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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Coherence in braided monoidal categories

Let ($\mathcal{C}$,c) be a braided monoidal (tensor) category. Then c is compatible with the morphisms l,r associated with the unit object 1 of $\mathcal{C}$, in the sense that: $l_X \circ ...
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Which are the natural morphisms between binary relations?

Which properties should morphisms $\alpha$ between binary relations have? $$ R \overset \alpha \longrightarrow R', \;R\subseteq X\times Y, \;R'\subseteq X'\times Y' $$ Can those properties be ...
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Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
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50 views

The category with binary relations as objects

I have reconsidered my ideas and remember how I thought ones upon a time. I will make a last try and delete if it doesn't work: Set is the category where sets are objects and functions are morphisms ...
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49 views

Product in the category of pointed sets..

I have the category $C$, where: objects are nonempty sets with one fixed element $Obj = \{(A,a)$, where $A$-nonemty sets, $a\in A\}$, morphisms are $Mor=\{ f:(A,a)\rightarrow (B,b)$; where $f$ - is ...
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What does it mean to categorify something?

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for?
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Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...
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21 views

Homotopy colimit of a 3x3

Hi I am wondering how you calculate homotopy colimits of a 3x3 diagram. In particular if we have (sorry not sure how to Tex these) Top/bottom row: * <-- * --> * Middle row: * <-- X --> ...
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39 views

Absolute coequalizers in $\mathbf {Set} $

Let $ A $ be a set and let $ R\subseteq A\times A $ be an equivalence relation on $ A $. Denote by $ p, q $ the projections $ R\longrightarrow A $ on the first and second factor, respectively. The ...
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Reference for understanding coalgebra

I am trying to read this paper, but I have no knowledge of coalgebra and have just started to learn Category Theory so I am struggling to understand it. Are there any references that can explain ...
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24 views

Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
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28 views

Defining the category of groups and center-preserving homomorphisms by a universal property; does this actually work?

Motivating Question. How can we define the category of groups and center-preserving homomorphisms by a universal property? Discussion. Given a diagram (call it Diagram 1) shaped like so, ...
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40 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
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62 views

What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
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42 views

Are there any stable $(\infty,1)$-topoi?

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
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77 views

Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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Building a function $p : \mathcal{D} \rightarrow \mathbf{Ord}$ from a faithful functor $U : \mathcal{C} \rightarrow \mathcal{D}.$

For simplicity, I will ignore size issues in this question. Let $\mathcal{D}$ and $\mathcal{O}$ denote categories. By a function $\mathcal{D} \rightarrow \mathcal{O},$ let us mean a functor from the ...
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51 views

If for every pair of objects $a_1, a_2$ we have $fa_{1}=fa_{2}\implies ha_{1}=ha_{2}$, then $h$ factors through $f$

I've just started to explore the category theory, so my question and reasoning might be trivial... Anyway im reading Conceptual Mathemathics by Lawvere and Schanuel. There is an excersise (session 5, ...
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Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
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40 views

Reference for $F$-algebras and induction?

I've been learning about $F$-coalgebras and coinduction from this fantastic paper, which has really helped me get a feel with its many examples. I'm starting to struggle with reconciling the ...
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33 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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2answers
46 views

Canonical Map in a Category with Finite Products, Coproducts and a Zero Object.

Let $C$ be a category as advertised in the title, $\left \{ A_{i} \right \}_{1\leqslant i\leqslant n}$ a finite collection of objects in $C$. Why is there then is a canonical map $f:\coprod ...
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“Structure of a measure space is the coarsest among all substantial structures on a set…”

In the book Lectures and Exercises on Functional Analysis by Helemskii I have stumbled upon the following note: The Rohlin theorem and similar results (see e.g., [19],[20]) show that the structure ...
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53 views

Somewhat Confused about Notation in Category Theory

I'm working through the exercises in "Category Theory for Computing Science" by Barr and Wells, and I'm a little confused about an early problem (though it could just come down to notation). Question ...
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30 views

In an abelian category,every morphism can be written as composition of epi and mono.

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
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What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
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138 views

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that ...
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Category theoretic view of coupling measures/RVs

Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, ...
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2answers
66 views

Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
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The category of theorems and proofs

On a philosophy website, it said that you could have a category with theorems as objects and proofs as arrows. This sounds awesome, but I couldn't find anything on the web that has both "category" and ...
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38 views

manipulation of internal hom [closed]

Let $\text{hom}(-,-):\mathcal{C}^{op}\times\mathcal{C}\rightarrow\mathcal{C}$ denote the internal Hom functor associated to a closed category $\mathcal{C}$, and let $X$, $Y$, and $Z$ be objects in ...
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47 views

Standard Notation For The Set of All the Morphisms Of A Category

Let $\mathscr C$ be a category. Let $\text{Ob}(\mathscr C)$ be the set of all the objects of $\mathscr C$. Is there a standard notation for $\bigcup_{A,B\in\text{Ob}(\mathscr C)}\text{Mor}(A,B)$? ...
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Joke explanation: “a comathematician is a device for turning cotheorems into ffee”

Ok, so apparently there is an old joke (which I DO get) that says that in Hungary a mathematician is a device for turning coffee into theorems. I found a post by Qiaochu Yuan that has the following ...
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What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
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Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
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Epimorphisms in two directions

Let $\mathcal{C}$ be a category. Consider the following statement: (S1) Whenever $A,B$ are objects and $\iota_1:A\to B$ and $\iota_2:B\to A$ are monomorphisms, then there is an isomorphism $\phi:A\to ...
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Is Category Theory geometric?

In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone ...
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Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
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Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
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Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
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50 views

Help with exercise from the Category Theory Wikibook

Reading through the Category Theory Wikibook, I came across the following exercise: (Harder.) If we add another morphism to the above example, it fails to be a category. Why? Hint: think about ...
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Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...
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Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
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Every closed (not-necessarily symmetric) monoidal category is canonically self-enriched, right?

Here it is stated that: A closed symmetric monoidal category is canonically self-enriched. This makes sense, but I don't see why it has to be symmetric. Every closed (not-necessarily symmetric) ...
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Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
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Exponential objects and hom-sets.

Let $C$ be a cartesian closed category and $X, Y$ two objects of $C$. Is it the case that $\text{Hom}(X,Y) = Y^X$?
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Showing that morphisms transforms as claimed, Borceux and enriched natural transformations.

I am having trouble following the proof given in the images below of lemma 6.3.3. More specifically, it is claimed that diagram 6.22 is equivalent to diagram 6.23, but I can't see it. Here, the object ...
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Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
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“Nice proof” that the unit of the left Kan extension of $F$ is an isomorphism, if $F$ is fully faithful

Let $F: \mathbf C \to \mathbf D, G: \mathbf C \to \mathbf E$ be functors. Assume that $\mathbf C$ is small, $\mathbf D$ is locally small and $\mathbf E$ is cocomplete. Then, I can compute the left Kan ...