen. proj. modules, then is $S$ stably finite?

(This is the culmination of my recent efforts to understand this problem, and it will be the last post of this type I think. I have a feeling that it might not allow for an answer that fits a forum ...
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### Is there a functor $F$ of left modules preserving $\oplus$ by arbitrary isomorphism, but its restriction to fin. gen. proj. modules isn't additive?

Are there rings $R$, $S$ and a functor $F:{_R\textbf{Mod}}\to{_S\textbf{Mod}}$ such that For all left $R$-modules $M,N$, we have $F(M\oplus N)\cong F(M)\oplus F(N)$ via an arbitrary isomorphism, and ...
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### Does a functor which is additive by arbitrary isomorphism restrict and corestrict to an actually additive functor?

(I appologize for asking a more and more refined version of the same problem for the third time in two days. This is just due to the fact that with every answer to a previous question, I realize that ...
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### Is there a functor $F$ preserving finite direct sums but not split exact sequences, for which $F\mathbb{Z}$ is free and finitely generated?

Is there a functor $F$ from the category of abelian groups to itself, such that $F$ preserves finite direct sums (as well as the empty direct sum) up to (arbitrary) isomorphism $F$ doesn't in general ...
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### The “Right” Concrete Category

One thinks of cardinality as being a property of groups, topological spaces and the objects of many other common categories. This is typically expressed through affixing to the category $\mathcal{C}$ ...
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### Is the product bifunctor uniquely determined by how it acts on objects?

Saunders MacLane states that "The product objects provide, by $<a, b> \mapsto a \times b$, a bifunctor $C \times C \to C.$ " I know that this is not in fact a bifunctor, as the product is unique ...
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### Lifting of a morphism in a hom-functor when hom-sets are empty

I'm having trouble in unserstanding a hom-functor. Suppose we have a hom-functor $Hom(X,$_$)$ for some Category $\mathcal{C}$. Suppose further that the hom-sets Hom$_{Set}(X,A)$ and Hom$_{Set}(X,B)$ ...
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### Is this end-like property expressible in a nicer way?

I have recently stumpbled upon a situation that I am trying to describe categorically, and it seems both familiar and not quite exactly what it should be. After idealizing the situation and peeling ...
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### Details of the criterion for representability of a functor of S-schemes

I've come across a problem that's made me look back over representabiilty of scheme functors and I'm having a lot of trouble piecing together some categorical details that I used to think I ...
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### Fully faithful functors preserve and reflect isomorphisms

Here are my attempts to prove the following lemma: (a) Let $f:A\to A'$ be an isomorphism. Let $g:A'\to A$ be its inverse. Consider $$\alpha_{A,A'}: \mathscr A(A,A')\to \mathscr B(J(A),J(A'))$$ This ...