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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Pullback stability?

Suppose the following square is a pullback. $$\require{AMScd} \begin{CD} E\times _BA @>{\pi_2}>> A\\ @V{\pi_1}VV @VV{\alpha}V\\ E @>>{p}> B \end{CD}$$ The following is ...
2
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1answer
43 views

limit functors as adjoints

Let $I$ be a small category and $\mathcal{C}$ an $I$-complete category. Denote $\iota : I \rightarrow \hat I$ the inclusion into the category obtained from $I$ by “adding an initial object”. A cone ...
3
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1answer
52 views

Left adjoint to pullback functor ($\Sigma$) vs coproduct in cartesian categories

Let A be an object in a cartesian category $\mathscr C$. Let $I$ denote the terminal object of $\mathscr C$. Consider the forgetful functor $\Sigma_A:\mathscr C/A\to\mathscr C$, which is left adjont ...
2
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1answer
74 views

Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
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44 views

Examples of free objects (beginner question)

I am trying to understand what are the free objects in the category of topological spaces. The definition on Wikipedia is clear to me, that is: Given a concrete category C and a functor F such that ...
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1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
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1answer
60 views

Representable Functors and Upper Sets (Final Segments)

I was looking around Wikipedia and came across this for representable functors: From another point of view, representable functors for a category $\mathcal{C}$ are the functors given with ...
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1answer
30 views

universal property of the direct colimit

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism ...
8
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1answer
126 views

Are categories larger than classes?

In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms. What ...
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1answer
79 views

Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
2
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1answer
42 views

Why are left/right proper model categories called so?

A model category is called left proper if weak equivalences are preserved by pushouts along cofibrations, and right proper if they are preserved by pullbacks along fibrations. It is called proper if ...
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1answer
50 views

Mobius functions for acyclic categories, question about formula in “Combinatorial algebraic topology” by Kozlov,

Let $C$ be an acyclic category with a terminal object $t$, in "Combinatorial algebraic topology" by Kozlov he defines Mobius functions for acyclic categories, he first starts by defining a function ...
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43 views

How is the existence of several different morphisms between two objects generalized in therms of the axioms of cathegory theory.

TL;DR I was confused: I viewed commutative diagrams in therms of objects, while in reality they express relationships between morphisms. According to the axioms of category theory, all we need to ...
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1answer
56 views

Algebra study of mathematical structure or algebraic structure [closed]

Algebra can be used to study mathematical structures such as rings, fields but they are called algebraic structures. Algebra is defined as study of structures. Can algebra be used to study any ...
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2answers
979 views

Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
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0answers
36 views

Equalizers in abelian categories

I'm trying to prove that hom-sets in an abelian category have a canonical abelian group structure, working with this definition of an abelian category: A category is abelian if It has a ...
12
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0answers
132 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
3
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1answer
39 views

Sufficient conditions for the category of group objects to have coproducts

For a category $\mathbf{C}$ with finite products, denote by $\mathbf{C}_{\text{Grp}}$ the category of group objects in $\mathbf{C}$. Using the fact that $G\in \operatorname{Obj}(\mathbf{C})$ is a ...
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1answer
38 views

A question about filtered colimits in a category of representations

For $k$ a field, are filtered colimits exact in the category $\mathbf{Rep}_k(G)$ of (finite-dimensional) $k$-representations of a group $G$? I can neither prove it nor find a counterexample.
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34 views

Characterization of fully faithful functors as objects in a functor category

Let $F:C\to D$ be fully faithful. Is it possible for the category $D^C$ to "detect" this property? That is, given an equivalence $\theta : D^C \to B^A$, must $\theta F : A \to B$ also be fully ...
2
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0answers
44 views

“If and only if” condition for imageability of functors

Let's call a functor $T\colon\mathcal{C}\to\mathcal{D}$ imageable, if its image on objects and morphisms forms a subcategory in $\mathcal{D}$. More formally, a functor ...
2
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3answers
65 views

Category theory coproduct beginner question

I'm reading Jeremy Gibbons's Chapter 5 "Calculating Functional Programs" (online at http://www.cs.ox.ac.uk). He uses some basic category theory, which is new to me. He introduces product and coproduct ...
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123 views

Injections between distinct models of the simply typed lambda calculus

Let a model of the simply typed lambda calculus be a Cartesian-closed functor from $C_T$ to Set, where $C_T$ is a free CCC (as in e.g. this reference, p. 83.) The simple case of one or two primitive ...
2
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1answer
57 views

Projective sequence of C*Algebras by factors of embedded ideals isomorphic to algebra

Let $A$ be a $C^*$-algebra and $$A = I_1 \supset I_2 \supset I_3 \supset\ldots$$ be a sequence of embedded ideals in $A$ such that $\bigcap_{i=1}^\infty I_i = 0$. Is it true, that the projective limit ...
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1answer
31 views

Special adjoint functor theorem (proof)

I'm currently going through the proof of the $\mathbf {Special ~ adjoint ~ functor ~ theorem}$ (SAFT) in Saunders Mac Lane's "Categories for the Working Mathematician" and I'm having trouble with the ...
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1answer
52 views

Motivation behind the definition of equivalence of categories.

