From 4bbb3e2d0bbde5ed5b2c9cbc18b10e4e0b38c15a Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Tue, 5 May 2026 09:25:10 -0400 Subject: [PATCH 1/5] Fix the details of a proof which will be referred to in CompHaus proofs --- databases/catdat/data/categories/Haus.yaml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/databases/catdat/data/categories/Haus.yaml b/databases/catdat/data/categories/Haus.yaml index b92dffc7..7f0737d7 100644 --- a/databases/catdat/data/categories/Haus.yaml +++ b/databases/catdat/data/categories/Haus.yaml @@ -68,7 +68,7 @@ unsatisfied_properties: reason: 'Recall the counterexample for sets: The unique maps $\IN_{\geq n} \to 1$ are surjective, but their limit $0 = \bigcap_{n \geq 0} \IN_{\geq n} \to 1$ is not. This also works in $\Haus$ by using discrete topologies. We could also apply a variant of (the dual of) this lemma to the discrete topology functor $\Set \to \Haus$, which does not preserve all cofiltered limits, but does preserve intersections.' - property_id: filtered-colimit-stable monomorphisms - reason: 'The proof is similar to $\Met$. For $n \geq 1$ let $X_n$ be the pushout of $[-1/n,+1/n] \hookrightarrow \IR$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \leq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \leq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is no monomorphism.' + reason: 'The proof is similar to $\Met$. For $n \geq 1$ let $X_n$ be the pushout of $(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is no monomorphism.' special_objects: initial object: From f40d37384cac046f8a14118b2992e76138cb9cee Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Tue, 5 May 2026 09:25:56 -0400 Subject: [PATCH 2/5] Add a general result to help deducing CompHaus is well-copowered --- .../catdat/data/category-implications/congruences.yaml | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/databases/catdat/data/category-implications/congruences.yaml b/databases/catdat/data/category-implications/congruences.yaml index a6f0b3ea..def9ee3a 100644 --- a/databases/catdat/data/category-implications/congruences.yaml +++ b/databases/catdat/data/category-implications/congruences.yaml @@ -8,6 +8,16 @@ reason: This holds by definition of a regular category. is_equivalence: false +- id: regular_well-powered_well-copowered + assumptions: + - regular + - epi-regular + - well-powered + conclusions: + - well-copowered + reason: The regularity condition gives a bijection between the collection of quotients of $X$ and the collection of effective congruences on $X$, where the latter is a subcollection of the collection of subobjects of $X\times X$. + is_equivalence: false + - id: congruence_quotients_are_reflexive_coequalizers assumptions: - reflexive coequalizers From 9bde5f2f466e1dcb5f60ba597c8baafd9fd2fabf Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Mon, 4 May 2026 19:11:51 -0400 Subject: [PATCH 3/5] Add category of compact Hausdorff spaces --- .cspell.json | 8 +- .../catdat/data/categories/CompHaus.yaml | 125 ++++++++++++++++++ databases/catdat/data/categories/Haus.yaml | 6 +- databases/catdat/data/macros.yaml | 1 + static/pdf/.gitignore | 2 + static/pdf/comphaus_copresentable.bib | 78 +++++++++++ static/pdf/comphaus_copresentable.pdf | Bin 0 -> 249665 bytes static/pdf/comphaus_copresentable.tex | 118 +++++++++++++++++ 8 files changed, 336 insertions(+), 2 deletions(-) create mode 100644 databases/catdat/data/categories/CompHaus.yaml create mode 100644 static/pdf/comphaus_copresentable.bib create mode 100644 static/pdf/comphaus_copresentable.pdf