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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When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
3
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3answers
196 views

In category theory, how to make sense of $(X \times Y) \times Z$ in a way that makes use of the natural projections associated with $X \times Y$?

Define that a binary diagram in a category $\mathcal{C}$ is a functor $2 \rightarrow \mathcal{C},$ where $2$ is the discrete category with two objects. Now if $X$ and $Y$ are objects of a category ...
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1answer
83 views

What is an object classifier and how does give a natural numbers object?

According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object. What is an ...
10
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3answers
377 views

Is there a category whose internal logic is paraconsistent?

The internal language of topoi is higher-order typed intuitionistic logic. Now according to wikipedia, the dual of intuitionistic logic, in some sense is paraconsistent. They say Intuitionistic ...
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1answer
181 views

Category on integers with the usual product and coproduct?

I've just been introduced to category theory. I understand the basic definitions, and I'm trying to get some intuition on how categories tick. I'm wondering: is there a category $C$ such that: Its ...
4
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1answer
87 views

Kan extensions for linear categories

Let $C$ be a cocomplete category. Suppose that $X : A \to B$ is a functor, where $A$ is small. Then every functor $F : A \to C$ admits a left Kan extension $\mathrm{Lan}_X(F) : B \to C$, defined by ...
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3answers
126 views

What is the nature of the identity mapping in categories.

A property of any category $\mathfrak{C}$ is that there exists a morphism $1_B:B\to B$, where $B$ is any object in $\mathfrak{C}$, such that if $f:A\to B$ and $g:B\to A$, then $1_B\circ f=f$ and ...
6
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1answer
137 views

What's the dual of a binary operation?

I have a binary operation: $ \diamond : M\times M \to M $ . I want to dualize the binary operation by flipping the arrow, giving me: $$ f : M \to M\times M $$ Now, I can define a coassociativity law ...
2
votes
2answers
47 views

Exactness and Products of Categories

A functor is left exact (resp. right exact) if it preserves finite limits (resp. finite colimits). Let $\mathcal \otimes \colon \mathcal A \times \mathcal B \longrightarrow \mathcal C$ be a bifunctor. ...
2
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1answer
33 views

Local smallness of Lawvere theories

Reading this blog post, I'm trying to care about foundational matters. To summarize the first part of the article, living in a univers $\mathcal V$ of sets, one defines a Lawvere theory as follow : ...
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2answers
48 views

Is there a convention for precedence of operators in an additive category?

The laws for an additive category are that there must be a zero object, binary products, that every Hom-set is an abelian group, and that the morphism addition distributes over composition. My ...
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58 views

What are the non-degenerate faces of $N\mathbb{Z}_2$

I don't understand the nerve construction. For $\mathbb{Z}_2$, Wikipedia says $\bullet \overset{1}\longrightarrow \bullet \overset{1}\longrightarrow \bullet$ should produce a nondegenerate 2-simplex, ...
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votes
1answer
112 views

Morphism of automata category

What is the typed morphism in category $\mathcal{A}$ of finite automata? Let $\mathcal{G}$ be a category of oriented graphs. Does $\mathcal{A}$ equivalent to $\mathcal{C}$
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4answers
103 views

Clarification of definition of category

Need the set of objects and the set of non-id morphisms be disjoint? If not, can the morphisms be a subset of the objects?
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1answer
74 views

Unnecessary hypothesis for a pullback in Hungerford

Reading Hungerford's Algebra I encountered the following statement Consider the diagram $$\require{AMScd} \begin{CD} A @>{\alpha}>>B @>{\beta}>> C\\ @V{\gamma}VV @V{\delta}VV ...
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0answers
103 views

When does a pushout of monics induces a monic arrow?

There are lots of arguments in the homotopy theory of simplicial sets (I refer bascially to the first chapter of Goerss-Jardine) which exploit the fact that in certain cases the map induced in a ...
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vote
1answer
53 views

On the definition of a 2-category

Wikipedia begins the list of ingredients in the definition of a 2-category as follows: A class of 0-cells (or objects) $A$, $B$, .... For all objects $A$ and $B$, a category $\mathbf{C}(A,B)$. The ...
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1answer
66 views

Epimorphisms and Algebras

Just a quick question. Suppose a monad on $\mathbf{Set}$ (in particular monad's endofunctor preserves epimorphisms), are epimorphisms in the category of algebras also surjective? Thanks
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1answer
156 views

When colimit of subobjects is still a subobject?

What are the conditions on a category (or on a certain object) that will guarantee that the colimit of a family of subobjects of a given object is a subobject of the same object? Update: To clarify ...
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votes
3answers
276 views

Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there ...
6
votes
1answer
96 views

Free $\Phi$-cocompletion of a category

let $\Phi$ be a class of small categories (for example, finite categories). A category $D$ is called $\Phi$-cocomplete if it admits all $\Phi$-colimits, i.e. for diagrams of the form $I \to D$ with $I ...
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2answers
115 views

What duality can you quote that says supremum always exists $\implies$ infimum always exists of a bounded set?

Say you've proven that for a subset of the reals bounded above, there exists a supremum of the set in the reals. How do you prove the dual version for infimum without going through all the steps ...
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2answers
97 views

Monoids with left common multiples

So this is a result which I think is true but have yet to find a quick proof for. Say $M$ is a monoid which is left cancellative ($xy=xz\Rightarrow y=z$) and admits left common multiples ($\forall ...
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1answer
136 views

Localization of modules as adjunction

Usually, the localization of a $R$-module $M$ by a multiplicative subset $S \subseteq R$ with $1 \in S$ is categorically defined as the initial object of the full subcategory $\mathbf C$ of $M \, ...
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67 views

Is $\Delta$ dense in $\bf Cat$?

