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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$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
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Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...
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Resources for learning functional programming/Haskell for the mathematically inclined.

I am a math student wanting to learn some functional programming with Haskell. From what I understand, many type theory concepts are analogous, even equivalent, to category theory concepts (e.g. ...
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Is it “okay practice” to exclude the epicness from the definition of extremal / strong epis?

Basically I wonder, whether I "should" include the property of being epi in the definition of extremal epis / strong epis (/...) (dually for extremal monos etc.). One hand it is terminology-wise a ...
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46 views

A question about colimits in enriched categories

I am just starting to learn about enriched categories, so excuse me if I am asking something trivial. Suppose $\mathcal{C}$ is a $\mathcal{V}$-enriched category $\mathcal{C}$, with $\mathcal{V}$ very ...
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58 views

Category theoretic definition of topological spaces.

We know that for algebraic structures like groups and rings, the axioms in their definition can be written in terms of objects and morphisms in the category of sets and hence can be generalised to ...
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Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups

Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct ...
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30 views

Free monoidal category over a set

From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$. I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which ...
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43 views

Homotopy of chain complexes (category theoretic proof)

I know the usual proof of the fact that if a morphism between chain complexes $f$ is homotopic to zero then it induces the $0$ map on cohomology. I was wondering if there is an easy proof of this ...
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52 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self ...
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36 views

Limit-preserving functor that factors through representable functor.

This should be easy, but for some reason I'm not figuring it out right now. Suppose you have a locally small, complete category $\mathcal{C}$ and a functor $\mathcal{B} \xrightarrow{F} \mathcal{C}$. ...
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A trivial slice category

I am clumsily trying to read my way through Algebra: Chapter 0. Among the first examples of categories presented in the text is the following. Let $\mathcal{C}$ be a category and fix one of its ...
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Natural way of looking at projective transformations.

Let $k$ be a field and let $V$ and $W$ be finite-dimensional $k$-vector spaces, where $\dim(V)\ge1$ and $\dim(W)\ge1$. Let $q:V\to\mathbb{P}(V)$, $u\to[u]$ be the quotient map. By my teacher, a map ...
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31 views

how do contravariant 2-functors preserve adjunctions?

I know that covariant 2-functors preserve adjunctions. Do contravariant 2-functors preserve the order left-right of the adjoints, or do they reverse it?
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Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
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Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...
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2answers
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Dividing left adjoints

If I have categories $C,D,E$ with "forgetful functors" $G_1:D\to C$ and $G_2:E\to D$, with $G_3=G_1\circ G_2:E\to C$, and left adjoints $F_1:C\to D$ and $F_3:C\to E$, is it possible to deduce a left ...
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Group theoretic meaning of natural isomorphisms between certain functors

So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to $G$ specified by $\forall g$ $\in G$ $\exists$ $\phi(1)=g$. Each of these homomorphisms is in ...
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52 views

Is isomorphism defined between large categories?

By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...
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57 views

Why are invertible objects reflexive in a tensor category?

I am reading Deligne and Milne's notes on Tannakian categories. I'm only just getting used to the idea of an abstract tensor category, and I've encountered a very believable statement that I'm ...
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84 views

Why this intuition about natural transformations corresponds to its formal definition?

Almost everywhere people introduce the notion of natural transformations between two functors $ F$, $ G$ : $ \textbf C \Rightarrow \textbf D$ by examples like what follows: This is the intuition ...
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modularizing category theory

I have made the experience that proofs using category theory often look very elegant and short but when it comes down to verifying the details there is quite a list of commutativities etc. to check. A ...
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Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
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37 views

Maximal subcategory inside a multicategory

Let $\mathcal M$ be a multicategory. Let $C(\mathcal M)$ be a category consisting of all objects and all unary multimorphisms of $\mathcal M$. Is there a standard name for $C(\mathcal M)$?
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Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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Connection between cobar construction of DG-coalgebra and cobar construction from monad

