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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1answer
18 views

Categories with some but not all exponentials

The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials. Are there not-too-esoteric examples of categories of ...
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Realization of simplicial sets

Consider a simplicial set $X$, i.e. a contravariant functor from $\mathbf{\Delta} \to \mathbf{Set}$ where $\mathbf{\Delta}$ has as objects $[n]:=\{0, \cdots, n \}$ for all $n \in \mathbb{N}$ and as ...
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3answers
156 views

In the definition of a functor, why is it necessary that $F(id_{A})=id_{F(A)}$?

A functor $F$ is defined to be a mapping from category $\mathcal{C}$ to $\mathcal{D}$ such that: (1) $F(f\circ_{\mathcal{C}} g)=F(f)\circ_{\mathcal{D}} F(g)$ (say, for a covariant functor). (2) ...
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1answer
39 views

Topological Version of First Isomorphism Theorem

Given a set $X$ and an equivalence relation $\sim$ on $X$, we can define the set $X_\sim=\left\lbrace\left[x\right]:x\in X\right\rbrace$ of equivalence classes, and we can define a projection map ...
3
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1answer
54 views

Commutative Diagram for group structure

I remember seeing once a commutative diagram that explained group structure. Where the associativity, identity element, inverse, multiplication and all was shown in a singular diagram, it is trivial ...
7
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1answer
66 views

How to recover multiplication of group elements from category of groups?

Motivating question: If we know everything about $\mathbf{Grp}$, do we know everything about groups? Background: Consider the category $\mathbf{Grp}$ abstractly, rather than as a concrete category. ...
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0answers
58 views

Existence of tensor product via category theory

In my class of category theory, my professor stated (without prove it) that the existence of tensor products between modules over commutative rings follows from the following result: a category ...
3
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1answer
158 views

Similarity of Infinite Direct Sums Vs. Infinite Direct Products Accross Categories

Let $|R|=|S|=\infty$. In many concrete categories, I know $R^S$ can be identified as the set of all functions from S to R, and the much "smaller" $R^{\oplus S}$ can be identified as the subset of ...
3
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1answer
78 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
17
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4answers
319 views

Why are particular combinations of algebraic properties “richer” than others?

Pedagogically, when students are exposed to algebraic structures it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are rich ...
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1answer
26 views

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 ...
4
votes
1answer
70 views

An endomorphism $f$ such that $f\circ f=1$ [duplicate]

What is the name for an endomorphism $f$ of a category such that $f\circ f=1$? Note that I work with category $\mathbf{Set}$.
3
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1answer
53 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 $\mathcal A$. ...
4
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1answer
35 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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0answers
37 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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0answers
45 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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2answers
578 views

Why is every category not isomorphic to its opposite?

This is a beginner category theory question: I'm trying to wrap my head around the fact that we do not have $\mathbf{Sets} \cong \mathbf{Sets}^\mathsf{op}$, i.e., the category of sets is not ...
4
votes
1answer
137 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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5answers
408 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 ...
2
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2answers
60 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 ...
2
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0answers
33 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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1answer
40 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 ...
9
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1answer
84 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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0answers
23 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 ...
9
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1answer
139 views

When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
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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 [closed]

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?
3
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0answers
45 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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0answers
77 views

May Algebraic Geometry be appropriate for me? [closed]

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 ...
2
votes
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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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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0answers
44 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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about the proof of proposition 2.2.1 in grothendieck's tohoku paper [closed]

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 ...
5
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1answer
121 views

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 ...
3
votes
3answers
65 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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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
votes
1answer
57 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 ...
9
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2answers
142 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 ...
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vote
2answers
115 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
5
votes
1answer
52 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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1answer
34 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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vote
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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2answers
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Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a ...
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Relation between categorical operations (limits and co-limits)

Suppose I have a diagram $B \longleftarrow A \longrightarrow C$ in a category, and I execute a push-out operation and get $B \longrightarrow D \longleftarrow C$. If I execute a pull-back over $B ...
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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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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 ...
4
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1answer
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Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?

Call a Lawvere theory $T$ dimensive iff, letting $F_T : \mathbf{Set} \rightarrow \mathbf{Mod}(T)$ denote the free functor, we have the following. Every finitely generated $T$-algebra is free. From ...
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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 ...