bio | website | dpmms.cam.ac.uk/~zll22 |
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visits | member for | 4 years, 7 months |
seen | 7 hours ago | |
stats | profile views | 3,494 |
Jul 30 |
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Why are Delta-generated spaces locally presentable?
It appears to me that indiscrete spaces of cardinality $\le \mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, can be obtained as quotients of $\Delta^1$. Thus we can construct an increasing sequence of ($\Delta$-generated) subspaces of cardinality $< \mathfrak{c}$ of a $\Delta$-generated space of cardinality $\mathfrak{c}$ whose union is the whole space. In particular, the presentability rank of a simplex is at least $\mathfrak{c}$. |
Jul 28 |
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Does every Lawvere theory arise in this way?
@goblin There's a problem with your question, though – the Lawvere theory of boolean algebras is not generated by $2$ as a boolean algebra but rather $2$ as a set (as you say). This is in contrast to vector spaces; after all, the opposite of the category of finite boolean algebras is the category of finite sets, whereas the opposite of the category of f.d. vector spaces is itself. |
Jul 28 |
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Does every Lawvere theory arise in this way?
What the two examples have in common is a good theory of dualisation. After all, the opposite of any Lawvere theory embeds in the category of algebras in a canonical way. |
Jul 22 |
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“Generalized theory of polynomials” for a given commutative Lawvere Theory
Yes, that is the procedure I am thinking of. It is a fact that the category of abelian group objects in the category of algebras for a Lawvere theory is itself equivalent to the category of algebras for a Lawvere theory (and, in particular, equivalent to the category of modules for a ring). |
Jul 22 |
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“Generalized theory of polynomials” for a given commutative Lawvere Theory
There is a general procedure that derives the theory of $R$-modules from the theory of commutative $R$-algebras, but I do not know of a way to do the reverse. |
Jul 18 |
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Exponential locales and a pointless version of the compact-open topology?
So you know, for instance, that there is a notion of locally compact locale, and that these are precisely the exponentiable locales? |
Jul 18 |
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Exponential locales and a pointless version of the compact-open topology?
Have you tried looking up exponentiable locales? |
Jul 15 |
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Stabilization of a generic pointed model category
Yes, Quillen's original definition only calls for finite limits and finite colimits. But I also have my own reasons for thinking about small model categories and universe enlargement... |
Jul 14 |
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Stabilization of a generic pointed model category
One can perfectly well have small model categories. There are no interesting small combinatorial model categories, however. |
Jul 14 |
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Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?
Sometimes, though, it turns out that naturality with respect to isomorphisms is trivial; for instance, the only isomorphisms in $\mathbf{\Delta}$ are the identities. I suppose this is no different to by-accident natural transformations... |
Jul 13 |
answered | Equivalent definition of a Kan fibration |
Jul 12 |
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$(\infty,1)$-categories and model categories
I would speculate that one might have to restrict to combinatorial model categories to get a fully faithful embedding. |
Jul 6 |
answered | What is descent data (of higher categories), conceptually? |
Jul 6 |
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What is descent data (of higher categories), conceptually?
I'm not sure I agree with your description of the category of descent data as a (homotopy) fibre product – you surely need to involve $\mathcal{F} (U \times_X U \times_X U)$. |
Jul 5 |
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a topos without an underlying boolean topos
What is an "underlying" boolean topos? Every topos has a boolean cover. Incidentally, $[\mathbf{Set}, \mathbf{FinSet}]$ is a topos. |
Jul 4 |
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Is the $\infty$-category of presentable $\infty$-categories presentable?
@YonatanHarpaz I would rather think of the $(\infty, 2)$-category of $(\infty, 1)$-categories as the archetypical $(\infty, 2)$-topos, just as the 1-category of 0-categories is the archetypical 1-topos. |
Jul 3 |
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Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?
I expect that it is a typographical error. |
Jul 1 |
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Explicit computation of a limit of a cosimplicial object
Yes, you can make $F_*$ into a simplicial algebra that way, but do you still mean $\varprojlim_{\mathbf{\Delta}^\mathrm{op}} F_*$? There is a very easy formula for that, namely $F_0$. |
Jul 1 |
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Explicit computation of a limit of a cosimplicial object
Do you mean $(F_*, d, s)$ is a cosimplicial algebra? Do you mean $\varprojlim_{\mathbf{\Delta}} F_*$? |
Jun 30 |
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What is the most transparent, rigorous definition of the Univalence Axiom?
You really must understand type theory first – at minimum the notion of identity types. I think it would be enough to read Chapter 1 of the HoTT book, then skip to §2.10. |