# Tagged Questions

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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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### how would you define the term “elementary” in the context of categories and sets?

I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I ...
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### Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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### Strange monomorphism of sheaves

What is an example of a Grothendieck topology $J$ on a small category $\mathcal{C}$ such that the category of sheaves $\mathsf{Sh}(\mathcal{C},J)$ has a monomorphism which is not a monomorphism of ...
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### Showing $f:a\to b$ is epic iff there exists some iso $g: f(a)\cong b$ with $g\circ f^*=f$.

This is Exercise 5.2.4 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". In any topos, given $f:a\to b$, define $p, q: b\to r$ using the pullback of $f$ along itself like so ...
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### Verifying a Construction Satisfies the $\Omega$-axiom.

I'm stuck on Exercise 4.5.1 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". It's in the topos $\mathbf{Bn}(I)$ of bundles over a set $I$. Goldblatt asks the reader to verify that $\tag{1}$ ...
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### Presheaves in a quasi-topos.

I do believe it is a trivial question. But unfortunately I don't know where I can find an answer. Where could I find the answer to the following question? If $S$ is a small category and $X$ is a ...
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### Motivation of Strong Monics

Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where ...
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### Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
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### A few questions concerning “universal colimits” and dense generating sets

My questions are triggered by Borceux, Vol 1, Proposition 4.5.6. The relevant part of the book is browsable on Google Books, but i'll go ahead and reproduce at least the statement here anyway: Let ...
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### Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
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### Is there a topos in which the natural numbers object are the finite dimensional vector spaces?

I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions. Now vector spaces are prominently characterised by dimensionality. This prompted the ...
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### 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 ...