fix: Remove nix cache step as it's GitHub only

.github/FUNDING.yml

@@ -1 +0,0 @@
-github: mrcjkb

.github/workflows/check.yml

@@ -0,0 +1,42 @@
+name: Perform checks
+
+on:
+  push:
+  pull_request:
+  workflow_dispatch:
+
+jobs:
+  formatting:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install nix
+        uses: DeterminateSystems/nix-installer-action@main
+      - name: Check Nix flake inputs
+        uses: DeterminateSystems/flake-checker-action@v4 # This action
+      - name: Run nix fmt
+        run: nix fmt -- --check .
+
+  nix_flake_check:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install nix
+        uses: DeterminateSystems/nix-installer-action@main
+      - name: Check Nix flake inputs
+        uses: DeterminateSystems/flake-checker-action@v4 # This action
+      - name: Run nix flake check
+        run: nix flake check
+
+  nix_build:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install nix
+        uses: DeterminateSystems/nix-installer-action@main
+      - name: Check Nix flake inputs
+        uses: DeterminateSystems/flake-checker-action@v4 # This action
+      - name: Build dzgui
+        run: nix build .\#default
+      - name: Build dzgui-testing
+        run: nix build .\#nvim

.github/workflows/update.yml

@@ -0,0 +1,26 @@
+name: Update `flake.lock`
+on:
+  workflow_dispatch:
+  schedule:
+    - cron: '0 0 * * 0' # runs weekly on Sunday at 00:00
+
+jobs:
+  update_lockfile:
+    runs-on: ubuntu-latest
+    steps:
+      - name: Checkout repository
+        uses: actions/checkout@v4
+      - name: Install Nix
+        uses: DeterminateSystems/nix-installer-action@main
+      - name: Update flake.lock
+        uses: DeterminateSystems/update-flake-lock@main
+        with:
+          # inputs: dzgui dzgui-testing # We'll see whether I want to limit the updates to dzgui
+          token: ${{ secrets.GH_TOKEN_FOR_UPDATES }}
+          pr-title: "Update flake.lock"
+          pr-assignees: jiriks74
+          pr-labels: |
+            flake.lock
+            automated
+      - name: Print PR number
+        run: echo Pull request number is ${{ steps.update.outputs.pull-request-number }}.