create mode 100644 static/pdf/comphaus_copresentable.tex diff --git a/.cspell.json b/.cspell.json index 16ef8c31..ef09b91f 100644 --- a/.cspell.json +++ b/.cspell.json @@ -14,7 +14,8 @@ "ndash", "emptyset", "varnothing", - "mdash" + "mdash", + "comphaus" ], "words": [ "abelian", @@ -36,6 +37,7 @@ "Catabase", "catdat", "Catégories", + "Čech", "clopen", "Clowder", "coaccessible", @@ -71,6 +73,7 @@ "colimits", "comonad", "comonadic", + "compactification", "conormal", "copower", "copowers", @@ -229,10 +232,13 @@ "surjectivity", "Tarski", "tensoring", + "Tietze", "topoi", "Turso", + "Tychonoff", "unital", "unitalization", + "Urysohn", "vercel", "Vite", "Wedderburn", diff --git a/databases/catdat/data/categories/CompHaus.yaml b/databases/catdat/data/categories/CompHaus.yaml new file mode 100644 index 00000000..d38cd8a3 --- /dev/null +++ b/databases/catdat/data/categories/CompHaus.yaml @@ -0,0 +1,125 @@ +id: CompHaus +name: category of compact Hausdorff spaces +notation: $\CompHaus$ +objects: compact Hausdorff spaces +morphisms: continuous functions +description: This is the full subcategory of $\Top$ consisting of those spaces that are compact and Hausdorff. +nlab_link: https://ncatlab.org/nlab/show/compact+Hausdorff+space + +tags: + - topology + +related_categories: + - Haus + - Top + +satisfied_properties: + - property_id: locally small + reason: It is a full subcategory of $\Top$, which is locally small. + + - property_id: semi-strongly connected + reason: This is already true for $\Top$. + + - property_id: products + reason: By the Tychonoff product theorem, a product in $\Top$ of compact Hausdorff spaces is compact; it is also clearly Hausdorff. Since the forgetful functor from $\CompHaus$ to $\Top$ is fully faithful, this limit is reflected in $\CompHaus$ as well. + check_redundancy: false + + - property_id: equalizers + reason: 'The equalizer in $\Top$ of two continuous functions $f, g : X \rightrightarrows Y$ between compact Hausdorff spaces is a closed subspace of $X$, and therefore it is also compact Hausdorff. Since the forgetful functor from $\CompHaus$ to $\Top$ is fully faithful, this limit is reflected in $\CompHaus$ as well.' + check_redundancy: false + + - property_id: cocomplete + reason: $\CompHaus$ is a reflective subcategory of $\Top$, with the reflector being the Stone-Čech compactification functor. See nLab for example. Therefore, as usual, we can form colimits in $\CompHaus$ by forming colimits in $\Top$ and then applying Stone-Čech compactification. + + - property_id: generator + reason: The one-point space is a generator because it represents the forgetful functor to $\Set$, which is faithful. + + - property_id: cogenerator + reason: 'The unit interval $[0, 1]$ is a cogenerator: Suppose we have $f, g : X \rightrightarrows Y$ with $f \ne g$. Choose $x\in X$ such that $f(x) \ne g(x)$. Then by Urysohn''s lemma, there is a continuous function $h : Y \to [0, 1]$ such that $h(f(x)) = 0$ and $h(g(x)) = 1$. Therefore, $h\circ f \ne h\circ g$.' + + - property_id: effective congruences + # TODO: rework this when Barr-exact is added + reason: The forgetful functor from $\CompHaus$ to $\Set$ is monadic; see for example nLab. Therefore, by this result, $\CompHaus$ is Barr-exact, and in particular it has effective congruences. + + - property_id: regular + # TODO: rework this when Barr-exact is added + reason: The forgetful functor from $\CompHaus$ to $\Set$ is monadic; see for example nLab. Therefore, by this result, $\CompHaus$ is Barr-exact and in particular is regular. + + - property_id: