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 Characterization of Categories with a Conservative Forgetful Functor to SET

Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the ...
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0answers
104 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
2
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1answer
52 views

Reference request: Derived category of category with sufficiently many injectives

I'm studying derived categories and have encountered problem with references I have. Namely, proof of the following theorem: Theorem: Let $\mathcal A$ be Abelian category and $\mathcal I$ full ...
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0answers
24 views

Proof of Quillen's lemma about minimal fibrations

In Hovey's book $\textit{Model categories}$ (available online here http://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey--model-cats.pdf) one finds the classical result that any ...
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0answers
60 views

$\mathbb Z$ is not a dense generator in $\mathsf{Ab}$

Why is $\mathbb{Z}$ not a dense generator in $\mathsf{Ab}$, the category of abelian groups? This is exercise 4.8.6 in Borceux's Handbook of Categorical Algebra. There is a hint which says to consider ...
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40 views

Showing that the moduli stack of triangles is equivalent to the quotient stack $[\tilde{T}/S_3]$

$\require{AMScd}$ A good reference for this question is found in the first chapter in the unfinished book Algebraic Stacks by Behrend, Conrad, Edidin, Fulton, Fantechi, Göttsche, and Kresch. ...
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4answers
212 views

The free monoid functor is fully faithful?

For every set $A$, there is a free monoid $A^*$ and a function $i_A : A \rightarrow A^*$, such that for all monoids $Z$ and functions $j : A \rightarrow Z$, there is a unique monoid morphism $j^* : ...
3
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1answer
43 views

A monomorphism in the category of compact Hausdorff spaces is regular

Let $f \colon X \rightarrow Y$ be a monomorphism of compact Hausdorff spaces. This is just a continuous injection. I am trying to show that $f$ is regular, i.e. it is an equaliser. My first thought ...
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1answer
39 views

Right adjoint of a triangle functor is also unique

In general, right adjoint of a functor is unique. In triangulated categories, this is also true. My question is why the natural isomorphism between two right adjoints is compatible with the triangle ...
2
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0answers
33 views

Subobject classifier in $Sets^{Q}$

Let Q be the linearly ordered set of rational numbers considered as a category while $R^{+}$ is the set of reals with $\infty$. In $Sets^{Q}$,prove that the subobject classifier $\Omega$ has ...
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60 views

Topological insight on an algebraic theory.

I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights ...
2
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1answer
31 views

Coherence result for (braided) monoidal functors

Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories) What I have in mind is a theorem like the following: "Let F be a (braided) ...
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1answer
86 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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1answer
55 views

Examples of certain types of toposes

I'm looking for examples of (non-degenerate) categories $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^{op}$ are toposes (assuming that such categories even exist). On a related note, ...
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1answer
40 views

Is Rel a topos?

Is the category Rel of sets and relations a topos? I've done a few Google searches about this question but I haven't found any answers either way. And I can't recall any answers either way in any of ...
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1answer
123 views

commutes homology

I am trying to prove the following: Let $R,A$ be rings and $\mathrm T:$$\mathscr M_R$ $\to $$\mathscr A_R$ such that $\mathscr M_R$ is category of left R modules and $\mathscr A_R$ is category of ...
4
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2answers
66 views

Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
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2answers
45 views

Category Group has unique identity morphism

Let $\mathcal C$ be a category which as finite products and a terminal element $\ast$. A monoid is a pair $(G,e,m)$, where $m: G \times G \rightarrow G$ and $e: \ast \rightarrow G$ are morphisms ...
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0answers
27 views

Filtered colimits are exact in abelian categories

It is well known that filtered colimits commute with finite limits in $\mathsf{Set}$, and hence in every algebraic category - $R\mathsf{Mod}$ in particular. Unless I'm wrong, from the Mitchell ...
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1answer
40 views

Examples of thick subcategory

I'd like to know several examples of thick/Serre subcategory of an abelian category, I have no one in mind now. Help me please!
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3answers
62 views

Direct products, direct sums and coproducts in category of groups

I have couple questions about terms I mentioned in the title. Why we don't define direct sum of non-abelian groups (subset of direct products which consists of elements with almost every component ...
5
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1answer
239 views

What does “hom” stand for in hom-sets and hom-functors?

With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor ...
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1answer
42 views

What are the objects and morphisms of the category $\operatorname{Vect}$?

What are the objects and morphisms of the category $\operatorname{Vect}$? I am trying to learn category theory, and it seems we have infinite objects in $\operatorname{Vect}$ being all of the finite ...
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1answer
57 views

List of universal properties

At the moment I am looking into category theory and I am wondering if there exists a list of universal properties? I couldn't seem to find one.
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1answer
30 views

Inductive limit of A-algebras

I try to compute a pushout in the category of commutative $A$-algebras, where $A$ is a commutative ring with unity. My question is if there is some abstract nonsense which gives me a simple ...
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1answer
70 views

About the definition of Day's convolution

I'm struggling with the definition of Day's convolution. Given a monoidal category $(\mathcal C,\otimes, I)$, there is a way to define a monoidal product on the category of presheaves ...
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2answers
19 views

Kernel of biproduct projection is the other biproduct injection

For some reason I'm unable to figure out what should be a trivial step in a proof.. Let $A\oplus B$ be a biproduct with injections $i_1,i_2$ and projections $p_1,p_2$. I aim to prove $i_1=\ker p_2$. ...
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0answers
45 views

Do functors that send direct sums to tensor products have a name?

