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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Can two smooth categories be equivalent if their object manifolds aren't diffeomorphic or homotopy equivalent?

There is a category of "smooth categories", where the objects and the morphisms don't form sets, but manifolds (and there are some other conditions that I won't repeat here). Important examples are ...
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Modules in Morita Equivalence

In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3). How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
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Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
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Can a general version of the covariant powerset monad be derived from the universal property of power objects?

As the title asks, I'm wondering if one can generally squeeze a "covariant power object monad" out of a topos (following the usual example in $\mathcal{Set}$ with functor part the direct image ...
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51 views

Multiplication with category theory

Using category theory why is 3*2=6?. In book conceptual mathematics this is explained as : So there are 3 object 6,3 & 2 with two maps , level & shadow. There are 6 mapping from object 6 ...
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How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
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An equivalence of categories which looks like Voevodsky's Univalence Axiom

Let $\mathcal{C}$ be a category. Consider the full subcategory $\mathrm{Isom}(\mathcal{C})$ of $\mathrm{Mor}(\mathcal{C})$ whose objects are isomorphisms $A \xrightarrow{\cong} B$. It has a full ...
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Characterization of Projective Objects

In which categories is an object $P$ projective if and only if every short exact sequence ending with it splits? $$0\longrightarrow A\longrightarrow B\longrightarrow P \longrightarrow 0$$
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Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
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50 views

why natural transformatoins are also called “morphisms” of functors?

I know a category can be described only with arrows, composition, domain and codomain operators.(without objects!) and a functor just a "morphism" between two categories, that is, it commutes with ...
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79 views

Why can I treat a category that isn't small as if it were?

I am working with a "combinatoric definition" of the zeroth and first homotopy groups for small categories. Let $C$ be a small category, the definitions are the following: The zeroth homotopy group ...
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49 views

Coproduct of bounded distributive lattices given as lattices of subsets

Let $X$ be a set. A lattice of subsets of $X$ is a subset of $\mathcal{P}(X)$ containing $\emptyset$ and $X$ and closed under finite intersection and finite union. Such a lattice is therefore a ...
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Perfect complexes and the derived category of a smooth projective variety

I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then ...
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81 views

Reconciling three definitions of 'surjective'

A book on Category Theory for programmers defines surjection as such $f: A \to B$ that for any $y: T \to B$ there exists such $x: T \to A$ that $f\circ x = y.$ Then it illustrates it with ...
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Does this prove an equivalence of categories?

The definition I am working with (I know there is a stronger notion) states that a functor is an $\textbf{equivalence of categories}$ if it is fully faithful and essentially surjective. I was reading ...
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88 views

Intuition for dinatural and extranatural transformations

Conceptually, natural transformations make perfect sense. What is the intuition behind dinatural and extranatural transformations? Added: I'm looking for conceptual intuition, not something alone the ...
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38 views

How do I prove that $X$ is the pullback of this cospan?

Say we have a category $C$ with products and pullbacks. Let $X$ be an object of $C$. There's a canonical comonoid with multiplication $\Delta\colon X \to X\times X$, and it satisfies the ...
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Can we sensibly universally quantify over a category? (Not a question about size issues.)

You can get pretty far in mathematics without lambda abstraction. I guess this is because any formula of the form $\lambda^{x : X} f(x)=\lambda^{x : X}g(x)$ can always be replaced by the equivalent ...
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Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
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Equalizers by pullbacks and products

I'm trying to solve exercise 5.6 in Steve Awodey's "Category Theory": Show that a category with pull-backs and products has equalizers as follows: given arrows $f, g: A \to B$, take the pullback ...
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Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
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64 views

Categories of relations over a fixed category $\mathcal{C}$

Let $\bf{Set}$ be the category of sets and functions. We have an associated category $\bf{Set}_\bf{Rel}$, whose objects are also sets but whose morphisms are relations, i.e. a morphism ...
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Equivalence of categories involving graded modules and sheaves.

Let $S$ be a graded ring with $S_0=A$ a finitely generated $\mathbb{K}$-algebra and $S_1$ a finitely generated $A$-module. Let $M$ be a graded $S$-module and $\tilde{M}$ the corresponding sheaf on ...
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56 views

Cartesian Product characterization in Rel Category (Category of sets and relations) in simpler terms

The question here explains how a cartesian product of sets are specified. It is difficult for me to follow it; for example, I do not know about "symmetric monoidal category". Can anybody please ...
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Currying a continuous function

There is a mapping which I will call the "currying operator" $\hat\square:Z^{X\times Y}\to(Z^Y)^X$ which maps $f:X\times Y\to Z$ to $\hat f:X\to Z^Y$ defined as $x\mapsto y\mapsto f(x,y)$, or ...
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The first lemma in Auslander's functors and morphisms determined by objects

[lemma 1.1] Let $\mathcal{C}$ be a preadditive category. Suppose G is in ($\mathcal{C^{op}}$, $\mathcal{Ab}$). If for each $X$ in $\mathcal{C}$ we are given a subgroup $A_x$ of $G(X)$ such that ...
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49 views

