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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Is the arrow category of cartesian closed category cartesian closed? [duplicate]

in a cartesian closed category $\mathcal{C}$. if we have $f: A\to X$ and $g: B\to Y$ then because the functor $(\_)^A$ is continuous, we have $g':B^A\to Y^A$ composing $\text{id}\times f: Y^X\times ...
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dense generators and left Kan extensions

Here is an exercise from Borceux, Handbook of Categorical Algebra I, p 174: Consider a category $\mathfrak{C}$, a family $(G_i)_{i \in I}$ of objects of $\mathfrak{C}$ and the corresponding full ...
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93 views

On pushouts and mapping cylinders in exact categories

Let $\mathcal N$ be an exact category and $C\mathcal N$ be the category of chain complexes with its usual exact structure. We have here the usual notion of "mapping cylinder" of a chain map. If ...
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Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
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70 views

Are there any non-trivial finite elementary topoi?

Title basically says it all: are there any finite topoi (that is, finite set of objects, finite hom-objects) other than $\textbf{1}$ (the terminal category) and $\textbf{2}$ (the category $\ast ...
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Doubt about Yoneda Embedding as image of the hom functor

We can read in the nLab (here), that for $C$ a locally small category, the Yoneda Embedding $$ Y : C \to [C^{op}, Set] $$ is the image of the $hom$ functor $$ Hom : C^{op} \times C \to Set $$ ...
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Category of objects over $X$

The definition of the category of objects over $X$ is defined as: given a category $C$ and an object $X \in Ob(C)$ the category of objects over $X$ consists of the objects as morphisms $Y \to X$ for ...
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(Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, ...
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53 views

How to call such a morphism?

I'm bumping into a property that I would like the morphisms in my own favourite category to have, and I would like to know if it already has a name. Suppose we have a morphism $r : X \rightarrow Y$ ...
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41 views

example diagram of pullbacks and fiber products

I am going through Category Theory for Scientists. I am on section 2.5.1 Pullbacks. I am having trouble visualizing a pullback. Earlier in the book the author gives a nice diagram of an example of ...
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41 views

Object set of a clonal category.

I read the statement that "a clonal category has a small set of objects", which I don't quite agree about. In the definition of clonal category, at least as it is given in that context, it is required ...
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45 views

Do covariant functors preserve direct sums?

Suppose $T$ is a covariant functor from the category $R$-mod to Ab (the category of abelian groups) Is is necessary that $T(B \oplus C) \cong T(B) \oplus T(C) $ ? Does the answer change if we ...
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49 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
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62 views

Unit for Left Adjoint to the Inclusion Functor

I have the following construction, which seems too easy. Could you review and comment? Thanks in advance. Suppose $I$ is a set, and $P(I)$ is its power set, viewed as a category whose arrows are set ...
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27 views

Extending morphism between derived functors

Let N,M be objects in a left exact functor $F:A\rightarrow B$ between abelian categoire's source, most importantly say there is an isomorpihsm $\psi: F(M)\rightarrow R^dF(N)$ is it possible to extend ...
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44 views

Explanation on a particular direct limit

Given the direct system $$\mathbb{Z}^2 \xrightarrow{A} \mathbb{Z}^2 \xrightarrow{A}\mathbb{Z}^2 \xrightarrow{A}\cdots$$ with $$A = \begin{pmatrix} 1 & 1 \\ 2 & 0 \end{pmatrix},$$ the direct ...
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79 views

Categorical proof of Pontrjagin Duality?

I would like to ask if there is any reference in which Pontrjagin Duality is proved in a categorical context: I started reading the Pontrjagin Dual entry in nLab, ...
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206 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
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33 views

Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
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41 views

What's a coproduct in slice category?

We know that product in slice category $\mathcal{C}\downarrow x$ is pullback in $\mathcal{C}$, but what's a coproduct in $\mathcal{C}\downarrow x$ (described in $\mathcal{C}$)? I tried to picture it ...
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216 views

Functor of points and category theory

I am trying to read the section on Functor of points from Eisenbud - Harris (and I also referred to Mumford's book). They all motivate functor of points this way : In general, for any object $Z$ of a ...
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Abelization of symmetric groups and its subgroups of bounded support

For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the ...
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41 views

On recognizing equality against bijection of hom-sets (in a locally small category).

I am a bit confused about the use of the bijection symbol against the equality symbol when dealing with hom-sets. I will give an example. Suppose you want to prove that every functor ...
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52 views

A few questions concerning “universal colimits” and dense generating sets

My questions are triggered by Borceux, Vol 1, Proposition 4.5.6. The relevant part of the book is browsable on Google Books, but i'll go ahead and reproduce at least the statement here anyway: Let ...
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394 views

The Category of Small Categories: a Zoo of Functors.

Wouldn't it be great if there was some website or something that visualized (some small portion of) the category of small categories(*)? Imagine you click on some categories from a list, say, ...
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108 views

What really is a morphism in a category?

I have had quite some exposure with category theory this year. I have even completed a quite long course in category theory and did very well in it. So I thought I am quite good at it. But, something ...
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60 views

Name for categories in which isomorphic implies equal?