coregular + reason: + 'It suffices to show that pushouts preserve (regular) monomorphisms in $\CompHaus$. Thus, suppose we have a pushout square + $$\begin{CD} + A @> i >> B \\ + @V f VV @VV g V \\ + C @>> j > D, + \end{CD}$$ + with $i : A \hookrightarrow B$ a monomorphism. Then for any pair of distinct elements $c, c'' \in C$, by Urysohn''s lemma there exists $\gamma : C \to [0, 1]$ with $\gamma(c) = 0$ and $\gamma(c'') = 1$. Also, by Tietze''s extension theorem, there exists $\beta : B \to [0, 1]$ such that $\beta \circ i = \gamma \circ f$. By the pushout property, there is a unique $\delta : D \to [0, 1]$ such that $\delta \circ g = \beta$ and $\delta \circ j = \gamma$. Since $\delta(j(c)) \ne \delta(j(c''))$, we conclude that $j(c) \ne j(c'')$. This shows that $j$ is injective, so it is a regular monomorphism.' + + - property_id: extensive + reason: This follows as for $\Top$ or $\Haus$ since finite coproducts in $\CompHaus$ are formed as disjoint union spaces with the disjoint union topology. + + - property_id: epi-regular + reason: |- + First, any epimorphism $f : X \to Y$ is surjective: if not, its image would be a proper subset of $Y$, which is compact and hence closed. Then by Urysohn's lemma, there would be a non-zero continuous function $g : Y \to [0, 1]$ which is $0$ on the image; but then $g \circ f = 0 \circ f$, giving a contradiction. + Now the identity morphism from $Y$, with the quotient topology of $f$, to $Y$ with its given topology is a bijective continuous function between compact Hausdorff spaces, so it is a homeomorphism. In other words, $f$ is a quotient map. Therefore, we see that if $g, h : E \rightrightarrows X$ is the kernel pair of $f$, and $U : \CompHaus \to \Top$ is the forgetful functor, then $U(f)$ is the coequalizer of $U(g)$ and $U(h)$. Since $U$ is fully faithful, that implies $f$ is the coequalizer of $g$ and $h$. + + - property_id: cofiltered-limit-stable epimorphisms + reason: 'Suppose we have a cofiltered diagram of epimorphisms $(f_i : X_i \to Y_i)$, and $y = (y_i) \in \lim_i Y_i$. Then by lemma 1 here, the limit of $f_i^{-1}(\{ y_i \})$ is non-empty. If $x$ is in this limit, that implies that $(\lim_i f_i)(x) = y$.' + + - property_id: locally copresentable + reason: A proof can be found here. + +unsatisfied_properties: + - property_id: skeletal + reason: This is trivial. + + - property_id: Malcev + reason: This is clear since $\FinSet$ is not Malcev and can be interpreted as the subcategory of finite discrete spaces. + + - property_id: regular subobject classifier + reason: The proof is almost identical to the one for $\Haus$. + + - property_id: natural numbers object + reason: >- + Let $I := [0, 1]$. If a natural numbers object $(N, z : 1 \to N, s : N \to N)$ existed, then we could iterate the initial conditions $I\to I\times I$, $x \mapsto (x, x)$ and the recursive step function $I\times I \to I \times I$, $(x, y) \mapsto (x, xy)$ to get a continuous function $N \times I \to I \times I$ such that $(s^n(z), x) \mapsto (x, x^n)$ for $x\in I$, $n \in \IN$. The sequence $(s^n(z)) \in N$ has a convergent subnet $(s^{n_\lambda}(z))_{\lambda \in \Lambda}$, say with limit $y$. Thus, for any $x\in I$ and $\lambda \in \Lambda$, we have $(s^{n_\lambda}(z), x) \mapsto (x, x^{n_\lambda})$. Taking limits, we see $(y, x) \mapsto (x, 0)$ if $x \ne 1$ or $(y, x) \mapsto (x, 1)$ if $x = 1$. In other words, $(y, x) \mapsto (x, \delta_{x, 1})$ for all $x\in I$. However, that contradicts the fact that the composition + $$\begin{align*} + I & \overset{y \times \id}\longrightarrow