# Tagged Questions

Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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### A filtered poset and a filtered diagram (category)

Let $X$ be a poset. Statement: A subset $Y\subseteq X$ is filtered if an only if there exists a filtered diagram (category) $D$ with a functor $D\rightarrow X$ such that the image of $D$ is $Y$. How ...
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### question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
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### The Verdier Quotient

In A.Neeman's book and D.Murfet's notes I have been reading about the construction of the Verdier quotient of a triangulated category, $\mathscr{T}$, by some triangulated subcategory $\mathscr{C}$. In ...
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### Image under a left adjoint functor

Suppose $\psi: \mathbf{Groups} \rightarrow \mathbf{Sets}$ is a left adjoint functor. How would I go about evaluating $\psi(\mathbb{Z})$? Since $\psi$ is left adjoint, let $\psi$ be left adjoint to a ...
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### How to prove uniqueness of *wannabe* final object in a slice category?

I am beginning to study category theory, and I think I need your help to find my way in this sea of uncertainty (!). I have the following problem (n. $5.11$ from Aluffi's Algebra: Chapter $0$). Let ...
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### Colimits in the 2-category of partial functions (which is locally posetal)

I am interested in the category $\mathcal{Pfn}$ of partial functions (using sets as objects), which is well-known to be bicomplete. And it is also known that one can consider $\mathcal{Pfn}$ as a ...
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### Direct sum and direct product of infinitely many abelian groups are not isomorphic

Let $I$ be an infinite set, and for each $i$ let $A_i$ be an abelian group with order $o(A_i) \ge 2$. Prove that the direct product $\prod A_i$ and the direct sum (coproduct) $\bigoplus A_i$ are ...
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### Aluffi: submodule $\Longleftrightarrow$ cokernel?

Aluffi makes the following brief statement, in the context of modules: "The last sentence of Proposition 6.2 simply reiterates the slogan submodule $\Longleftrightarrow$ kernel and its mirror ...
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### If $\Phi: \mathbf{Vec} \rightarrow \mathbf{Vec}$ with $\Phi(V) = V^{\ast\ast}$ and $f: V \rightarrow W$, what is $\Phi(f)$?

Let $\Phi$ be an endofunctor of the category of vector spaces over a field which sends a vector space to its double dual. Let $V$ and $W$ be 2 vector spaces and let $f: V \rightarrow W$ be a morphism ...
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### Axioms of Abelian Category [duplicate]

I know that the one of the axioms of abelian categories is that the induced morphism $\text{coker}(\ker f ) \longrightarrow \ker ( \text{coker} f )$ for any morphism $f$ is an isomorphism. ...
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### Morita contexts without tears

My question is: Has anybody seen Morita contexts introduced as it is done below? I first intended this as an answer to the question "Reference request: Morita contexts" by Bey, but then decided to ...
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### What are the faces of $Id_{A}$?

I've been reading Awodey's Category Theory and I've seen the definition of the Identity function in it. As it's definition was always a function $f:A\to A$ I used to assume that the identity function ...
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### Dinatural transformation,constant functor,hom functor

Let $U,V$ be functors between categories $C$ and $X$ and let $Y\in Set$. Why a dinatural transformation $Y\xrightarrow{\cdot \cdot}\hom_X(U-,V-)$ is a function which assigns to $y\in Y$ a natural ...
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### Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
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### Understanding the significance of a functor being full/faithful, especially with adjoints

I'm working through "Basic Category Theory" by Tom Leinster and am trying to get clarity on how to reason about things... one thing I'm not sure about is how to think about what a functor being ...
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### Computation of adjoint functors (sheafification)

In a (complete) category, limits can be "computed" assuming one knows how to compute products and equalisers. I have seen it mentioned that adjoint functors can be found using certain ...
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### Equivalence of categories $\Pi_1:\mathbf{K^1_{CW,*}}\to \mathbf{Grp}$

Let $K(G,1)$ denote the Eilenberg-Maclane spaces with fundamental groups isomorphic to $G$. Consider the category $\mathbf{K^1_{CW,*}}$ where objects are pointed $K(G,1)$ CW complex, and morphisms ...
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### A question regarding Yoneda's lemma.

Suppose you have two objects $A$ and $A'$ in a category $\mathfrak{C}$, and morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ for any object $C\in\mathfrak{C}$. Show that the $i_C$ are induced ...
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### How does Yoneda lemma give that the natural isomorphism $\operatorname{hom}(A,-)\cong\operatorname{hom}(B,-)$ implies $A\cong B$?

I'm trying to work out an element free proof of the associativity of the tensor product, that $$(M\otimes_A N)\otimes_B P\cong M\otimes_A (N\otimes_B P).$$ Since $\operatorname{hom}$ and $\otimes$ ...
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### “Identity-free” definition of an isomorphism in a semigroupoid / semicategory

I am looking for a way to define "Isomorphism" in a semigroupoid (or semicategory), that is a "category", which does not necessarily have identities. To be more specific I am looking for a way to ...
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### Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
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### Lang Universal Objects

STATEMENT: Let $\mathcal{C}$ be a category. An object $P$ of $\mathcal{C}$ is called universally attracting if there exists a unique morphism of each object of $\mathcal{C}$ into $P$, and is called ...
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### Lang Category Theory

STATEMENT: Let $A,B$ beobjects of a category $\mathcal{A}$. Let Iso$(A,B)$ be the set of isomorphisms of Awith B. Then the group Aut$(B)$ opoerates on Iso(A,B) by composition; namely, if ...
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### examples of additive categories which have morphism that has no kernel and morphism has no cokernels.

can you tell me examples of additive categories which have morphism that has no kernel and morphism has no cokernels. if you tell me reference which provide this kind of examples it will be ...