A quick terminology question: Is there any particular name for a category in which each object is uniquely determined by its isomorphism class?
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How can I make peace with contravariance?

My question is a bit vague, but I hope it can be answered in a good way. Various arguments involving contravariance sometimes trip me up when coming up with proofs in algebraic geometry and related ...
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35 views

Finitely presentable objects

After introducing the notion of finitely presentable object as an object $A$ such that ${\sf Hom}(A, -)$ preserves directed colimits, an "explicit" form of it is given: $A$ is finitely presentable ...
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48 views

Rel: the category of relations

$\text{Rel}$ is the standard name for the category of sets and relations. Confusingly in "Abstract and concrete categories" (ACC), page 22, $\text{Rel}$ is defined as a category whose objects are ...
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43 views

Confusion about “horizontal composition” of natural transformations

I'm having trouble with an exercise from Rotman's Homological Algebra. It has to do with what Wikipedia calls "horizontal composition" of natural transformations. Namely, given $F, ...
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72 views

Limit Creating Functor

This is exercise 5.5.1 of Maclane's book "Categories for the Working Mathematician". Let $X$ be any category. Prove that the projection $P:X^2 \to X \times X$ sending each arrow $f:x \to y$ to the ...
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36 views

Category of accessible functors and its closedness

Is the category of $\sf{Set}$ accessible endofunctors right closed w.r.t. composition (as a monoidal structure)? Any hint on how to prove this? I think that this is true if one works with finitary ...
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Integrals in analysis and category theory

Are integrals in analysis special cases of coends in category theory? They are both seen as weighted sums, denoted by $\displaystyle\int $ and share the same formal properties (for example Fubini's ...
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29 views

Meaning of fibered product

I need a small explanation about the next. If we write $p: TM\to M$ for the natural projection and $F$ for the natural bundle with $FM=p^{*}(T^{*}\otimes T^{*})M\to M$, then ...
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66 views

The splitting lemma and uniqueness

For the sake of concreteness, let's restrict discussion to the category of abelian groups. Throughout, $$ 0 \to A \overset{q}{\to} B \overset{r}{\to}C \to 0$$ is a short exact sequence. One part of ...
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Sheaf condition for subcoverings

This is a refinement of my previous question. Let $X$ be a space and $\mathscr{F}$ a presheaf on $X$ with values in a complete category. Let $\mathscr{U} = \{ U_i \}$ be an open covering and suppose ...
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Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
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3answers
158 views

Does it factor through?

Let $f:F\to G$ and $g:F\to H$ be group homomorphism between groups. If $\ker f \subset \ker g$ then does there exists $h:G\to H$ such that $hf = g$? I know the the above is true for vector spaces by ...
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Sheaf condition for finite coverings

I'm interested in Proposition 3.5 in Milne's book "Etale Cohomology," which says that a presheaf on a noetherian site is a sheaf if it satisfies the sheaf axiom with respect to finite coverings. By ...
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66 views

Adjoints to Forgetful Functor

Suppose $C$ is a category, $X\in C$. I want to find minimal conditions on $C$ for which the forgetful functor $U:C/X\rightarrow C$ has a left adjoint. edit: As pointed out in the comments, $U$ has ...
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46 views

Characterize a full functor as a morphism

Suppose we have a category of categories, with the morphisms being functors between categories. Can we express the property that a functor is full purely in terms of its properties as a morphism? ...
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61 views

Duality 2-functor on adjunctions

This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...
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When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
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40 views

Why is chosen for intersection instead of union?

Constructing a commutive monoid having idempotent elements (the underlying monoid of a Boolean ring) free over a set $X$, I arrive on a very natural way at monoid $M$ having the finite subsets of $X$ ...
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45 views

What is the category $\mathscr{S}^B $ ?

I've been reading Freyd and Scedrov's "Categories, Allegories" I love it so far, but have an issue with a bit of it. It reads in section 1.261: For B in the category of sets $\mathscr S$, we may ...
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Additive inverse of a morphism in additive category, diagrammatically

I can't find anything useful through the pages of Freyd's Abelian Cats and Google is of little help. I would like to understand how is the (additive) inverse $-f$ of $f\colon A\to B$ defined, for $f$ ...
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In a Morita context ($A, B, P, Q, f, g$), why are $P$ and $Q$ projective if $f$ is surjective?

The title probably says it all :). This is probably a very very simple question, please bear with me. Let $(A, B, P, Q, f, g)$ be a Morita context (or pre-equivalence data) as defined by Hyman Bass in ...
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Natural transformation is a mono iff the components are.

Could someone please give an honest proof of the fact that a natural transformation in the functor category $[\mathcal{C},\mathbf{Set}]$ is a monomorpism if and only if each of the components are ...
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Notation for the set of monomorphisms in $\mathrm{Hom}(A,B)$

Let $\mathcal{C}$ be a small category, and let $A$ and $B$ be objects in $\mathcal{C}$. Is there any standard notation for the subset of all monomorphisms $A\hookrightarrow B$ in $\mathrm{Hom}(A,B)$? ...