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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Closed under extensions without zero object

Suppose A is an abelian category and $\mathcal{B}$ is a full subcategory of A. If $\mathcal{B}$ is closed under extensions, must it be closed under isomorphisms? We require that $\mathcal{B}$ contains ...
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1answer
46 views

Full subcategory of abelian category is abelian

I'm trying to understand a proof in Rotman's 'Introduction to Homological Algebra', Proposition 5.92, p.310. Proposition: Let $\mathcal S$ be a full subcategory of an abelian category $\mathcal A$. ...
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31 views

What is the “internal language of a topos”?

What does the sentence "[...] these statements should be interpreted, of course, in the internal language of the topos $\mathcal{E}$" mean, in the context of, say, the definition of a groupoid in ...
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33 views

Is it impossible to recover multiplication from the division lattice categorically?

In this question it was asked if the division lattice (i.e., the preorder category $(\Bbb Z_{>0}, \mid)$) contains enough information categorically to recover the relation ...
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36 views

generalized affine scheme

I'm thinking about following theorem. For a finitaly algebraic theory $\mathbb{T}$, $\text{FP}\mathbb{T}$ denotes the full subcategory of $\mathbb{T}\text{-Alg(Set)}$ consisting of finitely presented ...
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1answer
130 views

Splitting field as a terminal object?

Let $f(x)\in K[x]$ be a polynomial over field $K$ and let $E$ be a splitting field. I would like to prove that $E$ is unique up to isomorphism by expressing the inclusion $K\to E$ as a terminal object ...
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2answers
57 views

$\mathbf{Set} \not \simeq \mathbf{Set}^*$ by considering $\{1, 2 \} \to \{1\}$

This answer gives a nice way of seeing why the category of sets is not isomorphic to its dual. I would like to know whether there is a proof from a certain different direction. When considering the ...
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0answers
32 views

Finitely generated abelian groups form an abelian subcategory of $\mathbb{Z}$-Mod

According to Weibel's Homological Algebra book a subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ is called an abelian subcategory if it is abelian and an exact sequence in $\mathcal{B}$ ...
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20 views

Question about “immediate” observation about finitely presentable objects

The following is an excerpt from volume II of Borceux: Here $F$ is the left adjoint to the forgetful functor $U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$. I'm confused and can't manage ...
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39 views

Proof forgetful functor $U:\mathsf{Mod}_\mathcal{T}\rightarrow \mathsf{Set}$ has a left adjoint

I'm confused by a proof in volume II of Borceux. First, two facts: Proposition 3.3.3 $\;\;\;$ Let $\mathsf{T}$ be an algebraic theory. Consider the functor ...
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44 views

Squares of adjunctions / Galois correspondences

$ \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\PS}{\mathcal P} \newcommand{\Idl}{\operatorname{Idl}} $ There are many situations where one encounters a square of things which are related by ...
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reference for unsolved problems in ETCS [on hold]

I am looking for unsolved problems in the theory of "elementary theory of category of sets" are there references for (foundational) problems in the category of sets as foundation?
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44 views

“Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
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40 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
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33 views

about the proof of proposition 2.2.1 in grothendieck's tohoku paper [on hold]

I have a problem that I can't complete the proof, that is, for any short exact sequence, I don't know how to construct the natural transformation extanding degree 0. I have read Cartan and ...
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0answers
75 views

May Algebraic Geometry be appropriate for me? [on hold]

I am a student of Mathematics who have to choose its area of specialization. I am trying to obtain as more information as possible, by asking a lot of questions to more experienced people, trying to ...
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1answer
78 views

How to give “categorical” specifications of categories like Grp?

Certain types of categories (like abelian categories) are specified by listing a set of "categorical" properties that the category must have. For example, we might demand the category has finite ...
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3answers
1k views

Motivation for different mathematics foundations

I've been studying an introductory book on set theory that uses the ZFC set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen ...
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1answer
54 views

Categories like $\mathsf{FinSet}$, but with elements of $\mathbb{Z}$ or $\mathbb{Q}$ as objects?

Suppose, we do universal algebra in a "non-evil" fashion, such that every algebra carries around an equivalence relation $\cong$ that poses as equality and such that size issues don't matter (we may ...
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3answers
64 views

Prove that in any category a product(if exists) is associative up to isomorphism.

I have a question regarding an exercise(p.38, ex.5.9) from Aluffi's Algebra textbook. Let C be a category with products. Find a reasonable candidate for the universal property that the product $A ...
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1answer
46 views

If $Y\to Z$ is a monomorphism then $X_1\times_Y X_2 \to X_1\times_Z X_2$ is an isomorphism.

This is an (easy) exercise from Vakil's textbook on Algebraic Geometry. We are working in an arbitrary category, let $Y\to Z$ be a monomorphism and we are given maps $X_1, X_2\to Y, X_1, X_2\to Z$. We ...
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1answer
62 views

Isomorphism between colimits.

I actually need something weaker than this but my hope is that this holds in its fullest generality. Let $I$ be a small diagram and $I'$ a full subcategory of $I$. Let $F: I\to {\rm vec}$ be a functor ...
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2answers
50 views

Is it possible that the inclusion functor does not preserve limits?

