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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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 ...
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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 ...
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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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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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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 ...
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“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 ...
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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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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 ...
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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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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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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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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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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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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 ...
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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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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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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 ...
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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, ...
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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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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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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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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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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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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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 $F$ ...
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Why is it bad to pick basis for a vector space?

Reading `This Week's Finds', http://math.ucr.edu/home/baez/week247.html, I'm informed that one should avoid picking coordinate systems and I'm unsure why that is the case. Any help on the matter is ...
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The monoid of integers is not free

I am reading the introductory lessons on Category Theory on wikiversity, and they discuss free monoids here: https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Monoids At the bottom ...
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Morphisms in the category of rings

We know that in the category of (unitary) rings, $\mathbb{Z}$ is the itinial object, i.e. it is the only ring such that for each ring $A,$ there exists a unique ring homomorphism $f:\mathbb{Z} \to A$. ...
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Why are contravariant functors called contravariant?

I'm just now learning a bit of category theory, and there often seems to be a certain notion, like limits for instance, and if you inverse certain arrows, you obtain a co-object related to that notion ...
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Requirement for having a left-adjoint functor

Let $G : \mathcal{C} \to \mathcal{D}$ be a functor between two categories. Suppose for each object $D \in \mathcal{D}$ there is a $C \in \mathcal{C}$ that is the "best approximation" of $D$ in the ...
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Direct Limits of Vector Spaces: Confusion about Definition of Mappings Given

First, I give the definitions I am using for the question. They are essentially those found on the Wikipedia page concerning Direct Limits. Let $\{V_i\}_{i\in I}$ be a family of vector spaces in ...
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Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
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On the definition of 2-rigs

I am reading the nLab entry on 2-rigs. In its list of definitions, it says that a 2-rig category can be defined as a $Ab$-enriched category which is enriched monoidal. Why is the enrichment in $Ab$? ...
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Proving Infinitely Ascending Chain of Subobjects

How would you prove that an infinitely ascending chain of subobjects of an object $X$ in $\mathcal{C}$ is stationary given that only finitely many preorders in the chain that are not isomorphisms in ...
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Having trouble proving natural transformation horizontal composition equality of two formulas using a diagram.

Let $C, C', C''$ be three categories. Let $C \xrightarrow{F_1, F_2} C'$ and $C' \xrightarrow{G_1, G_2} C''$, be four functors, and let $\theta : F_1 \Rightarrow F_2, \ \lambda : G_1 \Rightarrow G_2$, ...
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Etymology of transpose of morphisms in an adjunction

Let $F: \mathbf{C} \to \mathbf{D}$ be left adjoint to $G : \mathbf{D} \to \mathbf{C}$, witnessed by the family of bijections between hom-sets, natural in objects $X, Y$: ...
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classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
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How to define the Yoneda embedding

I'm missing in my lecture notes the definition of "Yoneda embedding". It starts by saying that for a category $\boldsymbol{A}$ the Yoneda embedding is a functor $$ \boldsymbol{A} \rightarrow ...
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How do I read this diagram?

I had my first category theory class today and the professor used these kind of diagrams, and terms like "the diagram commutes". I come from another university, and I have no idea what these kind of ...
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Matching the definition of hom-functor with how these are used when defining adjuncts

I have a problem matching the definition of a hom-functor (from nlab) with how this concept it used in the definition of adjunction (from nlab): The hom-functor is defined on $C^{\text{op}}\times C$, ...