An unexpected(?) way to identify a 2d vector space with its dual

Let $X$ be a finite-dimensional vector space over the field $\mathbb{R}$. Denote the dual of $X$ by $X^*$. Definition: Let us say that a (not necessarily linear) mapping $x \mapsto x^* : X \to ...
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Path to categorical realizability theory

I'm trying to understand the sorts of things found on this page: http://ncatlab.org/nlab/show/realizability In particular, I want to read Oosten's Realizability: An Introduction to the Categorical ...
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104 views

Is there a relationship between Turing's Halting theorem and Gödel Incompleteness

Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential ...
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159 views

Formalizing finitism in category theory

If we assume that finitism can be formalized by primitive recursive arithmetic (PRA), what category could it correspond to? In particular, which sort of a natural numbers object (NNO) may it contain? ...
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Is there a sleek categorical description of the obvious functor $(\mathrm{Mono}\,\mathbf{Set})^{\mathrm{op}} \rightarrow */\mathbf{Set}$?

Notation. Write $*/\mathbf{Set}$ for the category of pointed sets, and $F : \mathbf{Set} \rightarrow */\mathbf{Set}$ for the free functor (the "point-adjunction functor"). Given a category ...
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To what categorical concept is this proposition equivalent to…

In measure theory we have the following result (I know it's true for positive valued functions, so I'm taking a leap of faith assuming it's true for any measure space.) Proposition Let ...
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Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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56 views

If a functor is exact, does it always have an adjoint? If so, is the adjoint also exact?

For the first statement, if a functor is exact, can it admit both a left and right adjoint? (since it's both left and right exact). Under what conditions can these statements hold?
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Categories with basepoints from forgetful functor

The example of the forgetful functor: $$U: \text{Vect}_K \rightarrow Set$$ mapping the category of vector spaces over field $K$ to Set yields the category of elements consisting of based vector spaces ...
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Homotopy Extension Property as a pushout?

The usual diagram for the homotopy extension property is: where $i_t^X:X\rightarrow X\times I,x\mapsto(x,t)$. Isn't this the same as saying the following square is a pushout? $$\require{AMScd} ...
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100 views

Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure? ...
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72 views

How was the functorial factorization axiom “frequently misstated”?

In modern times, a model category is frequently required to have functorial factorizations of morphisms into (fibration/acyclic cofibration) and (acyclic fibration/cofibration). But the precise ...
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Is the pushforward measure a categorical-theoretic pushout?

Given two measurable spaces $(X,\mathscr{F}),(Y,\mathscr{G}),$ $f:X \to Y$ measurable and $\mu:\mathscr{F} \to [0,\infty)$ a measure, the pushforward of $f_*(\mu):\mathscr{G} \to [0, \infty)$ is ...
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Does “modular category” make sense without saying “abelian” or “linear”?

I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
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64 views

Deformation retracts are closed under pushouts

I'm having trouble with the proof of Proposition 2.4.9 in Hovey's Model Categories. Proposition. Deformation retracts are closed under pushouts. Proof. Suppose we have a pushout diagram ...
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Matrix associated to a linear transformation

Today my linear algebra teacher explained what is the matrix associated to a linear transformation between two finitely generated $\mathbb{K}$-vector spaces. In particular, if we have $B = \{v_1, ...
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118 views

Categorification of geometry

I don't know if this idea is known, relevant or dumb, but I noticed that one could define abstract connectedness with groupoids. Let us forget about topology for a while, and let us think ...
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77 views

Epimorphisms of monoids

I know that in the category of groups, the epimorphisms are precisely the surjective homomorphisms. What about the category of monoids? One can easily see, that surjective homomorphisms are epic (even ...
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Projection morphisms of categorical product

We say that an object $X$ is the categorical product of $X_1$ and $X_2$ if there exist morphisms $\pi_1$ and $\pi_2$, called projection morphisms, such that for every object $Y$ with morphisms $f_1 : ...
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A question about morphisms in a Grothendieck topos

I am not very familiar with topos theory so please excuse me if this is completely trivial. Fix an object $K$ in a Grothendieck topos $\mathcal{G}$. Let $k:0\to K$ be the unique morphism from the ...
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What is meant by “A and B represent the same functor whence are isomorphic” in this solution?

While browsing some old questions I came across the following: tensor product of sheaves commutes with inverse image It seemed like something interesting was going on in the answer, but I don't quite ...
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Lawvere algebraic theories as presentation-invariant

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ...
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What is the motivation for direct product? (categories)

This is the definition of direct product: let $\{X_i\}$, $i\in I$, be a family of objects in category $C$. A product $(X; p_i)$ is an object $X$, together with morphisms $p_i: X\rightarrow X_i$, with ...
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stratification (typage) of logic and syntax at the same time: is such a dream feasible? [closed]

This post is more philosophical than formal, yet I think it's an important question. There's an idea I have for long times already that would consist, in some sense, in doing a "theory of theories". ...