Let $\mathcal{B}$ be a full subcategory of a category $\mathcal{C}$. Is it possible that the inclusion functor does not preserve the pullbacks (or any limits) that exist in $\mathcal{B}$? I read this ...
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111 views
+50

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
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31 views

Locally presentable categories of categories

Cat, the category of small categories, is locally finitely presentable. I want to relax the quality of presentability to find interesting categories as colimits. Has anyone seen a treatment of ...
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22 views

Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by: ...
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1answer
23 views

Natural isomorphism of a monoidal category

The definition of a Monoidal Category from "Categories for the working mathematician" says that it is a category equipped with tensor products, associative up to a natural isomorphism. What does ...
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1answer
34 views

A question on a property of geometric morphisms related to locales.

Is the "localic reflection" of a geometric morphism between topoi the same thing as its "localic part"?
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2answers
139 views

Can the identity $ab=\gcd(a,b)\text{lcm}(a,b)$ be recovered from this category?

Define the category $\mathcal{C}$ as follows. The objects are defined as $\text{Obj}(\mathcal{C})=\mathbb{Z}^+$, and a lone morphism $a\to b$ exists if and only if $a\mid b$. Otherwise ...
3
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1answer
41 views

Name of this 2-categorical structure?

Let $\mathscr C$ be a 2-category. For each object $X$ in $\mathscr C$ suppose there is a category $\mathcal R(X)$, and for each pair of objects $X,Y$ in $\mathscr C$ there is an "action" functor $$ ...
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1answer
35 views

Functor $M\otimes_B -$ is left and right adjoint to $\operatorname{Hom}_A(M,-)$ when there is a symmetrizing form for $A$?

Suppose $A$ is an $R$-algebra over a commutative ring $R$, which is finitely generated and projective as an $R$-module. A symmetrizing form is a map $t\in\operatorname{Hom}_R(A,R)$ such that ...
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What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
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1answer
34 views

Reconstruct a category : Forgetful fuctor to underlying graph, free functor of graph and then a quotient

There is a forgetful functor that takes a category to its underlying graph. There is then an adjoint to this that takes the graph to its free category. Can we then take a quotient of this free ...
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1answer
33 views

Braided Hopf algebra - properties of braiding

Let $(H,\nabla,\Delta,S)$ be a Hopf algebra in a braided category. I'm trying to simplify the following $(\nabla\otimes \mathrm{id}\otimes \mathrm{id})(\nabla \otimes \Psi \otimes ...
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33 views

Categorical terminology

For any monoidal category $(\mathscr{C},\otimes,I)$ with objects $A,B,...$: Does the monoidal product $\otimes$ always have the property: $A \otimes A \rightarrow A, B \otimes B \rightarrow B,...$ ? ...
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1answer
20 views

Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...
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2answers
66 views

Is there any example of a Category without generators?

I was looking of an example of a category (a non trivial example) without generators (seperators). Is there any nice example? Or every category that we are using has generators?
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27 views

If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
5
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1answer
68 views

References for the threefold categorical equivalence of compact Riemann surfaces?

A lot of the books I've found assert that there is a threefold categorical equivalence between (1) compact Riemann surfaces, (2) smooth projective algebraic curves, and (3) function fields of ...
5
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1answer
50 views

Cech Cohomology and the Dold-Kan Correspondence

Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. ...
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4answers
72 views

Naturality of bijection given by Yoneda Lemma

I'm reading through Wikipedia's proof of the Yoneda Lemma (https://en.wikipedia.org/wiki/Yoneda_lemma), and am having trouble understanding what naturality means in this context. The articles proves ...
2
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0answers
63 views

Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where: weak-equivalences are quasi-isomorphisms; cofibrations ...
2
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1answer
54 views

The direct limit of morphisms and the direct limit of tensor product functors

While reading these notes I had something of an existential crisis, after realizing that my understanding of direct limits might somehow be fundamentally insufficient. In particular, alarms started ...
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Identifying targets of epimorphisms as cokernels of some morphism

In Aluffi's Algebra: Chapter 0 text, on P. 167, when discussing the category of R-Mod, he writes that "... every epimorphism identifies its target with the cokernel of some morphism." For any ...
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Exponential objects of internal objects respecting evaluation (2-exponentials?)

Let $(F,+_F)$, and $(G,+_G)$ be two commutative internal monoids in Sets. Set being cartesian closed, I can form $G^F$ as a set. My question is simple: is there a canonical/universal way to enforce ...
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1answer
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Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm ...
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1answer
52 views

Braided Hopf algebra - properties

If $(H,\Delta,\nabla)$ is a Hopf algebra in the prebraided monoidal category $(\mathcal{C},\Psi)$ then $\Psi_{H,H}=\left(\nabla\otimes \nabla\right)\left(S\otimes\Delta \nabla\otimes ...
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2answers
78 views

Left adjoint of the forgetful functor $\mathsf{Grpd} \to \mathsf{Cat}$?

I've heard that there is a left adjoint to the forgetful functor $\mathsf{Grp} \to \mathsf{Mon}$. I wonder if there is also a left adjoint $F : \mathsf{Cat} \to \mathsf{Grpd}$ to the forgetful functor ...
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1answer
39 views

Is the reflective localization $L_WC$ of a category $C$ equivalent to $C$? What am I missing?

This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization). Let $C$ be a category, let $W ...
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1answer
38 views

Induced morphisms into pullback

The induced morphism by the universal property of pullback, when is it an epimophism ( I'm looking at it in a regular category, when it's induced from a coproduct )