I'm trying to prove that the functor $i\colon\Delta\to{\bf Cat}$ which regards $[n]\in\Delta$ as a category is dense, to deduce that the nerve $N\colon\bf Cat\to sSet$ is fully faithful: how can I do? ...
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193 views

Category of profinite groups

My question is simple: Is the category of profinite groups an accessible category? Thank you Edit: I will add the (hopefully simpler) question: Is the category of profinite groups complete and ...
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0answers
45 views

Category of profinite groups [duplicate]

My question is simple: Is the category of profinite groups an accessible category? Thank you
7
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1answer
253 views

Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?

Set theory is typically formalized as a one-sorted theory without urelements. Is it possible to do the same with category theory or higher category theory, formalizing the whole affair as a theory ...
4
votes
1answer
75 views

Complete abelian categories with projectieve generators are fully abelian.

This is my first time on stackexchange so if you need more detail from me , please ask. I was reading the book "Abelian Categories : An Introduction to the Theory of Functors" by Peter Freyd , and I ...
5
votes
1answer
145 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
8
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5answers
457 views

categorical interpretation of quantification

Many constructions in intuitionistic and classical logic have relatively simple counterparts in category theory. For instance, conjunctions, disjunctions, and conditionals have analogues in products, ...
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1answer
52 views

On the monoidal product of comodules

Let $\mathcal{V}$ a symmetric monoidal category. A commutative comonoid is a triple $(A, \delta, \epsilon)$ whit $\delta: A \to A\otimes A$, $\epsilon: A\to I$ by the dual of monoid axioms. Given two ...
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3answers
93 views

Why is the axiom $(h \circ g)\circ f=h\circ(g\circ f)$ used to define morphisms?

We know that $hom(A,B)$ is a set of morphisms from $A$ to $B$. If $f\in hom(A,B), g\in hom(B,C)$ and $h\in hom(C,D)$, then they have to satisfy the following axiom: $$(h \circ g)\circ f=h\circ(g\circ ...
12
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0answers
244 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
0
votes
1answer
32 views

How to show that a category with products and equalizers of at most size $k$ has all limits of at most size $k$?

I am reading a nice book called category theory by S. Awodey and on page 104 he proves the above statement, which unfortunately for me is a bit sterile. I was wishing to see a picture-like ...
6
votes
1answer
273 views

What was the Lawveres explanation of adjoint functors in terms of Hegelian Philosophy?

I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there. According to the wikipedia entry on Categorical Logic: ...
3
votes
2answers
96 views

Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
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0answers
80 views

Are pulation squares “weak equivalences”?

Let $\cal C$ a category where a square is a pullback if and only if it is a pushout. Is there a model structure on $\cal C^\to$ (the category of arrows of $\cal C$) where the class of weak ...
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vote
2answers
75 views

Proof that equality on categorical products is componentwise equality

I want to proof that in the categorical product as defined here it holds that for $x,y \in \prod X_i$ then $$ x = y \textrm{ iff } \forall i \in I : \pi_i(x) = \pi_i(y). $$ The direction from left to ...
4
votes
2answers
132 views

Monomorphisms and epimorphisms in the category of Boolean algebras

A Boolean algebra is a ring with unity all of whose elements are idempotent. We regard a zero ring $0$ as a Boolean algebra. Let $\mathcal{B}$ be the category of Boolean algebras. A morphism in ...
3
votes
1answer
51 views

Localisation of a binary product of categories

Let $ C, D $ be categories. Let $ S, T $ be subclasses of the morphisms of $ C, D $ respectively (maybe containing identities). Then $$ C\times D [S \times T^{-1}] \cong C [S^{-1} ] \times D [T^{-1}] ...
4
votes
2answers
117 views

Is there an algorithm for determining when two graphs are isomorphic?

The title says it all. Is there such an algorithm? More generally, is there an algorithm for deciding when two objects are isomorphic in a particular category?
7
votes
1answer
229 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
3
votes
3answers
139 views

Is there a term for the converse of “unique up to isomorphism”?

Many people say that an object satisfying a property is unique up to isomorphism if every such object belongs to a unique isomorphism equivalence class. Is there a term for an object that is both ...
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votes
2answers
54 views

Do $\operatorname{Hom}( - , R)$ and $ - \otimes_R B$ commute when applied to $A\cong R^d?$

Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that $$ \operatorname{Hom} (A \otimes_R B , R) \cong ...
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votes
1answer
361 views

Is the Laplace transform a functor?

I may be oversimplifying, as I know very little about category theory, but: Does the Laplace transform, which—to my limited recollection—is a morphism between differential equations and algebraic ...
6
votes
1answer
104 views

How does indexing work in EGA/ how to search for a result in EGA?

I am interested in a certain result which says that if we have an open cover $F_i$ of a sheaf $F$ with each $F_i$ representable, then $F$ is representable. The reason I am interested in this is ...
3
votes
1answer
179 views

Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
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votes
1answer
304 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
5
votes
1answer
118 views

What is a (-1)-morphism?

So, I read the John Baez essay "Lectures on n-categories and cohomology" and I understand the notion of a (-1)-category" and a (-2)-category" and how to derive them. However, I'm not totally clear on ...