N\times I \to I\times I \overset{p_2}\longrightarrow I, \\ + x & \mapsto (y, x) \mapsto (x, \delta_{x,1}) \mapsto \delta_{x,1}, + \end{align*}$$ + would have to be continuous. + + - property_id: filtered-colimit-stable monomorphisms + reason: 'The proof is similar to $\Haus$. For $n \geq 1$ let $X_n$ be the pushout of $[1/n, 1] \hookrightarrow [0, 1]$ with itself. That is, $X_n$ is the union of two unit intervals $[0, 1] \times \{ 1 \}$ and $[0, 1] \times \{ 2 \}$ where we identify $(x,1) \equiv (x,2)$ when $x \geq 1/n$. As in the construction for $\Haus$, we see that the colimit in $\Haus$ is $[0, 1]$ where all corresponding points of both unit intervals are identified. Since this is compact Hausdorff, it also provides the colimit in $\CompHaus$. Again, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to [0,1]$ in the colimit, which is not a monomorphism.' + + - property_id: exact cofiltered limits + reason: |- + Consider the $\IN$-codirected systems $X_n := [0, 1] \times [0, 1/n]$ with the maps $X_{n+1} \to X_n$ being inclusion maps, and $Y_n := [0, 1+1/n]$ with the maps $Y_{n+1} \to Y_n$ also being inclusion maps. We define $f_n : X_n \to Y_n$, $(x, y) \mapsto x$ and $g_n : X_n \to Y_n$, $(x, y) \mapsto x+y$. It is straightforward to check these give morphisms of $\IN$-codirected systems in $\CompHaus$. + Now for each $n$, we claim the coequalizer of $f_n$ and $g_n$ is a singleton space. To see this, we prove the more general result that for $r, s > 0$ the coequalizer of $f, g : [0, r] \times [0, s] \rightrightarrows [0, r+s]$, $f(x,y) = x$, $g(x,y) = x+y$ is a singleton. We must show that for any $h : [0, r+s] \to T$ with $h\circ f = h\circ g$, then $h$ is constant. To this end, we show by induction on $n$ that whenever $x \in [0, r+s]$ and $x \le ns$, we have $h(x) = h(0)$. The base case $n=0$ is trivial. For the inductive step, if $x \le s$, then $f(0,x) = 0$ and $g(0,x) = x$, so $h(0) = h(x)$. Otherwise, we have $x-s \in [0,r]$ and $x-s \le (n-1)s$, so by inductive hypothesis $h(x-s) = h(0)$. Also, $f(x-s, s) = x-s$ and $g(x-s, s) = x$, so $h(x-s) = h(x)$, completing the induction. With this established, the desired result follows from the case $n := \lceil r/s \rceil + 1$. + On the other hand, $\lim X_n \simeq [0, 1] \times \{ 0 \}$; $\lim Y_n \simeq [0, 1]$; and $\lim f_n = \lim g_n$, $(x, 0) \mapsto x$. Thus, the coequalizer of $\lim f_n$ and $\lim g_n$ is $[0, 1]$, showing that the limit does not preserve this coequalizer. + +special_objects: + initial object: + description: empty space + terminal object: + description: singleton space + coproducts: + description: Stone-Čech compactification of the disjoint union with the disjoint union topology (in the finite case, the disjoint union is already compact Hausdorff so Stone-Čech compactification is not necessary) + products: + description: direct product with the product topology (which is compact by the Tychonoff product theorem) + +special_morphisms: + isomorphisms: + description: homeomorphisms + reason: This is easy. + monomorphisms: + description: injective continuous maps (which are automatically closed embeddings) + reason: 'For the non-trivial direction, the forgetful functor to $\Set$ is representable (by the terminal object), hence preserves monomorphisms. To prove the parenthetical remark, given an injective continuous function $f : X \to Y$ between compact Hausdorff spaces, the image of $f$ is a closed subset. Also, the induced map from $X$ to $\im(f)$ with the subspace topology is a bijective continuous map between