# 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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### Functorial isomorphism with a direct sum

I need help on a exercise: Let $M$ be a $A$-module with finite presentation, and $(N_{\lambda})_{\lambda \in \Lambda}$ a family of $A$-modules. Show that exist a functorial isomorphism of $\mathbb{K}$-...
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### Restriction functor from sheaves on $X$ to sheaves on basis is an equivalence of categories

This is exercise 4 of chapter 2, Mac lane Moerdijk which I stuck in.... For a basis $B$ of the topology on a space $X$, the restriction functor $r: Sh(X)\rightarrow Sh(B)$ is an equivalence of ...
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### Having trouble with the definition of functors in terms of nerves

In a brief note nlab mentions functors can be thought of maps between nerves. https://ncatlab.org/nlab/show/functor#definition I don't really get this definition. I'm interested because I'm playing ...
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### Why doesn't right exactness of a functor (say tensor product) imply exact?

Let's work in $R$-mod for the remainder of this question. I know I'm probably missing something super basic, but here goes. Let $F$ be the functor $- \otimes_R M$ for some $R$-module $M$. Let's use ...
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### What does it mean that a functor preserves infinite limits?

What does it mean that a functor preservers infinite limits? Can you please give an example of a functor which preserves finite limits but not infinite ones?
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### Definition of functor through diagram

I am studying commutative algebra and right now I'm learning a bit of category theory. I have encountered many times diagrams (commutative ones as well: e.g. generally to express universal mapping ...
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### pair homotopic maps induce the same homology

I'd like to prove that given $f,g : (X,A) \longrightarrow (Y,B)$ homotopic as map of pair, i.e $H(A\ \times I) \subset B$ then they induce the same homology. I already know the theorem which states ...
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### Connection between universal properties and the existence of a left adjoint

I've noticed that in many cases, whenever a functor $G:\mathscr B\to\mathscr A$ has a left adjoint, there's some kind of universal property around (I don't know how to state this more precisely other ...
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### Universal Property of the functor $q:\mathbf{TOP}\rightarrow \mathbf{HO(TOP)}$

Let $\mathbf{TOP}$ be the category of topological spaces and $\mathbf{HO(TOP)}$ be the category whose objects are topological spaces and morphisms are equivalence classes of continuous maps. We have a ...
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### Polynomial functors, Type Theory and Homotopy

I am finding that there is a bit of a battle going on to provide a "foundation" of Type Theory, and perhaps for Mathematics, either with polynomial functors or Homotopy Type Theory. There ...
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### Limits in functor categories

Let $C,C’,D$ be categories and $u:C\to C’$ be a functor. The functor $u^*:\mathbf{Hom}(C’^\circ,D)\to\mathbf{Hom}(C^\circ,D)$ that sends a functor $G$ to $G\circ u$ commutes with limits and colimits, ...
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### What conditions ensure that a functor has a left / right inverse.

A functor $F : C \to D$ has a left inverse $G : D \to C$ if $G \circ F : C \to C$ is naturally isomorphic to $\mathrm{id}_C$ and ditto for a right inverse. Are there nice criteria for when a functor ...
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### Presheaves are the Free Cocompletion - Proving that the functor preserves colimits

I am trying to understand a proof that, for any small category $\mathcal{C}$, the category $\widehat{C} = [\mathcal{C}^\mathrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathcal{C}$. In ...
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### Exponentials in the category of graphs?

Let $\Gamma$ be the category $e\overset{s}{\underset{t}{\rightrightarrows}}v$ and $\mathbf{Graphs}=\mathbf{Sets}^{\Gamma}$ the category of (directed multi) graphs. For graphs $G$ and $H$, it's my ...
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### Sheaf morphism whose cokernel is not a sheaf

Let $\mathcal{F}$ and $\mathcal{G}$ be sheafs of vector spaces. I am looking for an example of a sheaf morphism $\theta:\mathcal{F}\longrightarrow\mathcal{G}$ whose cokernel $\text{Coker}(\theta)$ is ...
### Natural transformation $\mathbb{A}^n \setminus \{ 0 \} \to X$.
Let $\text{Ring}$ the category of commutative ring and $\text{Set}$ the category of set. Denote by $\Omega : \text{Ring} \to \text{Set}$, the functor $R \mapsto \{ \text{Ideal of$R$} \}$, for a ring ...