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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198 views

Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
2
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2answers
116 views

What's the name of a morphism the morphism category of the category of categories?

Let $Cat$ be the category of categories, then its morphism category consists of functors as objects and morphism between functors as morphisms. If we restrict to the case where the two functors ...
12
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2answers
221 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
5
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1answer
118 views

Which mathematical structures are particular cases of small categories?

In what follows, all categories are assumed to be small (classes of objects and morphisms are sets). Which mathematical structures $X$ can be seen as particular cases of small categories ...
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131 views

n-globular sets and n-categories

Is the forgetfull functor between the category of n-categories and the category of n-globular sets always monadic? It's seems so, but, in nLab, they are talking only about "2-globular sets and ...
2
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0answers
164 views

What are the prerequisites for reading SGA 1?

My question concerns, basically, scheme theory. If there is someone who has actually read SGA 1, I would really like to hear what their opinion is on that. For example, is EGA in its entirety a ...
4
votes
2answers
214 views

Examples of useful Category theory results?

I'm layman to Category theory. Trying to understand it, I just read a bit briefing on it and wiki pages of Category, Functor, Morphism. However I still could not see the merits of it. Category ...
3
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2answers
237 views

Category with no product?

Is there a family of objects in some category which has no product? If so is there a simple reason for it?
5
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1answer
98 views

Category Cat and set theory

My doubt is simple. I have some possible foundations for category theory. If i'm doing category with NBG as a foundation, and if i define Cat as the category of all small categories i have in ...
10
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1answer
211 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
8
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2answers
472 views

Trying to understand the use of the “word” pullback/pushforward.

Essentially, my question is the following : Is everything we call "pullback" or "pushforward" an actual categorical pullback/pushout? I have seen tons of pullbacks in differential geometry but we ...
4
votes
1answer
207 views

Compact subsets of an inverse limit of topological spaces

Denote by $\mathcal C(X)$ be the space of compact subsets of a topological space X. Let $(X_\alpha)_\alpha$ be an inverse system of topological spaces, then $(\mathcal C(X_\alpha))_\alpha$ is also an ...
6
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3answers
265 views

Why is the 'mapping space' between two objects in a quasi-category a Kan complex?

Let $X$ be a quasi-category. (i.e. a simplicial set satisfying the weak Kan extension condition). Given two 'objects' $a,b\in X_0$ in $X$, one defines the mapping space $X(a,b)$ to be the pullback of ...
1
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1answer
48 views

Determining Objects in a Semicategory

Suppose $S$ is a small semicategory (or semigroupoid, if that's your preferred term) and $\cdot$ is the binary operation on $S$. Implicit in this definition is the set $\operatorname{Ob}(S)$ and two ...
2
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0answers
47 views

Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex ...
3
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1answer
284 views

Do pullbacks commute with filtered limits in this sense?

Let $A_n, B_n, C_n$ be directed systems in some abelian category. Denote by $A \times_C B$ the fibre product of $A$ and $B$ over $C$. Is it true that $(\varprojlim A_n) \times_{\varprojlim C_n} ...
3
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1answer
113 views

A doubt regarding the Category Theory definition of a group.

My Algebra textbook "Chapter 0" by Aluffi states that the category of a group consists of groups as objects and homomorphisms between them as morphisms. Then it also gives a commutative diagram to ...
1
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0answers
72 views

automorphism group of forgetful functor $Vect_k^{gr} \to Vect_k$

Is the automorphism group of the forgetful functor $Vect_k^{gr} \to Vect_k$ (finite dimensional $\mathbb{Z}$-graded vector spaces) equal to $(k^\times)^{\mathbb{Z}} \wr Sym(\mathbb{Z})$ (wreath ...
1
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1answer
41 views

morphisms in $Vect_k^\otimes$

Let $Vect_k^\otimes$ be the tensor category of finite dimensional $k$-vector spaces with the tensor product. What is an example of a morphism $f: V \to W$ in $Vect_k$ which is no morphism in ...
2
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1answer
135 views

Exercise from Rotman, direct limit of quotient modules

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules. Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j ...
3
votes
1answer
33 views

Is the collection of dinatural transformations between two functors a category?

We can, of course, define the functor category $[F,G]$ by considering the natural transformations between two functors $F$ and $G$. But, suppose that instead of the natural transformations we ...
4
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1answer
167 views

Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
1
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1answer
81 views

Showing that a diagram commutes in the most economical way

Suppose that one had to consider (co)cones on a complicated diagram, with many arrows and objects and that one wished to prove that one of them is final/initial. Given another (co)cone, one would ...
2
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1answer
115 views

Foundations in category theory

I'm new in this study and I don't know much about the foundations of mathematics, so I have a question. If I'm doing category theory, and I need to talk about "small categories" , "locally small" and ...
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31 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
0
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64 views

Tensor structure on the category of algebras for a strong monad

Proposition 1.4 in "Monads on tensor categories" by I. Moerdijk says: "If $S$ is a Hopf monad on a tensor category $\mathcal{C}$, then the category of $S$-algebras is again a tensor category." ...
7
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2answers
208 views

The dense topology

The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z ...
5
votes
1answer
606 views

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
3
votes
2answers
213 views

Right adjoint to forgetful functor $\mathbf{Cat} \to \mathbf{Graph}$

There is a forgetful functor $U:\mathbf{Cat} \to \mathbf{Graph}$, which assigns a (small) category to its underlying (small) graph. Also, it has a left adjoint $F:\mathbf{Graph} \to \mathbf{Cat}$, ...
2
votes
1answer
105 views

Question about the univalence axiom versus skeleta.

Here, Dan Licata writes: [Univalence] can be used to build algebraic structures in such a way that isomorphic structures are equal (e.g. equality of groups is group isomorphism). He writes ...
2
votes
1answer
195 views

Notation of category theory

Let $F:I \to C$ be a functor, where $I$ is an index category and $C$ is a category. Show there is a natural bijection between $\operatorname{Mor}_C(T, \lim_{i \in I}F(i))$ and $\lim_{i \in I} ...
3
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2answers
115 views

What's the significance of defining group as a group object in category $\mathcal{Set}$?

At first sight, redefining group as a group object in the category of sets $\mathcal{Set}$ seems just like a meaningless restatement, but when we apply this definition to other categories, ...
2
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0answers
63 views

Should a left module be an enriched functor or enriched presheaf?

In this page http://ncatlab.org/nlab/show/module#InEnrichedCategory, They defined the left module over a monoid object A as an enriched presheaf. However, consider the modules over a ring, the left ...
1
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1answer
90 views

When will a pointed category / arrow category be cartesian closed / a model category if the base category is cartesian closed / a model category?

For a category $\mathcal{C}$ with terminal object we have some construction on it : define the pointed category to be $*\downarrow \text{Id}$ the coslice category relative to the terminal object ; ...
2
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1answer
208 views

Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
3
votes
2answers
134 views

Is the arrow category of cartesian closed category cartesian closed?

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 ...
2
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0answers
89 views

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 ...
2
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1answer
120 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 ...
4
votes
0answers
153 views

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 ...
5
votes
1answer
160 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 ...
2
votes
1answer
64 views

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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0answers
56 views

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 ...
3
votes
0answers
44 views

(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, ...
2
votes
1answer
61 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 $r : X \rightarrow Y$ ...
7
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1answer
414 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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1answer
46 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 ...
2
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1answer
217 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 ...
3
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0answers
62 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 ...
4
votes
1answer
110 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 ...
0
votes
1answer
36 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 ...