compact Hausdorff spaces, so it is a homeomorphism.' + epimorphisms: + description: surjective continuous maps (which are automatically quotient maps) + reason: For the non-trivial direction, and for a proof of the parenthetical remark, see the proof above that $\CompHaus$ is epi-regular. + regular monomorphisms: + description: same as monomorphisms + reason: This is because the category is mono-regular. + regular epimorphisms: + description: same as epimorphisms + reason: This is because the category is epi-regular. diff --git a/databases/catdat/data/categories/Haus.yaml b/databases/catdat/data/categories/Haus.yaml index 7f0737d7..931c4a85 100644 --- a/databases/catdat/data/categories/Haus.yaml +++ b/databases/catdat/data/categories/Haus.yaml @@ -12,6 +12,7 @@ tags: related_categories: - Met_c - Top + - CompHaus satisfied_properties: - property_id: locally small @@ -68,7 +69,10 @@ unsatisfied_properties: reason: 'Recall the counterexample for sets: The unique maps $\IN_{\geq n} \to 1$ are surjective, but their limit $0 = \bigcap_{n \geq 0} \IN_{\geq n} \to 1$ is not. This also works in $\Haus$ by using discrete topologies. We could also apply a variant of (the dual of) this lemma to the discrete topology functor $\Set \to \Haus$, which does not preserve all cofiltered limits, but does preserve intersections.' - property_id: filtered-colimit-stable monomorphisms - reason: 'The proof is similar to $\Met$. For $n \geq 1$ let $X_n$ be the pushout of $(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is no monomorphism.' + reason: |- + The proof is similar to $\Met$. For $n \geq 1$ let $X_n$ be the pushout of + $$(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$$ + with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is not a monomorphism. special_objects: initial object: diff --git a/databases/catdat/data/macros.yaml b/databases/catdat/data/macros.yaml index 149a871f..d33feabd 100644 --- a/databases/catdat/data/macros.yaml +++ b/databases/catdat/data/macros.yaml @@ -80,6 +80,7 @@ \PMet: \mathbf{PMet} \Top: \mathbf{Top} \Haus: \mathbf{Haus} +\CompHaus: \mathbf{CompHaus} \sSet: \mathbf{sSet} \Man: \mathbf{Man} \LRS: \mathbf{LRS} diff --git a/static/pdf/.gitignore b/static/pdf/.gitignore index 372e1325..662623f7 100644 --- a/static/pdf/.gitignore +++ b/static/pdf/.gitignore @@ -1,5 +1,7 @@ # Files from Latex Workshop *.aux +*.bbl +*.blg *.fdb_latexmk *.fls *.log diff --git a/static/pdf/comphaus_copresentable.bib b/static/pdf/comphaus_copresentable.bib new file mode 100644 index 00000000..c94f7425 --- /dev/null +++ b/static/pdf/comphaus_copresentable.bib @@ -0,0 +1,78 @@ +@misc{marra2020characterisationcategorycompacthausdorff, + title={A characterisation of the category of compact {H}ausdorff spaces}, + author={Vincenzo Marra and Luca Reggio}, + year={2020}, + eprint={1808.09738}, + archivePrefix={arXiv}, + primaryClass={math.CT}, + url={https://arxiv.org/abs/1808.09738}, +} +@article{HUSEK2019251, + title = {Factorization and local presentability in topological and uniform spaces}, + journal = {Topology and its Applications}, + volume = {259}, + pages = {251-266}, + year = {2019}, + note = {William Wistar Comfort (1933-2016): In Memoriam}, + issn = {0166-8641}, + doi = {https://doi.org/10.1016/j.topol.2019.02.033}, + url = {https://www.sciencedirect.com/science/article/pii/S0166864119300513}, + author = {M. Hušek and J. Rosický}, + keywords = {Realcompact space, Factorization, Locally presentable category}, + abstract = {Investigating dual local presentability of some topological and uniform classes, a new procedure is developed for factorization of maps defined on subspaces of products and a new characterization of local presentability is produced. The factorization is related to large cardinals and deals, mainly, with realcompact spaces. Instead of factorization of maps on colimits, local presentability is characterized by means of factorization on products.