The background: The naive notion of isomorphism of categories is that: a functor $F:\mathscr{C}\rightarrow\mathscr{D}$ is an isomorphism if there exists $F^{-1}:\mathscr{D}\rightarrow\mathscr{C}$ ...
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1answer
56 views

List of common and uncommon categories

I want to learn more about the category of "super commutative" graded $k$-algebra, for instance, its coproduct. However, I couldn't find anything related material. So, am I be able to get access to ...
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39 views

Left adjoint functor to forgetful functor from C*-algebras to *-algebras category [closed]

Does exist left adjoint functor of forgetful functor from category of C*-algebras to category *-algebras?
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34 views

Properties of the complexification functor for Lie algebras

The complexification of Lie algebras determines a functor from real Lie algebras to complex Lie algebras, whose right adjoint is the restriction of scalars functor. Thus, we know that complexification ...
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Does universal enveloped C*-algebra is continuous functor?

Let $K$ is category of *-algebras that have next property: for each $x \in B$ (where $B$ is *-algebra) $\sup_{\pi - bounded}||\pi(x)|| < \infty$ where $\pi : B \to B(H)$ - is some bounded ...
2
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0answers
32 views

Graphs and Krull-Schmidt Theorem

The book "Abstract Analytic Number Theory" of Knopfmacher states a similarity between Fundamental Theorem of Arithmetic and categorical Krull-Schmidt Theorem. Essentially, it states that, in some ...
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1answer
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Name of a particular category

I'd like to work with a certain category which seems classic to me, but I don't know its usual name. Let's define $$Ob(\mathcal{C}) = \{(Y,Y_1,Y_2,f) : Y = Y_1 \cup Y_2, f : Y_1 \to Y_2\},$$ where ...
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Flat Modules are Filtered Colimits of Free Modules

A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ...
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1answer
31 views

What is the internal hom functor in the context of an internally projective object?

I am trying to understand the definition of an internally projective object from nLab. It says that an object $E$ of a topos $\mathcal{T}$ is called internally projective if the internal hom functor ...
6
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2answers
119 views

Why a sheaf is an object that permits to get global information from local one?

Is there somebody who can explain/show me why a sheaf is something that can permit us to move from the local to the global? An explanation for the layman would be fine. Usually I tend to abhor them, ...
5
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1answer
108 views

Is there an functor without an adjoint?

So I'm doing some research into category theory, and I don't know whether this is a trivial question or not so I'll ask it anyway. Which functors don't have left adjoints? I know there must be some, ...
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1answer
32 views

finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
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2answers
52 views

How, intuitively, does commuting with filtered colimits capture “smallness”?

Definition. A compact object is an object representing a copresheaf which commutes with filtered colimits. In algebraic categories, the compact objects are the finitely presented ones, so commuting ...
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216 views

What is the opposite category of $\operatorname{Top}$?

My question is rather imprecise and open to modification. I am not entirely sure what I am looking for but the question seemed interesting enough to ask: The opposite category of rings is the ...
3
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0answers
43 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
4
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2answers
75 views

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of ...
5
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1answer
59 views

Examples of Waldhausen categories.

Waldhausen's wS construction of K-theory assigns K-groups to an arbitrary small Waldhausen category, my main goal in reading this construction was to apply it to the case of exact categories with weak ...
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1answer
63 views

Effective equivalence relations in a topos

I have a question about Johnstone's proof (in either Topos Theory or the Elephant; the accounts are essentially the same, so far as I can tell) that internal equivalence relations in a topos are ...
5
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2answers
53 views

S,R bimodules subcategory of the category of S+R modules?

If $R$ and $S$ are commutative rings, then does the category $R \oplus S$-modules encompase the category of $(S,R)$-bimodules? I was thinking we can accomplish this by defining the action to be: ...
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2answers
60 views

Opposite category trivial example

I have noticed that the basic notion of opposite category puzzles more than one person. I have also read many complete and motivated answers, as well as read definitions on books by Awodey, MacLane, ...
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1answer
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When are canonical maps between limits monomorphisms?

If $\mathbf{D}_1 \hookrightarrow \mathbf{D}_2$ is an inclusion of diagrams in a category $\mathbf{C}$, and $\mathbf{C}$ has $\varprojlim \mathbf{D}_1$ and $\varprojlim \mathbf{D}_2$, then the ...
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1answer
56 views

Concrete description of (co)limits in elementary toposes via internal language?

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any ...
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Completeness of Total Space of Fibration

The forgetful functor from category of ringed spaces to the category of topological spaces $F\colon RS\to Top$ is a bifibration. The fiber over each topological space $X$ is equivalent to the opposite ...
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2answers
63 views

Lemma 1.3.11 of Categories & Sheaves. Having trouble proving $F_0$ is unique up to unique isomorphism.

Lemma 1.3.11. Consider a functor $F: \mathcal{C} \to \mathcal{C'}$ and a full subcategory $\mathcal{C}_0'$ of $\mathcal{C}'$ such that for each $X \in \mathcal{C}$, there exists $Y \in ...
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1answer
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To prove something is a functor isn't it enough to prove that it commutes with composition?

The second thing you usually have to prove is that $F(\text{id}_X) = \text{id}_{FX}$ for all $X \in C$, where $F: C \to C'$ is the supposed functor. I think it's enough to just prove that $F$ ...