Suppose a category $C$ is given with direct sums and tensor products, and let $F:C\to C$ be a functor with the property that $F(A\oplus B)\cong F(A)\otimes F(B)$. It would be tempting to call $F$ an ...
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0answers
141 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
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0answers
54 views

Inverse limit of isomorphic objects

Is the inverse limit of isomorphic objects isomorphic to each one of them? I know there is this question: inverse limit of isomorphic vector spaces but it seems that there they have a total order ...
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3answers
72 views

Generators and relations for $\mathbb{R}_{\geq 0} \cup \{\infty\}$ involving infinite sums

A countably-complete semiring is basically a semiring with some additional structure making infinite sums possible. Formally, it is a tuple $R=(|R|,\Sigma,\cdot,1)$, where $|R|$ is a set, ...
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1answer
31 views

Constructing a $G$-equivariant map from given set map.

In my category theory text I've come across a lot of questions where my own background is lacking in some topics. I have the following question I'm trying to work through: Describe nontrivial ...
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1answer
57 views

Given $A : \mathcal{C} \rightarrow \mathcal{E}$, when does $\operatorname{colim} A = A(1)$?

Suppose $\mathcal{C}$ a small category and $A$ : $\mathcal{C} \rightarrow \mathcal{E} $ where $\mathcal{E}$ is a cocomplete category and $\mathcal{C}$ has a terminal object $1$ When is the colimit ...
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95 views

Yoneda-type lemma for compositions on the hom-functor

The Yoneda lemma basically rephrases a rigidity property of natural transformations out of a covariant hom functor: A natural transformation $\psi : \mathsf{Hom}(Z,-)\Rightarrow F$ is determined by ...
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0answers
41 views

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image? [duplicate]

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image ? that is if $X,Y$ are Hausdorff spaces and $f:X \to Y$ is continuous such that for any ...
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6answers
558 views

Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
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1answer
26 views

A morphism to the mapping cone?

In the second part of the proof for the Proposition 2. in Derived Categories by Daniel Mufet, one finds the following: A collection of morphisms $f^n:Q^n\to X^n\oplus Y^{n-1}$ with components ...
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0answers
39 views

What does the statement “x is a diagram of classical logics” mean?

From the Wikipedia entry on quantum logic A more modern approach to the structure of quantum logic is to assume that it is a diagram – in the sense of category theory – of classical logics (see ...
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1answer
47 views

Limit as universal arrow

I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book. Let us begin with some notation: Let $C$ be a category, and $J$ be an index ...
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1answer
39 views

Functor from category of group representations to space of $G$ invariants

For a representation $(V,\rho)$ of a group $G$, define the subspace of $G$-invariants by $$ V^G=\{v\in V: \rho(g)v=v\quad \forall g\in G\} $$ and want to prove the following: $V\mapsto V^G$ ...
7
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1answer
60 views

Intuitive explanation of Four Lemma

In the Short Five Lemma where the rows are exact, it is a fact that $$\alpha \text{ and }\gamma \text{ injective (surjective) }\implies \beta \text{ injective (surjective)}.$$ I've heard this fact ...
5
votes
2answers
76 views

Examples of functors that preserves products but not equalizers, and vice versa.

What are simple examples, for student consumption, of A functor which preserves products (or at least finite products) but not equalizers. A functor which preserves equalizers but not products. ...
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2answers
39 views

About the presheaf used to define the inverse image sheaf.

Let $f \colon X \to Y$ be continuous and $\mathcal{F}$ be a sheaf on $Y$. Then the inverse image sheaf $f^*\mathcal{F}$ is defined to be the sheafification of the presheaf on $X$ given by $$ U \mapsto ...
0
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1answer
52 views

Every epimorphism in Sets is split: why is it equivalent to axiom of choice?

Suppose that $f: A \rightarrow B$ is epic in Sets. One can construct a section $s: B \rightarrow A$ of $f$ as follow: Let us define an equivalence relation $R$ on $A$ as follow: $aRa'$ iff $a, a' \in ...
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1answer
36 views

Homotopy split monomorphisms [closed]

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose ...
2
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1answer
55 views

How to “lift” a path to the tangent bundle?

Given a path $c: (-\epsilon,\epsilon)=I \to M$ in a manifold. Define $\widetilde c:I \to TM$ (a kind of "lift") as the curve $t_0 \mapsto [c(t+t_0)] \in T_{c(t_0)}M$. Is there a nice categorical ...
2
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1answer
48 views

When is an object in a linear or abelian category simple? Or: How should I define fusion categories?

I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple ...
3
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1answer
86 views

If the cohomology of two objects in the derived category are equal, are the objects isomorphic?

Let $\mathcal{A}$ be an abelian category. Given objects $A^\bullet,B^\bullet$ in the derived category $D(\mathcal{A})$. Assume that $H^n(A^\bullet)=H^n(B^\bullet)$ for all $n\in\mathbb{Z}$. Can we ...
2
votes
1answer
44 views

What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
5
votes
2answers
225 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...