} +} +@article{Marra_2017, + title={Stone duality above dimension zero: Axiomatising the algebraic theory of {C(X)}}, + volume={307}, + ISSN={0001-8708}, + url={http://dx.doi.org/10.1016/j.aim.2016.11.012}, + DOI={10.1016/j.aim.2016.11.012}, + journal={Advances in Mathematics}, + publisher={Elsevier BV}, + author={Marra, Vincenzo and Reggio, Luca}, + year={2017}, + month=Feb, pages={253–287} } +@article{Isb82, + title = {Generating the algebraic theory of {C(X)}}, + journal = {Algebra Universalis}, + volume = {15 (2)}, + pages = {153-155}, + year = {1982}, + author = {Isbell, J.R.} +} +@inproceedings{Dus69, + title = {Variations on {B}eck's tripleability criterion}, + series = {Reports of the Midwest Category Seminar III}, + editor = {MacLane, S.}, + author = {Duskin, J.}, + pages = {74-129}, + year = {1969}, + publisher = {Springer Berlin Heidelberg} +} +@article{Hoff18, + title = {Generating the algebraic theory of {C(X)}: The case of partially ordered compact spaces}, + journal = {Theory and Applications of Categories}, + author = {Hoffman, Dirk and Neves, Renato and Nora, Pedro}, + pages = {276-295}, + year = {2018}, + volume = {33}, + number = {12}, + url = {http://www.tac.mta.ca/tac/volumes/33/12/33-12.pdf}} +@article{GU71, + title = {Lokal pr\"asentierbare {K}ategorien}, + author = {P. Gabriel and F. Ulmer}, + journal = {Lecture Notes in Mathematics}, + publisher = {Springer-Verlag, Berlin}, + volume = {221}, + year = {1971} +} +@book{SGL, + title={Sheaves in Geometry and Logic: A First Introduction to Topos Theory}, + author={Mac Lane, Saunders and Moerdijk, Ieke}, + year={1992}, + publisher={Springer-Verlag New York, Inc.}, + series={Universitext}, + isbn={978-0-387-97710-2}, + doi={10.1007/978-1-4612-0927-0}, + url={https://link.springer.com/book/10.1007/978-1-4612-0927-0} +} diff --git a/static/pdf/comphaus_copresentable.pdf b/static/pdf/comphaus_copresentable.pdf new file mode 100644 index 0000000000000000000000000000000000000000..96c164894fa4cac9655700988ed99b4b844dee91 GIT binary patch literal 249665 zcma&tQ+p=hvM}h_wr$(CZ9D07Z0C(_cWgTy+qP|YY`x!_Z)P8@gV}Xd&kv}2>bmPH zN)-tyMphAVE0RcE>1xp8OHyaWTRxS>b{~K`3QoronEL}*L zrR+@IEF~<>9W5;3goNN+-CQh9?cuyOR|vLTwzysdjSmDEInm9d_%oHPxMXt``F`b$ zI&c$mBgZOixlyXp%Q=6)`XYg-K|qdsEZ$uAW@6m%yyAZ!J}vAS%le4W(w102mt#Z! zL7m|dfD=O$pMAozZ#;TB#_uX2@nUaGK(yu(EF#3qH@`Y22(^bv zfB?j8UCej@(e#6r5rg+ukkqDR`VJ;Hw!FNdf3sT$rSc3lt0&#pYJcl;-@ICOge$GB zY8g{0Sg%y&bj)gUhEUNqy95zl)&1R>UJb~7V81SDwmRZBQBTwAeh?u(d~kGf+6+y`78_gtnl~6?CT?oIPH)uWAmMtD=cEa5jN=MAcGuHfB~Xu@ zaWiIh@xj>Q%-_haqX?F+2@}g%C9%yQz)gWjccx1Lms|SLyU9f zY#h%h!ka@-PsMCXl5Q6W6|hX12oy;QwI>wvxgBq2eCdFfmb1E2V{aan{{iY4Uy|AJ1!OfsMm6C#e9gYM3t`Be~$q^z*|8ccy|CC9LKE(wkg_6%P3gVD4VVz4n^>kE#SH!Uy98VikmmBe<2kLiHV=o1GQj5bCZ6Ek6>dt;q*3+gjzb=eTQCLT_jyzKY+0%q4Td35V z?^tF8&{;WNReg6NtXeRehlcv0YkOB_wNV>{gNv|`7)RxpG1=A~-5mxX6lJrz_QPf} zeHPY$rKyhH%z~P@Rj?&0v`_Bpl3tv27ptUZoM2?v8$5RFGS**u)ay8kN^2Ud%++ zR%y*1M+z_@%!d#d;2@CFI_&VC){hmpO?c+n9B~ZnrNDbq3l90q(mT3&w{%`^Ra|~+T@c0HHS*2dmjXCG?@xY zQS(-q#}DyaK3|e_sY)IyPGzlVs**jA8wHQ;}il`srZ6+pe0Eht(#5jyUL8AKk zX$$Nyd~Wb`9(v%6;7q=^^1K7&M6J5;ypeVVQwQ4Wp(??VTiWmT0iDbfvbO`v_s$P; 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