Given a monad $M:C\to C$ we can construct a cobar resolution from it directly as a functor $\Delta\to [C,C]$ Given a DG-coalgebra $(C,d)$ we can construct a cobar resolution $\Omega C$ of it as ...
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Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff ...
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Show that $R_{1} \times R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$

Let $\mathcal{R}$ denote the category of rings. Show that $R_{1} \times R_{2} \simeq R_{1} \oplus R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$. I know if $R_{1} \times R_{2}$ ...
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1answer
43 views

Defining natural isomorphism without the language of category theory

I was wondering, if it was possible to fully define the natural isomorphisms without the language of category theory, but only with that of set theory. I am not interested in natural transformations ...
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2answers
49 views

Categorical interpretation of equality type

Consider the Martin Lof type theory. It's know that: product type correspond to product of two obects; unit type correspond to terminal object; and so on. The equality type corresponds to some ...
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elementary question concerning definition of sifted colimit

I am reading a proof (in Algebraic Theories by Adamek et al, Theorem 2.15) for the fact that sifted categories $\mathcal{D}$ are precisely those for which the diagonal functor $\Delta : \mathcal{D} ...
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67 views

Where to find about the category theoretic study of manifolds?

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start? Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an ...
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1answer
37 views

Limits in a functor category do what precisely to morphisms?

So I know that if $\textbf{C}$ is a category and $\textbf{D}$ is a complete category then the functor category $\textbf{D}^\textbf{C}$ is complete and limits are "computed pointwise" in the sense that ...
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1answer
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Pullback stability?

Suppose the following square is a pullback. $$\require{AMScd} \begin{CD} E\times _BA @>{\pi_2}>> A\\ @V{\pi_1}VV @VV{\alpha}V\\ E @>>{p}> B \end{CD}$$ The following is ...
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limit functors as adjoints

Let $I$ be a small category and $\mathcal{C}$ an $I$-complete category. Denote $\iota : I \rightarrow \hat I$ the inclusion into the category obtained from $I$ by “adding an initial object”. A cone ...
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1answer
48 views

Left adjoint to pullback functor ($\Sigma$) vs coproduct in cartesian categories

Let A be an object in a cartesian category $\mathscr C$. Let $I$ denote the terminal object of $\mathscr C$. Consider the forgetful functor $\Sigma_A:\mathscr C/A\to\mathscr C$, which is left adjont ...
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1answer
70 views

Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
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44 views

Examples of free objects (beginner question)

I am trying to understand what are the free objects in the category of topological spaces. The definition on Wikipedia is clear to me, that is: Given a concrete category C and a functor F such that ...
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1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
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1answer
57 views

Representable Functors and Upper Sets (Final Segments)

I was looking around Wikipedia and came across this for representable functors: From another point of view, representable functors for a category $\mathcal{C}$ are the functors given with ...
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universal property of the direct colimit

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism ...
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Are categories larger than classes?

In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms. What ...
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1answer
77 views

Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
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1answer
39 views

Why are left/right proper model categories called so?

A model category is called left proper if weak equivalences are preserved by pushouts along cofibrations, and right proper if they are preserved by pullbacks along fibrations. It is called proper if ...
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Mobius functions for acyclic categories, question about formula in “Combinatorial algebraic topology” by Kozlov,

Let $C$ be an acyclic category with a terminal object $t$, in "Combinatorial algebraic topology" by Kozlov he defines Mobius functions for acyclic categories, he first starts by defining a function ...
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How is the existence of several different morphisms between two objects generalized in therms of the axioms of cathegory theory.

TL;DR I was confused: I viewed commutative diagrams in therms of objects, while in reality they express relationships between morphisms. According to the axioms of category theory, all we need to ...
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Algebra study of mathematical structure or algebraic structure [closed]

Algebra can be used to study mathematical structures such as rings, fields but they are called algebraic structures. Algebra is defined as study of structures. Can algebra be used to study any ...
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967 views

Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
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35 views

Equalizers in abelian categories

I'm trying to prove that hom-sets in an abelian category have a canonical abelian group structure, working with this definition of an abelian category: A category is abelian if It has a ...
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Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...