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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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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1answer
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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2answers
204 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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57 views

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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1answer
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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1answer
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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2answers
107 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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86 views

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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1answer
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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42 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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1answer
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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1answer
90 views

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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28 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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1answer
65 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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1answer
37 views

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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52 views

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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1answer
52 views

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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65 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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1answer
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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2answers
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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63 views

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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1answer
57 views

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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1answer
48 views

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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3answers
80 views

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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1answer
41 views

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 $\mathrm{Hom}(A,B)$? ...
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“twisted” powers in symmetric monoidal categories

Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power ...
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1answer
53 views

Quantificators vs pullbacks

Let $\mathscr C$ be a cartesian category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$. Consider the following pullback square: $\require{AMScd} \begin{CD} P @>{\bar ...
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2answers
76 views

The “magic diagram” is cartesian

I am trying to solve an exercise from Vakil's lecture notes on algebraic geometry, namely, I want to show that \begin{matrix} X_1\times_Y X_2 & \longrightarrow & X_1\times_Z X_2 & \\\\ ...
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Functorial first order theories interpretation

Question will be a bit naive, so please, be kind. Consider first order theories, $\Gamma, \Gamma'$ . Let $\mathcal{M}$ be the category of models for $\Gamma$ and $\mathcal{M}'$ be the category of ...
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1answer
57 views

Equivalent characterizations of group objects

Let $\mathcal C$ be a category. In our lecture, a group object in $\mathcal C$ is defined as an object $c ∈ \mathcal C$, interpreted by a contravariant functor $L \colon \mathcal C^\mathrm{op} → ...
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Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
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1answer
55 views

Inherited Well-Poweredness.

I am reading a proof of SAFT, where the following fact is used: If category $D$ is well-powered, and $G: D\rightarrow C$ is a functor, then $(A\downarrow G)$ is well-powered, for any $A$ in $C$. The ...
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1answer
81 views

Additive functor is exact $\iff$ quasi-ismorphisms preserved?

While reading Weibel's Homological Algebra, on pg. 391 he considers an additive functor $F:\mathcal{A}\to\mathcal{B}$ between abelian categories, and writes "If $F$ is not exact, then the induced ...
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1answer
164 views

Will learning category theory lead to a better and clearer understanding of mathematics?

I read the first chapter on a book about category theory Conceptual Mathematics:A first introduction to categories.In the preface the authors say: It has been the good fortune of the authors to live ...
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1answer
74 views

Do any familiar adjunctions arise from this construction?

I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide. For any set ...
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Check that this is a category

Assume we have a fixed field $F$. We define objects as homomorphisms $\phi:F\rightarrow G$. Then we define morphisms between $\phi:F\rightarrow G$ and $\psi:F\rightarrow L$ as ring homomorphism from ...
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53 views

Categories in which coproducts embed into products

Let $\mathcal{C}$ be a category with coproducts and zero morphisms. Then we have projections $\bigoplus_{i \in I} M_i \to M_i$. For every object $T$ they induce a map $\hom(T,\bigoplus_{i \in I} M_i) ...
4
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1answer
104 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
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67 views

Category In Which Not All Free Objects Exists

I am trying to think of a category in which not all free objects exists. I thought this might be the case in sets (I thought I might be able to violate the uniqueness ) but I couldn't get anywhere so ...
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1answer
37 views

Example of a functor on products

I am trying to come up with an example of a functor which maps products to products but not the same one. That is: Let $C,D$ be categories such that $C$ has all products. I need to define ...
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1answer
45 views

Why is sets not a path category

If we let $U$ be a class and let disjont sets $Q(A,B)$ be given for each pair $(A,B)\in U^2$ then the associated path category is the category $\mathfrak{C}$ with: $ob(\mathfrak{C})=U$ and morphisms ...
3
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1answer
67 views

Is the continuation monad terminal?

For each $R$ an object of a cartesian closed category, there is a monad $\mathrm{Cont}_R(A) = [[A,R],R]$, the continuation monad. If $M$ is a strong monad, we can find for each pair of objects $A$ ...
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Locally small category whose collection of isomorphism classes cannot be a set

For natural examples of locally small categories like the category $\mathbf{Grp}$ of groups, the isomorphism classes themselves are normally not sets. In a set theory like ZFC, even the collection of ...